allee-effect-induced instability in a reaction-diffusion...

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Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 487810, 10 pages http://dx.doi.org/10.1155/2013/487810 Research Article Allee-Effect-Induced Instability in a Reaction-Diffusion Predator-Prey Model Weiming Wang, 1 Yongli Cai, 2 Yanuo Zhu, 1 and Zhengguang Guo 1 1 College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China 2 School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China Correspondence should be addressed to Weiming Wang; [email protected] Received 25 December 2012; Revised 28 February 2013; Accepted 9 March 2013 Academic Editor: Lan Xu Copyright © 2013 Weiming Wang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the spatiotemporal dynamics induced by Allee effect in a reaction-diffusion predator-prey model. In the case without Allee effect, there is nonexistence of diffusion-driven instability for the model. And in the case with Allee effect, the positive equili- brium may be unstable under certain conditions. is instability is induced by Allee effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holes, stripes-holes mixture, stripes, stripes-spots mixture, and spots replication, which shows that the dynamics of the model with Allee effect is not simple, but rich and complex. 1. Introduction In 1952, Turing published one paper [1] on the subject called “pattern formation”—one of the central issues in ecology [2], putting forth the Turing hypothesis of diffusion-driven instability. Pattern formation in mathematics refers to the process that, by changing a bifurcation parameter, the spa- tially homogeneous steady states lose stability to spatially inhomogeneous perturbations, and stable inhomogeneous solutions arise [3]. Turing’s revolutionary idea was that passive diffusion could interact with the chemical reaction in such a way that even if the reaction by itself has no symmetry- breaking capabilities, diffusion can destabilize the symmetry, so that the system with diffusion can have them [4]. From then on, pattern formation has become a very active area of research, motivated in part by the realization that there are many common aspects of patterns formed by diverse physical, chemical, and biological systems and by cellular automata and reaction-diffusion equations [57]. And the appearance and evolution of these patterns have been a focus of recent research activity across several disciplines [815]. Segel and Jackson [16] were the first to call attention to the Turing’s ideas that would be also applicable in population dynamics. At the same time, Gierer and Meinhardt [17] gave a biologically justified formulation of a Turing model and studied its properties by employing numerical simulation. Levin and Segel [18, 19] suggested this scenario of spatial pattern formation as a possible origin of planktonic patchi- ness. e understanding of patterns and mechanisms of spatial dispersal of interacting species is an issue of significant cur- rent interest in conservation biology, ecology, and biochem- ical reactions [2022]. e spatial component of ecological interaction has been identified as an important factor in how ecological communities are shaped. Empirical evidence sug- gests that the spatial scale and structure of environment can influence population interactions [23]. A significant amount of work has been done by using this idea in the field of mathematical biology by Murray [20], Okubo and Levin [21], Cantrell and Cosner [23], and others [3, 2427]. In general, assume that the species prey and predator move randomly on spatial domain, and the spatial movement of the individuals is modeled by diffusion with diffusion coef- ficients 1 > 0, 2 > 0 for the prey and predator V, respectively. As an example, a prototypical predator-prey

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Page 1: Allee-Effect-Induced Instability in a Reaction-Diffusion ...emis.maths.adelaide.edu.au/.../Volume2013/487810.pdf · 2 AbstractandAppliedAnalysis interactionmodelwithlogisticgrowthrateofthepreyinthe

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 487810 10 pageshttpdxdoiorg1011552013487810

Research ArticleAllee-Effect-Induced Instability in a Reaction-DiffusionPredator-Prey Model

Weiming Wang1 Yongli Cai2 Yanuo Zhu1 and Zhengguang Guo1

1 College of Mathematics and Information Science Wenzhou University Wenzhou 325035 China2 School of Mathematics and Computational Science Sun Yat-Sen University Guangzhou 510275 China

Correspondence should be addressed to Weiming Wang weimingwang2003163com

Received 25 December 2012 Revised 28 February 2013 Accepted 9 March 2013

Academic Editor Lan Xu

Copyright copy 2013 Weiming Wang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We investigate the spatiotemporal dynamics induced byAllee effect in a reaction-diffusion predator-preymodel In the case withoutAllee effect there is nonexistence of diffusion-driven instability for the model And in the case with Allee effect the positive equili-briummay be unstable under certain conditionsThis instability is induced by Allee effect and diffusion together Furthermore vianumerical simulations the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holesstripes-holes mixture stripes stripes-spots mixture and spots replication which shows that the dynamics of the model with Alleeeffect is not simple but rich and complex

1 Introduction

In 1952 Turing published one paper [1] on the subject calledldquopattern formationrdquomdashone of the central issues in ecology[2] putting forth the Turing hypothesis of diffusion-driveninstability Pattern formation in mathematics refers to theprocess that by changing a bifurcation parameter the spa-tially homogeneous steady states lose stability to spatiallyinhomogeneous perturbations and stable inhomogeneoussolutions arise [3] Turingrsquos revolutionary idea was thatpassive diffusion could interact with the chemical reaction insuch away that even if the reaction by itself has no symmetry-breaking capabilities diffusion can destabilize the symmetryso that the system with diffusion can have them [4] Fromthen on pattern formation has become a very active areaof research motivated in part by the realization that thereare many common aspects of patterns formed by diversephysical chemical and biological systems and by cellularautomata and reaction-diffusion equations [5ndash7] And theappearance and evolution of these patterns have been a focusof recent research activity across several disciplines [8ndash15]

Segel and Jackson [16] were the first to call attention tothe Turingrsquos ideas that would be also applicable in population

dynamics At the same time Gierer and Meinhardt [17] gavea biologically justified formulation of a Turing model andstudied its properties by employing numerical simulationLevin and Segel [18 19] suggested this scenario of spatialpattern formation as a possible origin of planktonic patchi-ness

The understanding of patterns andmechanisms of spatialdispersal of interacting species is an issue of significant cur-rent interest in conservation biology ecology and biochem-ical reactions [20ndash22] The spatial component of ecologicalinteraction has been identified as an important factor in howecological communities are shaped Empirical evidence sug-gests that the spatial scale and structure of environment caninfluence population interactions [23] A significant amountof work has been done by using this idea in the field ofmathematical biology byMurray [20] Okubo and Levin [21]Cantrell and Cosner [23] and others [3 24ndash27]

In general assume that the species prey and predatormove randomly on spatial domain and the spatial movementof the individuals ismodeled by diffusionwith diffusion coef-ficients 119889

1gt 0 119889

2gt 0 for the prey 119906 and predator

V respectively As an example a prototypical predator-prey

2 Abstract and Applied Analysis

interaction model with logistic growth rate of the prey in theabsence of predation is of the following form [28 29]

d119906d119905= 119906 (120572 minus 120573119906) minus 119891 (119906) 119892 (V) + 119889

1Δ119906

dVd119905= 120590119891 (119906) 119892 (V) minus 119911 (V) + 119889

2ΔV

(1)

where 119906(119905) and V(119905) are the densities of the prey and predatorat time 119905 gt 0 respectively And Δ = 12059721205971199092 + 12059721205971199102 is theLaplacian operator in two-dimensional space

In recent years many studies for example [30ndash40] andthe references therein show that the reaction-diffusion pre-dator-prey model (eg model (1)) is an appropriate tool forinvestigating the fundamental mechanism of complex spa-tiotemporal predation dynamics Of them Alonso et al[30] studied how diffusion affects the stability of predator-prey coexistence equilibria and show a new difference bet-ween ratio- and prey-dependent models that is the prey-dependent models cannot give rise to spatial structuresthrough diffusion-driven instabilities however predator-dependent models with the same degree of complexity canBaurmann et al [31] investigated the emergence of spatiotem-poral patterns in a generalized predator-prey system derivedthe conditions for Hopf and Turing instabilities withoutspecifying the predator-prey functional responses discussedtheir biological implications identified the codimension-2 Turing-Hopf bifurcation and the codimension-3 Turing-Takens-Bogdanov bifurcation and found that these bifurca-tions give rise to complex pattern formation processes in theirneighborhood And Banerjee and Petrovskii [36] studiedpossible scenarios of pattern formation in a ratio-dependentpredator-prey system and found that the emerging patternsare stationary in the large time limit and exhibit only aninsignificant spatial irregularity and spatiotemporal chaoscan indeed be observed but only for parameters well insidethe Turing-Hopf parameter domain away from the bifurca-tion point Rodrigues et al [40] paid their attentions to systemproperties in a vicinity of the Turing-Hopf bifurcation ofthe predator-prey and found that the asymptotical stationarypattern arises as a sudden transition between two differentpatterns

On the other hand in the research of population dynam-ics Allee effect in the population growth has been studiedextensively Allee effect named after ecologist Allee [41] isa phenomenon in biology characterized by a positive corre-lation between population size or density and the mean indi-vidual fitness (often times measured as per capita populationgrowth rate) of a population or species [42] and may occurunder several mechanisms such as difficulties in findingmates when population density is low social dysfunctionat small population sizes and increased predation risk dueto failing flocking or schooling behavior [43ndash45] In anecological point of view Allee effect has been modeled intostrong and weak cases The strong Allee effect introduces apopulation threshold and the population must surpass thisthreshold to grow In contrast a population with a weak Alleeeffect does not have a threshold It has been attracting muchmore attention recently owing to its strong potential impact

on the population dynamics of many plants and animalspecies [46] Detailed investigations relating to Allee effectmay be found in [47ndash59]

In most predation models it has been considered thatAllee effect influences only the prey population For instancein model (1) corresponding to the function of prey growthrate of the prey 119906(120572 minus 120573119906) to express Allee effect the mostusual continuous growth of the equation that is given as

119866 (119906) = 119906 (120572 minus 120573119906 minus119902

119906 + 119887) (2)

is called additive Allee effect which was first deduced in [43]and applied in [60ndash62] Here 119902119906(119906+119887) is the termof additiveAllee effect and 119887 isin (0 1) and 119902 isin (0 1) are Allee-effectconstants If 119902 lt 119887120572 then 119866(0) = 0 1198661015840(0) gt 0 and 119866(119906) iscalled weak Allee effect if 119902 gt 119887120572 then 119866(0) = 0 1198661015840(0) lt 0and 119866(119906) is strong Allee effect

Corresponding tomodel (1) a prototypical predator-preyinteraction model with Allee effect on the prey is given by

d119906d119905= 119906 (120572 minus 120573119906 minus

119902

119906 + 119887) minus 119891 (119906) 119892 (V) + 119889

1Δ119906

dVd119905= 120590119891 (119906) 119892 (V) minus 119911 (V) + 119889

2ΔV

(3)

According to Turingrsquos idea [1] for model (1)mdashthe specialcase of model (3) without Allee effect (ie 119902 = 0)mdashif thepositive equilibrium point 119864lowast = (119906lowast Vlowast) is stable in the case1198891= 1198892= 0 (the nonspatial model) but unstable with

respect to solutions in the cases 1198891gt 0 and 119889

2gt 0 (the

spatial model) then 119864lowast is called diffusion-driven instability(ie Turing instability or Turing bifurcation) and model (1)may exhibit Turing pattern formation In contrast if 119864lowast =(119906lowast Vlowast) is stable in the cases 119889

1gt 0 and 119889

2gt 0 then there

is nonexistence of diffusion-driven instability for model (1)and the model cannot exhibit any pattern formation And inthis situation for model (3) with Allee effect on the preythere comes a question is there any instability of the positiveequilibrium occurring Or is there any diffusion-driveninstability of the positive equilibrium occurring In additiondoes model (3) exhibit any pattern formation controlled byAllee effect

The goal of this paper is to make an insight into theinstability induced by the Allee effect in model (3) Our maininterest is to check whether the Allee effect is a plausiblemechanism of developing spatiotemporal pattern in themodel

The paper is organized as follows In the next section wegive the model and stability of the equilibria In Section 3we discuss the stabilityinstability of the spatial model withwithout Allee effect derive the conditions for the occurrenceof Allee-diffusion-driven instability of the case with Alleeeffect and illustrate typical Turing patterns via numericalsimulations Finally conclusions and remarks are presentedin Section 4

Abstract and Applied Analysis 3

2 The Model System

Inmodel (1) the product119891(119906)119892(V) gives the rate atwhich preyis consumed The prey consumed per predator 119891(119906)119892(V)Vwas termed as the functional response by Solomon [63]These functions can be defined in differentways In this paperfollowing Lotka [64] we adopt

119891 (119906) = 119888119906 (4)

which is a linear functional response without saturationwhere 119888 gt 0 denotes the capture rate [65] And followingHar-rison [28 29] we set

119892 (V) =V

119898V + 1 (5)

where 119898 gt 0 represents a reduction in the predation rate athigh predator densities due tomutual interference among thepredators while searching for food

The proportionality constant 120590 is the rate of prey con-sumption And the function 119911(V) is given by

119911 (V) = 120574V + 119897V2 120574 gt 0 119897 ge 0 (6)

where 120574 denotes the natural death rate of the predator and119897 gt 0 can be used tomodel predator intraspecific competitionthat is not the direct competition for food such as some typeof territoriality [28] In this paper we will discuss the case119897 = 0 which is used in a much more traditional case

Based on the previous discussions we can establish thefollowing predation model of two partial differential equa-tions with additive Allee effect on prey

120597119906

120597119905= 119906 (120572 minus 120573119906 minus

119902

119906 + 119887) minus

119888119906V

119898V + 1+ 1198891Δ119906

120597V

120597119905= V (minus120574 +

119904119906

119898V + 1) + 1198892ΔV

(7)

with the positive initial conditions

119906 (119909 119910 0) gt 0 V (119909 119910 0) gt 0

(119909 119910) isin Ω = (0 119871) times (0 119871)

(8)

and the zero-flux boundary conditions

120597119906

120597120592=120597V

120597120592= 0 (119909 119910) isin 120597Ω (9)

where 119904 denotes conversion rate and Ω is a bounded opendomain in R2

+with boundary 120597Ω 120592 is the outward unit

normal vector on 120597Ω and zero-flux conditions reflect thesituationwhere the population cannotmove across the boun-dary of the domain

Themain purpose of this paper is to focus on the impactsof diffusion orandAllee effect on themodel system about thepositive equilibrium especially for the instability and patternformation

3 Stability Analysis

31The Case without Allee Effect Wefirst consider the stabil-ity of the positive equilibria of model (7) without Allee effectthat is 119902 = 0 and the model is given by

120597119906

120597119905= 119906 (120572 minus 120573119906) minus

119888119906V

119898V + 1+ 1198891Δ119906

120597V

120597119905= V (minus120574 +

119904119906

119898V + 1) + 1198892ΔV

(10)

Easy to know that model (10) has a unique positive equi-librium 119864lowast = (119906lowast Vlowast) with 119904120572 gt 120573120574 where

119906lowast=119898119904120572 minus 119888119904 + radic1199042(119898120572 minus 119888)

2+ 4119888119898119904120573120574

2119898119904120573

Vlowast=

s119906lowast minus 120574119898120574

(11)

which is locally asymptotically stable Next we will discussthe effect of diffusion on 119864lowast

Set 1198801= 119906 minus 119906

lowast 1198811= V minus Vlowast and the linearized system

(10) around 119864lowast = (119906lowast Vlowast) is as follows

1205971198801

120597119905= 1198891Δ1198801minus 120573119906lowast1198801minus

119888119906lowast

(119898Vlowast + 1)21198802

1205971198802

120597119905= 1198892Δ1198802+

119904Vlowast

119898Vlowast + 11198801minus119898119904119906lowastVlowast

(119898Vlowast + 1)21198802

1205971198801

120597]

10038161003816100381610038161003816100381610038161003816120597Ω=1205971198802

120597]

10038161003816100381610038161003816100381610038161003816120597Ω= 0

(12)

FollowingMalchow et al [66] we know that any solutionof system (12) can be expanded into a Fourier series as follows

1198801(r 119905) =

infin

sum

119899119898=0

119906119899119898(r 119905) =

infin

sum

119899119898=0

120572119899119898(119905) sin kr

1198802(r 119905) =

infin

sum

119899119898=0

V119899119898(r 119905) =

infin

sum

119899119898=0

120573119899119898(119905) sin kr

(13)

where r = (119909 119910) and 0 lt 119909 lt 119871 0 lt 119910 lt 119871 k = (119896119899 119896119898) and

119896119899= 119899120587119871 119896

119898= 119898120587119871 are the corresponding wavenumbers

Having substituted 119906119899119898

and V119899119898

into (12) we obtain

119889120572119899119898

119889119905= (minus120573119906

lowastminus 11988911198962) 120572119899119898+ minus

119888119906lowast

(119898Vlowast + 1)2120573119899119898

119889120573119899119898

119889119905=

119904Vlowast

119898Vlowast + 1120572119899119898+ (minus

119898119904119906lowastVlowast

(119898Vlowast + 1)2minus 11988921198962)120573119899119898

(14)

where 1198962 = 1198962119899+ 1198962

119898

A general solution of (14) has the form 1198621exp(120582

1119905) +

1198622exp(120582

2119905) where the constants 119862

1and 119862

2are determined

4 Abstract and Applied Analysis

by the initial conditions (8) and the exponents 1205821 1205822are the

eigenvalues of the following matrix

119869119864lowast = (

minus120573119906lowastminus 11988911198962

minus119888119906lowast

(119898Vlowast + 1)2

119904Vlowast

119898Vlowast + 1minus119898119904119906lowastVlowast

(119898Vlowast + 1)2minus 11988921198962

) (15)

Correspondingly 120582119894(119894 = 1 2) arises as the solution of fol-

lowing equation

1205822

119894minus tr (119869

119864lowast) 120582119894+ det (119869

119864lowast) = 0 (16)

where the trace and determinant of 119869119864lowast are respectively

tr (119869119864lowast) = minus (119889

1+ 1198892) 1198962minus 120573119906lowastminus119898119904119906lowastVlowast

(119898Vlowast + 1)2

det (119869119864lowast) = 119889

111988921198964+ (1198892120573119906lowast+1198891119898119904119906lowastVlowast

(119898Vlowast + 1)2)1198962

+119887119898120573119906

lowast2Vlowast

(119898Vlowast + 1)2+

119887119888119906lowastVlowast

(119898Vlowast + 1)3

(17)

It is easy to know that tr(119869119864lowast) lt 0 and det(119869

119864lowast) gt 0

Hence the positive equilibrium119864lowast ofmodel (10) is uniformlyasymptotically stable

Obviously there is no effect on the stability of the positiveequilibrium whether model (10) with diffusion or not Thatis to say there is nonexistence of diffusion-driven instabilityin model (10) which is the special case of model (7) withoutAllee effect

32 The Case with Allee Effect

321 Allee-Diffusion-Driven Instability In this subsectionwe restrict ourselves to the stability analysis of spatial model(7) which is in the presence of Allee effect on prey

For the sake of learning the effect of Allee effect on thepositive equilibrium of model (7) we first give a definitioncalled Allee-diffusion-driven instability as follows

Definition 1 If a positive equilibrium is uniformly asymptot-ically stable in the reaction-diffusion model without Allee-effect (eg model (10)) but unstable with respect to solutionsof the reaction-diffusion model with Allee effect (eg model(7)) then this instability is called Allee-diffusion-driveninstability

Next we will only investigate the stability of the positiveequilibrium of model (7) For simplicity we take the weakAllee effect case (0 lt 119902 lt 119887120572) as an example and the uniquepositive equilibrium is named 119864

119908= (119906119908 V119908) = (119906

119908 (119904119906119908minus

120574)119898120574) We first give the stability of 119864119908in the case without

diffusion as follows that is 1198891= 1198892= 0 in model (7)

d119906d119905= 119906 (120572 minus 120573119906 minus

119902

119906 + 119887) minus

119888119906V

119898V + 1≜ 119891 (119906 V)

dVd119905= V (minus120574 +

119904119906

119898V + 1) ≜ 119892 (119906 V)

(18)

The Jacobianmatrix of (18) evaluated in the positive equi-librium 119864

119908takes the form

119869119864119908

= (

minus120573119906119908+

119902119906119908

(119906119908+ 119887)2

minus1198881205742

1199042119906119908

119904119906119908minus 120574

119898119906119908

(120574 minus 119904119906119908) 120574

119904119906119908

) (19)

Suppose that (119906119908+ 119887)2(119888120574 + 119898119904120573119906

2

119908) minus 119898119902119904119906

2

119908gt 0 and set

119902[119906119908]= (120573119906

119908minus(120574 minus 119904119906

119908) 120574

119904119906119908

)(119906119908+ 119887)2

119906119908

(20)

By some computational analysis we obtain tr(119869119864119908

) lt 0det(119869119864119908

) gt 0 Hence 119864119908= (119906119908 (119904119906119908minus 120574)119898120574) is locally

asymptotically stableAnd the Jacobian matrix of model (7) at 119864

119908= (119906119908 V119908) is

given by

119864119908

= (

(minus120573 +119902

(119906119908+ 119887)2)119906119908minus 11988911198962

minus1198881205742

1199042119906119908

119904119906119908minus 120574

119898119906119908

minus120574 (119904119906119908minus 120574)

119904119906119908

minus 11988921198962

)

(21)

and the characteristic equation of 119864119908

at 119864119908is

1205822minus tr (

119864119908

) 120582 + det (119864119908

) = 0 (22)

where

tr (119864119908

) = tr (119869119864119908

) minus (1198891+ 1198892) 1198962

det (119864119908

) = det (119869119864119908

) + 119889111988921198964

+ (1198891120574 (119904119906119908minus 120574)

119904119906119908

+(120573 minus119902

(119906119908+ 119887)2)1198892119906119908)1198962

(23)

And the instability sets in when at least tr(119864119908

) gt 0 ordet(119864119908

) lt 0 is violatedSince tr(119869

119864119908

) lt 0

tr (119864119908

) = tr (119869119864119908

) minus (1198891+ 1198892) 1198962lt 0 (24)

is always true Hence only violation of det(119864119908

) lt 0 gives riseto Allee-diffusion-driven instability which leads to

1198891120574 (119904119906119908minus 120574)

119904119906119908

+ (120573 minus119902

(119906119908+ 119887)2)1198892119906119908≜ Θ lt 0 (25)

otherwise det(119864119908

) gt 0 for all 119896 if det(119869119864119908

) gt 0

Abstract and Applied Analysis 5

03305

033

03295

0329

03285

0328

03275

(a)

0336

0334

0332

033

0328

0326

0324

0322

032

(b)

045

04

035

03

025

02

015

01

(c)

045

04

035

03

025

02

015

01

(d)

Figure 1 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 1751198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

Notice that det(119864119908

) achieves its minimum

min119896

det (119864119908

) =411988911198892det (119869

119864119908

) minus Θ2

411988911198892

(26)

at the critical value 119896lowast2 gt 0 where

119896lowast2

= minusΘ

211988911198892

(27)

And Θ lt 0 is equivalent to

(1198891120574 (119904119906119908minus 120574)

11988921199041199062119908

+ 120573) (119906119908+ 119887)2lt 119902 lt 119887120572 (28)

where min119896det(119864119908

) lt 0 is equivalent to 411988911198892det(119869119864119908

) minus

Θ2lt 0 which is equivalent to

119902 gt (119906119908+ 119887)2

(120573 +1198891120574 (119904119906119908minus 120574)

11988921199041199062119908

+

2radic11988911198892det (119869

119864119908

)

1198892119906119908

)

(29)

And from det(119864119908

) = 0 we can determine 1198961and 1198962as

1198962

1=

minusΘ + radicΘ2 minus 411988911198891det (119869

119864119908

)

211988911198892

1198962

2=

minusΘ minus radicΘ2 minus 411988911198891det (119869

119864119908

)

211988911198892

(30)

In conclusion if 11989621lt 1198962lt 1198962

2 then det(

119864119908

) lt 0 and thepositive equilibrium 119864

119908of model (7) is unstableThatrsquos to say

6 Abstract and Applied Analysis

02835

0283

02825

0282

02815

0281

02805

(a)

032

031

03

029

028

027

026

025

024

023

022

(b)

05

045

04

035

03

025

02

015

01

005

(c)

05

045

04

035

03

025

02

015

01

005

(d)

Figure 2 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 21198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

Allee-diffusion-driven instability occurs and model (7) mayexhibit Turing pattern formation

322 Pattern Formation In this subsection in two-dimen-sional space we perform extensive numerical simulations ofthe spatially extended model (7) in the case with weak Alleeeffect and show qualitative results All of the numerical simu-lations employ the zero-flux boundary conditions (9) with asystem size of 200times 200 Other parameters are fixed as 120572 = 1120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 0351198891= 0015 and 119889

2= 1

The numerical integration of model (7) is performed byusing an explicit Euler method for the time integration [67]with a time step size Δ119905 = 1100 and the standard five-pointapproximation [68] for the 2119863 Laplacian with the zero-fluxboundary conditionsThe initial conditions are always a smallamplitude random perturbation around the positive constant

steady state solution119864119908 After the initial period during which

the perturbation spreads the model goes into either a time-dependent state or an essentially steady state solution (time-independent state)

In the numerical simulations different types of dynamicscan be observed and it is found that the distributions ofpredator and prey are always of the same type Consequentlywe can restrict our analysis of pattern formation to onedistribution We only show the distribution of prey 119906 as aninstance

In Figure 1 with 119904 = 175 there is a pattern consisting ofblue hexagons (minimum density of 119906) in a red (maximumdensity of 119906) background that is isolated zones with lowpopulation densities We call this pattern as ldquoholesrdquo

When increasing 119904 to 119904 = 2 the model dynamics exhibitsa transition from stripes-holes growth to stripes replicationthat is holes decay and the stripes pattern emerges (cfFigure 2)

Abstract and Applied Analysis 7

0177

01765

0176

01755

0175

01745

0174

01735

(a)

03

025

02

015

01

(b)

045

04

035

03

025

02

015

01

005

(c)

045

04

035

03

025

02

015

01

005

(d)

Figure 3 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 301198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

When 119904 increasing to 119904 = 30 the later random pertur-bations make these stripes decay end with the time-indepen-dent regular spots (cf Figure 3) which is isolated zones withhigh prey densities

In Figure 4 we show patterns of time-independentstripes-holes and stripes-spots mixture obtained with model(7) These two patterns are similar to each other With 119904 =19 (cf Figure 4(a)) the stripes-holes mixture pattern is atrelatively low prey densities while 119904 = 245 (cf Figure 4(b))at high prey densities

From Figures 1 to 4 one can see that on increasing thecontrol parameter 119904 the pattern sequence ldquoholes rarr stripes-holes mixture rarr stripes rarr stripes-spots mixture rarr spotsrdquois observed

From the viewpoint of population dynamics ldquospotsrdquo pat-tern (cf Figure 3) shows that the prey population is driven bypredator to a very low level in those regions The final resultis the formation of patches of high prey density surroundedby areas of low prey densities [30] That is to say under the

control of these parameters the prey is predominant in thedomain In contrast ldquoholesrdquo pattern (cf Figure 1) indicatesthat the predator is predominant in the domain

4 Conclusions and Remarks

In summary in this paper we have investigated the spa-tiotemporal dynamics of a predator-prey model that involvesAllee effect on prey analytically and numerically

For model (7) in the case without Allee effect there isno effect on the stability of the positive equilibrium whetherwith diffusion or not That is to say there is nonexistence ofdiffusion-driven instability in the model without Allee effectMore precisely the distribution of species converge to a spa-tially homogeneous steady state which varies in time

And in the case with Allee effect the positive equilibriummay be unstableThis instability is induced byAllee effect anddiffusion together so we give a new definition called ldquoAllee-diffusion-driven instabilityrdquo and present the analysis of this

8 Abstract and Applied Analysis

05

045

04

035

03

025

02

015

01

005

(a)

05

045

04

035

03

025

02

015

01

005

(b)

Figure 4 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 1198891= 0015

and 1198892= 1 (a) 119904 = 19 (b) 119904 = 245

instability of the model in details To the best of our knowl-edge this is the first reported case Furthermore via numer-ical simulations it is found that the model dynamics exhibitsboth Allee effect and diffusion controlled pattern formationgrowth to holes stripes-holes mixtures stripes stripes-spotsmixtures and spots replicationThat is to say the distributionof species is aggregation This indicates that the pattern for-mation of the model with Allee effect is not simple but richand complex

In fact for a predator-prey system Okubo and Levin [21]noted Allee effect on the functional response and a density-dependent death rate of the predator is necessary to generatespatial patterns And in this paper we show that a predator-prey system with Allee effect on prey can generate complexTuring spatial patterns which may be a supplementary to[21]

It is needed to note that in this paper we investigate thedynamics of localized patterns inmodel (7) Such patterns arecharacterized by a highly spatially heterogeneous solutionsand are far from the spatially uniform state These patternsoccur in two-component systems when the ratio of the twodiffusion coefficients are very large In the numerical simula-tions we take the diffusivity ratio as 10015 ≫ 1 and so weare close to the regime of localized patterns And the existenceof these spatial patterns can be rigorously proved usingtools from nonlinear functional analysis such as Liapunov-Schmidt reduction and fixed-point theorems [13 14] this isdesirable in future studies

Acknowledgments

The authors would like to thank the anonymous refereefor very helpful suggestions and comments which led toimprovements of our original paper This research was sup-ported by Natural Science Foundation of Zhejiang Province(LY12A01014 and LQ12A01009) and the National BasicResearch Program of China (2012 CB 426510)

References

[1] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society B vol 237 pp 37ndash72 1952

[2] S Levin ldquoThe problem of pattern and scale in ecologyrdquo Ecologyvol 73 no 6 pp 1943ndash1967 1992

[3] Z A Wang and T Hillen ldquoClassical solutions and pattern for-mation for a volume filling chemotaxis modelrdquo Chaos vol 17no 3 Article ID 037108 2007

[4] N F Britton Essential Mathematical Biology Springer 2003[5] M C Cross and P H Hohenberg ldquoPattern formation outside of

equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp 851ndash1112 1993

[6] K J Lee W D McCormick Q Ouyang and H L SwinneyldquoPattern formation by interacting chemical frontsrdquo Science vol261 no 5118 pp 192ndash194 1993

[7] A B Medvinsky S V Petrovskii I A Tikhonova H Malchowand B-L Li ldquoSpatiotemporal complexity of plankton and fishdynamicsrdquo SIAM Review vol 44 no 3 pp 311ndash370 2002

[8] W-M Ni andM Tang ldquoTuring patterns in the Lengyel-Epsteinsystem for the CIMA reactionrdquo Transactions of the AmericanMathematical Society vol 357 no 10 pp 3953ndash3969 2005

[9] R B Hoyle Pattern Formation An Introduction to MethodsCambridge University Press Cambridge UK 2006

[10] M JWard and JWei ldquoThe existence and stability of asymmetricspike patterns for the Schnakenberg modelrdquo Studies in AppliedMathematics vol 109 no 3 pp 229ndash264 2002

[11] M J Ward ldquoAsymptotic methods for reaction-diffusion sys-tems past and presentrdquo Bulletin of Mathematical Biology vol68 no 5 pp 1151ndash1167 2006

[12] J Wei ldquoPattern formations in two-dimensional Gray-Scottmodel existence of single-spot solutions and their stabilityrdquoPhysica D vol 148 no 1-2 pp 20ndash48 2001

[13] J Wei and M Winter ldquoStationary multiple spots for reaction-diffusion systemsrdquo Journal of Mathematical Biology vol 57 no1 pp 53ndash89 2008

[14] W Chen and M J Ward ldquoThe stability and dynamics of local-ized spot patterns in the two-dimensional Gray-Scott modelrdquo

Abstract and Applied Analysis 9

SIAM Journal on Applied Dynamical Systems vol 10 no 2 pp582ndash666 2011

[15] R McKay and T Kolokolnikov ldquoStability transitions anddynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimensionrdquo Discrete and Con-tinuous Dynamical Systems B vol 17 no 1 pp 191ndash220 2012

[16] L Segel and J Jackson ldquoDissipative structure an explanationand an ecological examplerdquo Journal of Theoretical Biology vol37 no 3 pp 545ndash559 1972

[17] A Gierer and H Meinhardt ldquoA theory of biological patternformationrdquo Biological Cybernetics vol 12 no 1 pp 30ndash39 1972

[18] S Levin and L Segel ldquoHypothesis for origin of planktonic pat-chinessrdquo Nature vol 259 no 5545 p 659 1976

[19] S A Levin and L A Segel ldquoPattern generation in space andaspectrdquo SIAM Review A vol 27 no 1 pp 45ndash67 1985

[20] J D MurrayMathematical Biology Springer 2003[21] A Okubo and S A Levin Diffusion and Ecological Problems

Modern Perspectives Springer NewYorkNYUSA 2nd edition2001

[22] C Neuhauser ldquoMathematical challenges in spatial ecologyrdquoNotices of the AmericanMathematical Society vol 48 no 11 pp1304ndash1314 2001

[23] R S Cantrell and C Cosner Spatial Ecology via Reaction-Diffusion Equations JohnWiley amp SonsWest Sussex UK 2003

[24] T K Callahan and E Knobloch ldquoPattern formation in three-dimensional reaction-diffusion systemsrdquo Physica D vol 132 no3 pp 339ndash362 1999

[25] A Bhattacharyay ldquoSpirals and targets in reaction-diffusionsystemsrdquo Physical Review E vol 64 no 1 Article ID 0161132001

[26] T Leppanen M Karttunen K Kaski and R Barrio ldquoDimen-sionality effects inTuring pattern formationrdquo International Jour-nal of Modern Physics B vol 17 no 29 pp 5541ndash5553 2003

[27] J He ldquoAsymptotic methods for solitary solutions and compac-tonsrdquoAbstract andAppliedAnalysis vol 2012 Article ID 916793130 pages 2012

[28] G W Harrison ldquoMultiple stable equilibria in a predator-preysystemrdquo Bulletin of Mathematical Biology vol 48 no 2 pp 137ndash148 1986

[29] G W Harrison ldquoComparing predator-prey models to Luckin-billrsquos experiment with didinium and parameciumrdquo Ecology vol76 no 2 pp 357ndash374 1995

[30] D Alonso F Bartumeus and J Catalan ldquoMutual interferencebetween predators can give rise to Turing spatial patternsrdquo Ecol-ogy vol 83 no 1 pp 28ndash34 2002

[31] M Baurmann T Gross and U Feudel ldquoInstabilities in spatiallyextended predator-prey systems spatio-temporal patterns inthe neighborhood of Turing-Hopf bifurcationsrdquo Journal ofTheoretical Biology vol 245 no 2 pp 220ndash229 2007

[32] WWang Q-X Liu and Z Jin ldquoSpatiotemporal complexity of aratio-dependent predator-prey systemrdquo Physical Review E vol75 no 5 Article ID 051913 p 9 2007

[33] WWang L Zhang HWang and Z Li ldquoPattern formation of apredator-prey systemwith Ivlev-type functional responserdquo Eco-logical Modelling vol 221 no 2 pp 131ndash140 2010

[34] R K Upadhyay W Wang and N K Thakur ldquoSpatiotemporaldynamics in a spatial plankton systemrdquoMathematicalModellingof Natural Phenomena vol 5 no 5 pp 102ndash122 2010

[35] JWang J Shi and JWei ldquoDynamics and pattern formation in adiffusive predator-prey system with strong Allee effect in preyrdquo

Journal of Differential Equations vol 251 no 4-5 pp 1276ndash13042011

[36] M Banerjee and S Petrovskii ldquoSelf-organised spatial patternsand chaos in a ratiodependent predatorndashprey systemrdquoTheoret-ical Ecology vol 4 no 1 pp 37ndash53 2011

[37] M Banerjee ldquoSpatial pattern formation in ratio-dependentmodel higher-order stability analysisrdquo Mathematical Medicineand Biology vol 28 no 2 pp 111ndash128 2011

[38] M Banerjee and S Banerjee ldquoTuring instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner modelrdquoMathematical Biosciences vol 236 no 1 pp 64ndash76 2012

[39] S Fasani and S Rinaldi ldquoFactors promoting or inhibitingTuring instability in spatially extended preyndashpredator systemsrdquoEcological Modelling vol 222 no 18 pp 3449ndash3452 2011

[40] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[41] W C Allee Animal Aggregations A Study in General SociologyAMS Press 1978

[42] F Courchamp J Berec and J GascoigneAllee Effects in Ecologyand Conservation Oxford University Press New York NYUSA 2008

[43] BDennis ldquoAllee effects population growth critical density andthe chance of extinctionrdquoNatural Resource Modeling vol 3 no4 pp 481ndash538 1989

[44] MAMcCarthy ldquoTheAllee effect findingmates and theoreticalmodelsrdquo Ecological Modelling vol 103 no 1 pp 99ndash102 1997

[45] GWang X G Liang and F ZWang ldquoThe competitive dynam-ics of populations subject to an Allee effectrdquo Ecological Mod-elling vol 124 no 2 pp 183ndash192 1999

[46] MA Burgman S FersonH RAkcakaya et alRiskAssessmentin Conservation Biology Chapman amp Hall London UK 1993

[47] M Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993

[48] P A Stephens andW J Sutherland ldquoConsequences of the Alleeeffect for behaviour ecology and conservationrdquo Trends in Ecol-ogy and Evolution vol 14 no 10 pp 401ndash405 1999

[49] P A Stephens W J Sutherland and R P Freckleton ldquoWhat isthe Allee effectrdquo Oikos vol 87 no 1 pp 185ndash190 1999

[50] T H Keitt M A Lewis and R D Holt ldquoAllee effects invasionpinning and Speciesrsquo bordersrdquoAmericanNaturalist vol 157 no2 pp 203ndash216 2002

[51] S R Zhou Y F Liu and G Wang ldquoThe stability of predatorndashprey systems subject to the Allee effectsrdquoTheoretical PopulationBiology vol 67 no 1 pp 23ndash31 2005

[52] S Petrovskii A Morozov and B-L Li ldquoRegimes of biologicalinvasion in a predator-prey system with the Allee effectrdquo Bul-letin of Mathematical Biology vol 67 no 3 pp 637ndash661 2005

[53] J Shi and R Shivaji ldquoPersistence in reaction diffusion modelswith weak Allee effectrdquo Journal of Mathematical Biology vol 52no 6 pp 807ndash829 2006

[54] A Morozov S Petrovskii and B-L Li ldquoSpatiotemporal com-plexity of patchy invasion in a predator-prey system with theAllee effectrdquo Journal of Theoretical Biology vol 238 no 1 pp18ndash35 2006

[55] L Roques A Roques H Berestycki and A KretzschmarldquoA population facing climate change joint influences of Alleeeffects and environmental boundary geometryrdquoPopulation Eco-logy vol 50 no 2 pp 215ndash225 2008

10 Abstract and Applied Analysis

[56] C Celik andODuman ldquoAllee effect in a discrete-time predator-prey systemrdquo Chaos Solitons and Fractals vol 40 no 4 pp1956ndash1962 2009

[57] J Zu andMMimura ldquoThe impact of Allee effect on a predator-prey system with Holling type II functional responserdquo AppliedMathematics and Computation vol 217 no 7 pp 3542ndash35562010

[58] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[59] E Gonzalez-Olivares H Meneses-Alcay B Gonzalez-Yanez JMena-Lorca A Rojas-Palma and R Ramos-Jiliberto ldquoMultiplestability and uniqueness of the limit cycle in a Gause-typepredator-prey model considering the Allee effect on preyrdquoNon-linear Analysis Real World Applicationsl vol 12 no 6 pp 2931ndash2942 2011

[60] P Aguirre E Gonzalez-Olivares and E Saez ldquoTwo limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo Nonlinear Analysis Real World Applications vol 10 no3 pp 1401ndash1416 2009

[61] PAguirre EGonzalez-Olivares andE Saez ldquoThree limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo SIAM Journal on Applied Mathematics vol 69 no 5 pp1244ndash1262 2009

[62] Y Cai W Wang and J Wang ldquoDynamics of a diffusivepredator-prey model with additive Allee effectrdquo InternationalJournal of Biomathematics vol 5 no 2 Article ID 1250023 2012

[63] M E Solomon ldquoThenatural control of animal populationsrdquoTheJournal of Animal Ecology vol 18 no 1 pp 1ndash35 1949

[64] A J Lotka Elements of Physical Biology Williams and Wilkins1925

[65] C Jost Comparing PredatorndashPrey Models Qualitatively andQuantitatively with Ecological Time-Series Data InstitutNational Agronomique Paris-Grignon 1998

[66] H Malchow S V Petrovskii and E Venturino SpatiotemporalPatterns in Ecology and Epidemiology Chapman amp Hall BocaRaton Fla USA 2008

[67] M R Garvie ldquoFinite-difference schemes for reaction-diffusionequations modeling predator-prey interactions in MATLABrdquoBulletin of Mathematical Biology vol 69 no 3 pp 931ndash9562007

[68] A Munteanu and R Sole ldquoPattern formation in noisy self-rep-licating spotsrdquo International Journal of Bifurcation and Chaosvol 16 no 12 pp 3679ndash3683 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014

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Stochastic AnalysisInternational Journal of

Page 2: Allee-Effect-Induced Instability in a Reaction-Diffusion ...emis.maths.adelaide.edu.au/.../Volume2013/487810.pdf · 2 AbstractandAppliedAnalysis interactionmodelwithlogisticgrowthrateofthepreyinthe

2 Abstract and Applied Analysis

interaction model with logistic growth rate of the prey in theabsence of predation is of the following form [28 29]

d119906d119905= 119906 (120572 minus 120573119906) minus 119891 (119906) 119892 (V) + 119889

1Δ119906

dVd119905= 120590119891 (119906) 119892 (V) minus 119911 (V) + 119889

2ΔV

(1)

where 119906(119905) and V(119905) are the densities of the prey and predatorat time 119905 gt 0 respectively And Δ = 12059721205971199092 + 12059721205971199102 is theLaplacian operator in two-dimensional space

In recent years many studies for example [30ndash40] andthe references therein show that the reaction-diffusion pre-dator-prey model (eg model (1)) is an appropriate tool forinvestigating the fundamental mechanism of complex spa-tiotemporal predation dynamics Of them Alonso et al[30] studied how diffusion affects the stability of predator-prey coexistence equilibria and show a new difference bet-ween ratio- and prey-dependent models that is the prey-dependent models cannot give rise to spatial structuresthrough diffusion-driven instabilities however predator-dependent models with the same degree of complexity canBaurmann et al [31] investigated the emergence of spatiotem-poral patterns in a generalized predator-prey system derivedthe conditions for Hopf and Turing instabilities withoutspecifying the predator-prey functional responses discussedtheir biological implications identified the codimension-2 Turing-Hopf bifurcation and the codimension-3 Turing-Takens-Bogdanov bifurcation and found that these bifurca-tions give rise to complex pattern formation processes in theirneighborhood And Banerjee and Petrovskii [36] studiedpossible scenarios of pattern formation in a ratio-dependentpredator-prey system and found that the emerging patternsare stationary in the large time limit and exhibit only aninsignificant spatial irregularity and spatiotemporal chaoscan indeed be observed but only for parameters well insidethe Turing-Hopf parameter domain away from the bifurca-tion point Rodrigues et al [40] paid their attentions to systemproperties in a vicinity of the Turing-Hopf bifurcation ofthe predator-prey and found that the asymptotical stationarypattern arises as a sudden transition between two differentpatterns

On the other hand in the research of population dynam-ics Allee effect in the population growth has been studiedextensively Allee effect named after ecologist Allee [41] isa phenomenon in biology characterized by a positive corre-lation between population size or density and the mean indi-vidual fitness (often times measured as per capita populationgrowth rate) of a population or species [42] and may occurunder several mechanisms such as difficulties in findingmates when population density is low social dysfunctionat small population sizes and increased predation risk dueto failing flocking or schooling behavior [43ndash45] In anecological point of view Allee effect has been modeled intostrong and weak cases The strong Allee effect introduces apopulation threshold and the population must surpass thisthreshold to grow In contrast a population with a weak Alleeeffect does not have a threshold It has been attracting muchmore attention recently owing to its strong potential impact

on the population dynamics of many plants and animalspecies [46] Detailed investigations relating to Allee effectmay be found in [47ndash59]

In most predation models it has been considered thatAllee effect influences only the prey population For instancein model (1) corresponding to the function of prey growthrate of the prey 119906(120572 minus 120573119906) to express Allee effect the mostusual continuous growth of the equation that is given as

119866 (119906) = 119906 (120572 minus 120573119906 minus119902

119906 + 119887) (2)

is called additive Allee effect which was first deduced in [43]and applied in [60ndash62] Here 119902119906(119906+119887) is the termof additiveAllee effect and 119887 isin (0 1) and 119902 isin (0 1) are Allee-effectconstants If 119902 lt 119887120572 then 119866(0) = 0 1198661015840(0) gt 0 and 119866(119906) iscalled weak Allee effect if 119902 gt 119887120572 then 119866(0) = 0 1198661015840(0) lt 0and 119866(119906) is strong Allee effect

Corresponding tomodel (1) a prototypical predator-preyinteraction model with Allee effect on the prey is given by

d119906d119905= 119906 (120572 minus 120573119906 minus

119902

119906 + 119887) minus 119891 (119906) 119892 (V) + 119889

1Δ119906

dVd119905= 120590119891 (119906) 119892 (V) minus 119911 (V) + 119889

2ΔV

(3)

According to Turingrsquos idea [1] for model (1)mdashthe specialcase of model (3) without Allee effect (ie 119902 = 0)mdashif thepositive equilibrium point 119864lowast = (119906lowast Vlowast) is stable in the case1198891= 1198892= 0 (the nonspatial model) but unstable with

respect to solutions in the cases 1198891gt 0 and 119889

2gt 0 (the

spatial model) then 119864lowast is called diffusion-driven instability(ie Turing instability or Turing bifurcation) and model (1)may exhibit Turing pattern formation In contrast if 119864lowast =(119906lowast Vlowast) is stable in the cases 119889

1gt 0 and 119889

2gt 0 then there

is nonexistence of diffusion-driven instability for model (1)and the model cannot exhibit any pattern formation And inthis situation for model (3) with Allee effect on the preythere comes a question is there any instability of the positiveequilibrium occurring Or is there any diffusion-driveninstability of the positive equilibrium occurring In additiondoes model (3) exhibit any pattern formation controlled byAllee effect

The goal of this paper is to make an insight into theinstability induced by the Allee effect in model (3) Our maininterest is to check whether the Allee effect is a plausiblemechanism of developing spatiotemporal pattern in themodel

The paper is organized as follows In the next section wegive the model and stability of the equilibria In Section 3we discuss the stabilityinstability of the spatial model withwithout Allee effect derive the conditions for the occurrenceof Allee-diffusion-driven instability of the case with Alleeeffect and illustrate typical Turing patterns via numericalsimulations Finally conclusions and remarks are presentedin Section 4

Abstract and Applied Analysis 3

2 The Model System

Inmodel (1) the product119891(119906)119892(V) gives the rate atwhich preyis consumed The prey consumed per predator 119891(119906)119892(V)Vwas termed as the functional response by Solomon [63]These functions can be defined in differentways In this paperfollowing Lotka [64] we adopt

119891 (119906) = 119888119906 (4)

which is a linear functional response without saturationwhere 119888 gt 0 denotes the capture rate [65] And followingHar-rison [28 29] we set

119892 (V) =V

119898V + 1 (5)

where 119898 gt 0 represents a reduction in the predation rate athigh predator densities due tomutual interference among thepredators while searching for food

The proportionality constant 120590 is the rate of prey con-sumption And the function 119911(V) is given by

119911 (V) = 120574V + 119897V2 120574 gt 0 119897 ge 0 (6)

where 120574 denotes the natural death rate of the predator and119897 gt 0 can be used tomodel predator intraspecific competitionthat is not the direct competition for food such as some typeof territoriality [28] In this paper we will discuss the case119897 = 0 which is used in a much more traditional case

Based on the previous discussions we can establish thefollowing predation model of two partial differential equa-tions with additive Allee effect on prey

120597119906

120597119905= 119906 (120572 minus 120573119906 minus

119902

119906 + 119887) minus

119888119906V

119898V + 1+ 1198891Δ119906

120597V

120597119905= V (minus120574 +

119904119906

119898V + 1) + 1198892ΔV

(7)

with the positive initial conditions

119906 (119909 119910 0) gt 0 V (119909 119910 0) gt 0

(119909 119910) isin Ω = (0 119871) times (0 119871)

(8)

and the zero-flux boundary conditions

120597119906

120597120592=120597V

120597120592= 0 (119909 119910) isin 120597Ω (9)

where 119904 denotes conversion rate and Ω is a bounded opendomain in R2

+with boundary 120597Ω 120592 is the outward unit

normal vector on 120597Ω and zero-flux conditions reflect thesituationwhere the population cannotmove across the boun-dary of the domain

Themain purpose of this paper is to focus on the impactsof diffusion orandAllee effect on themodel system about thepositive equilibrium especially for the instability and patternformation

3 Stability Analysis

31The Case without Allee Effect Wefirst consider the stabil-ity of the positive equilibria of model (7) without Allee effectthat is 119902 = 0 and the model is given by

120597119906

120597119905= 119906 (120572 minus 120573119906) minus

119888119906V

119898V + 1+ 1198891Δ119906

120597V

120597119905= V (minus120574 +

119904119906

119898V + 1) + 1198892ΔV

(10)

Easy to know that model (10) has a unique positive equi-librium 119864lowast = (119906lowast Vlowast) with 119904120572 gt 120573120574 where

119906lowast=119898119904120572 minus 119888119904 + radic1199042(119898120572 minus 119888)

2+ 4119888119898119904120573120574

2119898119904120573

Vlowast=

s119906lowast minus 120574119898120574

(11)

which is locally asymptotically stable Next we will discussthe effect of diffusion on 119864lowast

Set 1198801= 119906 minus 119906

lowast 1198811= V minus Vlowast and the linearized system

(10) around 119864lowast = (119906lowast Vlowast) is as follows

1205971198801

120597119905= 1198891Δ1198801minus 120573119906lowast1198801minus

119888119906lowast

(119898Vlowast + 1)21198802

1205971198802

120597119905= 1198892Δ1198802+

119904Vlowast

119898Vlowast + 11198801minus119898119904119906lowastVlowast

(119898Vlowast + 1)21198802

1205971198801

120597]

10038161003816100381610038161003816100381610038161003816120597Ω=1205971198802

120597]

10038161003816100381610038161003816100381610038161003816120597Ω= 0

(12)

FollowingMalchow et al [66] we know that any solutionof system (12) can be expanded into a Fourier series as follows

1198801(r 119905) =

infin

sum

119899119898=0

119906119899119898(r 119905) =

infin

sum

119899119898=0

120572119899119898(119905) sin kr

1198802(r 119905) =

infin

sum

119899119898=0

V119899119898(r 119905) =

infin

sum

119899119898=0

120573119899119898(119905) sin kr

(13)

where r = (119909 119910) and 0 lt 119909 lt 119871 0 lt 119910 lt 119871 k = (119896119899 119896119898) and

119896119899= 119899120587119871 119896

119898= 119898120587119871 are the corresponding wavenumbers

Having substituted 119906119899119898

and V119899119898

into (12) we obtain

119889120572119899119898

119889119905= (minus120573119906

lowastminus 11988911198962) 120572119899119898+ minus

119888119906lowast

(119898Vlowast + 1)2120573119899119898

119889120573119899119898

119889119905=

119904Vlowast

119898Vlowast + 1120572119899119898+ (minus

119898119904119906lowastVlowast

(119898Vlowast + 1)2minus 11988921198962)120573119899119898

(14)

where 1198962 = 1198962119899+ 1198962

119898

A general solution of (14) has the form 1198621exp(120582

1119905) +

1198622exp(120582

2119905) where the constants 119862

1and 119862

2are determined

4 Abstract and Applied Analysis

by the initial conditions (8) and the exponents 1205821 1205822are the

eigenvalues of the following matrix

119869119864lowast = (

minus120573119906lowastminus 11988911198962

minus119888119906lowast

(119898Vlowast + 1)2

119904Vlowast

119898Vlowast + 1minus119898119904119906lowastVlowast

(119898Vlowast + 1)2minus 11988921198962

) (15)

Correspondingly 120582119894(119894 = 1 2) arises as the solution of fol-

lowing equation

1205822

119894minus tr (119869

119864lowast) 120582119894+ det (119869

119864lowast) = 0 (16)

where the trace and determinant of 119869119864lowast are respectively

tr (119869119864lowast) = minus (119889

1+ 1198892) 1198962minus 120573119906lowastminus119898119904119906lowastVlowast

(119898Vlowast + 1)2

det (119869119864lowast) = 119889

111988921198964+ (1198892120573119906lowast+1198891119898119904119906lowastVlowast

(119898Vlowast + 1)2)1198962

+119887119898120573119906

lowast2Vlowast

(119898Vlowast + 1)2+

119887119888119906lowastVlowast

(119898Vlowast + 1)3

(17)

It is easy to know that tr(119869119864lowast) lt 0 and det(119869

119864lowast) gt 0

Hence the positive equilibrium119864lowast ofmodel (10) is uniformlyasymptotically stable

Obviously there is no effect on the stability of the positiveequilibrium whether model (10) with diffusion or not Thatis to say there is nonexistence of diffusion-driven instabilityin model (10) which is the special case of model (7) withoutAllee effect

32 The Case with Allee Effect

321 Allee-Diffusion-Driven Instability In this subsectionwe restrict ourselves to the stability analysis of spatial model(7) which is in the presence of Allee effect on prey

For the sake of learning the effect of Allee effect on thepositive equilibrium of model (7) we first give a definitioncalled Allee-diffusion-driven instability as follows

Definition 1 If a positive equilibrium is uniformly asymptot-ically stable in the reaction-diffusion model without Allee-effect (eg model (10)) but unstable with respect to solutionsof the reaction-diffusion model with Allee effect (eg model(7)) then this instability is called Allee-diffusion-driveninstability

Next we will only investigate the stability of the positiveequilibrium of model (7) For simplicity we take the weakAllee effect case (0 lt 119902 lt 119887120572) as an example and the uniquepositive equilibrium is named 119864

119908= (119906119908 V119908) = (119906

119908 (119904119906119908minus

120574)119898120574) We first give the stability of 119864119908in the case without

diffusion as follows that is 1198891= 1198892= 0 in model (7)

d119906d119905= 119906 (120572 minus 120573119906 minus

119902

119906 + 119887) minus

119888119906V

119898V + 1≜ 119891 (119906 V)

dVd119905= V (minus120574 +

119904119906

119898V + 1) ≜ 119892 (119906 V)

(18)

The Jacobianmatrix of (18) evaluated in the positive equi-librium 119864

119908takes the form

119869119864119908

= (

minus120573119906119908+

119902119906119908

(119906119908+ 119887)2

minus1198881205742

1199042119906119908

119904119906119908minus 120574

119898119906119908

(120574 minus 119904119906119908) 120574

119904119906119908

) (19)

Suppose that (119906119908+ 119887)2(119888120574 + 119898119904120573119906

2

119908) minus 119898119902119904119906

2

119908gt 0 and set

119902[119906119908]= (120573119906

119908minus(120574 minus 119904119906

119908) 120574

119904119906119908

)(119906119908+ 119887)2

119906119908

(20)

By some computational analysis we obtain tr(119869119864119908

) lt 0det(119869119864119908

) gt 0 Hence 119864119908= (119906119908 (119904119906119908minus 120574)119898120574) is locally

asymptotically stableAnd the Jacobian matrix of model (7) at 119864

119908= (119906119908 V119908) is

given by

119864119908

= (

(minus120573 +119902

(119906119908+ 119887)2)119906119908minus 11988911198962

minus1198881205742

1199042119906119908

119904119906119908minus 120574

119898119906119908

minus120574 (119904119906119908minus 120574)

119904119906119908

minus 11988921198962

)

(21)

and the characteristic equation of 119864119908

at 119864119908is

1205822minus tr (

119864119908

) 120582 + det (119864119908

) = 0 (22)

where

tr (119864119908

) = tr (119869119864119908

) minus (1198891+ 1198892) 1198962

det (119864119908

) = det (119869119864119908

) + 119889111988921198964

+ (1198891120574 (119904119906119908minus 120574)

119904119906119908

+(120573 minus119902

(119906119908+ 119887)2)1198892119906119908)1198962

(23)

And the instability sets in when at least tr(119864119908

) gt 0 ordet(119864119908

) lt 0 is violatedSince tr(119869

119864119908

) lt 0

tr (119864119908

) = tr (119869119864119908

) minus (1198891+ 1198892) 1198962lt 0 (24)

is always true Hence only violation of det(119864119908

) lt 0 gives riseto Allee-diffusion-driven instability which leads to

1198891120574 (119904119906119908minus 120574)

119904119906119908

+ (120573 minus119902

(119906119908+ 119887)2)1198892119906119908≜ Θ lt 0 (25)

otherwise det(119864119908

) gt 0 for all 119896 if det(119869119864119908

) gt 0

Abstract and Applied Analysis 5

03305

033

03295

0329

03285

0328

03275

(a)

0336

0334

0332

033

0328

0326

0324

0322

032

(b)

045

04

035

03

025

02

015

01

(c)

045

04

035

03

025

02

015

01

(d)

Figure 1 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 1751198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

Notice that det(119864119908

) achieves its minimum

min119896

det (119864119908

) =411988911198892det (119869

119864119908

) minus Θ2

411988911198892

(26)

at the critical value 119896lowast2 gt 0 where

119896lowast2

= minusΘ

211988911198892

(27)

And Θ lt 0 is equivalent to

(1198891120574 (119904119906119908minus 120574)

11988921199041199062119908

+ 120573) (119906119908+ 119887)2lt 119902 lt 119887120572 (28)

where min119896det(119864119908

) lt 0 is equivalent to 411988911198892det(119869119864119908

) minus

Θ2lt 0 which is equivalent to

119902 gt (119906119908+ 119887)2

(120573 +1198891120574 (119904119906119908minus 120574)

11988921199041199062119908

+

2radic11988911198892det (119869

119864119908

)

1198892119906119908

)

(29)

And from det(119864119908

) = 0 we can determine 1198961and 1198962as

1198962

1=

minusΘ + radicΘ2 minus 411988911198891det (119869

119864119908

)

211988911198892

1198962

2=

minusΘ minus radicΘ2 minus 411988911198891det (119869

119864119908

)

211988911198892

(30)

In conclusion if 11989621lt 1198962lt 1198962

2 then det(

119864119908

) lt 0 and thepositive equilibrium 119864

119908of model (7) is unstableThatrsquos to say

6 Abstract and Applied Analysis

02835

0283

02825

0282

02815

0281

02805

(a)

032

031

03

029

028

027

026

025

024

023

022

(b)

05

045

04

035

03

025

02

015

01

005

(c)

05

045

04

035

03

025

02

015

01

005

(d)

Figure 2 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 21198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

Allee-diffusion-driven instability occurs and model (7) mayexhibit Turing pattern formation

322 Pattern Formation In this subsection in two-dimen-sional space we perform extensive numerical simulations ofthe spatially extended model (7) in the case with weak Alleeeffect and show qualitative results All of the numerical simu-lations employ the zero-flux boundary conditions (9) with asystem size of 200times 200 Other parameters are fixed as 120572 = 1120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 0351198891= 0015 and 119889

2= 1

The numerical integration of model (7) is performed byusing an explicit Euler method for the time integration [67]with a time step size Δ119905 = 1100 and the standard five-pointapproximation [68] for the 2119863 Laplacian with the zero-fluxboundary conditionsThe initial conditions are always a smallamplitude random perturbation around the positive constant

steady state solution119864119908 After the initial period during which

the perturbation spreads the model goes into either a time-dependent state or an essentially steady state solution (time-independent state)

In the numerical simulations different types of dynamicscan be observed and it is found that the distributions ofpredator and prey are always of the same type Consequentlywe can restrict our analysis of pattern formation to onedistribution We only show the distribution of prey 119906 as aninstance

In Figure 1 with 119904 = 175 there is a pattern consisting ofblue hexagons (minimum density of 119906) in a red (maximumdensity of 119906) background that is isolated zones with lowpopulation densities We call this pattern as ldquoholesrdquo

When increasing 119904 to 119904 = 2 the model dynamics exhibitsa transition from stripes-holes growth to stripes replicationthat is holes decay and the stripes pattern emerges (cfFigure 2)

Abstract and Applied Analysis 7

0177

01765

0176

01755

0175

01745

0174

01735

(a)

03

025

02

015

01

(b)

045

04

035

03

025

02

015

01

005

(c)

045

04

035

03

025

02

015

01

005

(d)

Figure 3 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 301198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

When 119904 increasing to 119904 = 30 the later random pertur-bations make these stripes decay end with the time-indepen-dent regular spots (cf Figure 3) which is isolated zones withhigh prey densities

In Figure 4 we show patterns of time-independentstripes-holes and stripes-spots mixture obtained with model(7) These two patterns are similar to each other With 119904 =19 (cf Figure 4(a)) the stripes-holes mixture pattern is atrelatively low prey densities while 119904 = 245 (cf Figure 4(b))at high prey densities

From Figures 1 to 4 one can see that on increasing thecontrol parameter 119904 the pattern sequence ldquoholes rarr stripes-holes mixture rarr stripes rarr stripes-spots mixture rarr spotsrdquois observed

From the viewpoint of population dynamics ldquospotsrdquo pat-tern (cf Figure 3) shows that the prey population is driven bypredator to a very low level in those regions The final resultis the formation of patches of high prey density surroundedby areas of low prey densities [30] That is to say under the

control of these parameters the prey is predominant in thedomain In contrast ldquoholesrdquo pattern (cf Figure 1) indicatesthat the predator is predominant in the domain

4 Conclusions and Remarks

In summary in this paper we have investigated the spa-tiotemporal dynamics of a predator-prey model that involvesAllee effect on prey analytically and numerically

For model (7) in the case without Allee effect there isno effect on the stability of the positive equilibrium whetherwith diffusion or not That is to say there is nonexistence ofdiffusion-driven instability in the model without Allee effectMore precisely the distribution of species converge to a spa-tially homogeneous steady state which varies in time

And in the case with Allee effect the positive equilibriummay be unstableThis instability is induced byAllee effect anddiffusion together so we give a new definition called ldquoAllee-diffusion-driven instabilityrdquo and present the analysis of this

8 Abstract and Applied Analysis

05

045

04

035

03

025

02

015

01

005

(a)

05

045

04

035

03

025

02

015

01

005

(b)

Figure 4 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 1198891= 0015

and 1198892= 1 (a) 119904 = 19 (b) 119904 = 245

instability of the model in details To the best of our knowl-edge this is the first reported case Furthermore via numer-ical simulations it is found that the model dynamics exhibitsboth Allee effect and diffusion controlled pattern formationgrowth to holes stripes-holes mixtures stripes stripes-spotsmixtures and spots replicationThat is to say the distributionof species is aggregation This indicates that the pattern for-mation of the model with Allee effect is not simple but richand complex

In fact for a predator-prey system Okubo and Levin [21]noted Allee effect on the functional response and a density-dependent death rate of the predator is necessary to generatespatial patterns And in this paper we show that a predator-prey system with Allee effect on prey can generate complexTuring spatial patterns which may be a supplementary to[21]

It is needed to note that in this paper we investigate thedynamics of localized patterns inmodel (7) Such patterns arecharacterized by a highly spatially heterogeneous solutionsand are far from the spatially uniform state These patternsoccur in two-component systems when the ratio of the twodiffusion coefficients are very large In the numerical simula-tions we take the diffusivity ratio as 10015 ≫ 1 and so weare close to the regime of localized patterns And the existenceof these spatial patterns can be rigorously proved usingtools from nonlinear functional analysis such as Liapunov-Schmidt reduction and fixed-point theorems [13 14] this isdesirable in future studies

Acknowledgments

The authors would like to thank the anonymous refereefor very helpful suggestions and comments which led toimprovements of our original paper This research was sup-ported by Natural Science Foundation of Zhejiang Province(LY12A01014 and LQ12A01009) and the National BasicResearch Program of China (2012 CB 426510)

References

[1] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society B vol 237 pp 37ndash72 1952

[2] S Levin ldquoThe problem of pattern and scale in ecologyrdquo Ecologyvol 73 no 6 pp 1943ndash1967 1992

[3] Z A Wang and T Hillen ldquoClassical solutions and pattern for-mation for a volume filling chemotaxis modelrdquo Chaos vol 17no 3 Article ID 037108 2007

[4] N F Britton Essential Mathematical Biology Springer 2003[5] M C Cross and P H Hohenberg ldquoPattern formation outside of

equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp 851ndash1112 1993

[6] K J Lee W D McCormick Q Ouyang and H L SwinneyldquoPattern formation by interacting chemical frontsrdquo Science vol261 no 5118 pp 192ndash194 1993

[7] A B Medvinsky S V Petrovskii I A Tikhonova H Malchowand B-L Li ldquoSpatiotemporal complexity of plankton and fishdynamicsrdquo SIAM Review vol 44 no 3 pp 311ndash370 2002

[8] W-M Ni andM Tang ldquoTuring patterns in the Lengyel-Epsteinsystem for the CIMA reactionrdquo Transactions of the AmericanMathematical Society vol 357 no 10 pp 3953ndash3969 2005

[9] R B Hoyle Pattern Formation An Introduction to MethodsCambridge University Press Cambridge UK 2006

[10] M JWard and JWei ldquoThe existence and stability of asymmetricspike patterns for the Schnakenberg modelrdquo Studies in AppliedMathematics vol 109 no 3 pp 229ndash264 2002

[11] M J Ward ldquoAsymptotic methods for reaction-diffusion sys-tems past and presentrdquo Bulletin of Mathematical Biology vol68 no 5 pp 1151ndash1167 2006

[12] J Wei ldquoPattern formations in two-dimensional Gray-Scottmodel existence of single-spot solutions and their stabilityrdquoPhysica D vol 148 no 1-2 pp 20ndash48 2001

[13] J Wei and M Winter ldquoStationary multiple spots for reaction-diffusion systemsrdquo Journal of Mathematical Biology vol 57 no1 pp 53ndash89 2008

[14] W Chen and M J Ward ldquoThe stability and dynamics of local-ized spot patterns in the two-dimensional Gray-Scott modelrdquo

Abstract and Applied Analysis 9

SIAM Journal on Applied Dynamical Systems vol 10 no 2 pp582ndash666 2011

[15] R McKay and T Kolokolnikov ldquoStability transitions anddynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimensionrdquo Discrete and Con-tinuous Dynamical Systems B vol 17 no 1 pp 191ndash220 2012

[16] L Segel and J Jackson ldquoDissipative structure an explanationand an ecological examplerdquo Journal of Theoretical Biology vol37 no 3 pp 545ndash559 1972

[17] A Gierer and H Meinhardt ldquoA theory of biological patternformationrdquo Biological Cybernetics vol 12 no 1 pp 30ndash39 1972

[18] S Levin and L Segel ldquoHypothesis for origin of planktonic pat-chinessrdquo Nature vol 259 no 5545 p 659 1976

[19] S A Levin and L A Segel ldquoPattern generation in space andaspectrdquo SIAM Review A vol 27 no 1 pp 45ndash67 1985

[20] J D MurrayMathematical Biology Springer 2003[21] A Okubo and S A Levin Diffusion and Ecological Problems

Modern Perspectives Springer NewYorkNYUSA 2nd edition2001

[22] C Neuhauser ldquoMathematical challenges in spatial ecologyrdquoNotices of the AmericanMathematical Society vol 48 no 11 pp1304ndash1314 2001

[23] R S Cantrell and C Cosner Spatial Ecology via Reaction-Diffusion Equations JohnWiley amp SonsWest Sussex UK 2003

[24] T K Callahan and E Knobloch ldquoPattern formation in three-dimensional reaction-diffusion systemsrdquo Physica D vol 132 no3 pp 339ndash362 1999

[25] A Bhattacharyay ldquoSpirals and targets in reaction-diffusionsystemsrdquo Physical Review E vol 64 no 1 Article ID 0161132001

[26] T Leppanen M Karttunen K Kaski and R Barrio ldquoDimen-sionality effects inTuring pattern formationrdquo International Jour-nal of Modern Physics B vol 17 no 29 pp 5541ndash5553 2003

[27] J He ldquoAsymptotic methods for solitary solutions and compac-tonsrdquoAbstract andAppliedAnalysis vol 2012 Article ID 916793130 pages 2012

[28] G W Harrison ldquoMultiple stable equilibria in a predator-preysystemrdquo Bulletin of Mathematical Biology vol 48 no 2 pp 137ndash148 1986

[29] G W Harrison ldquoComparing predator-prey models to Luckin-billrsquos experiment with didinium and parameciumrdquo Ecology vol76 no 2 pp 357ndash374 1995

[30] D Alonso F Bartumeus and J Catalan ldquoMutual interferencebetween predators can give rise to Turing spatial patternsrdquo Ecol-ogy vol 83 no 1 pp 28ndash34 2002

[31] M Baurmann T Gross and U Feudel ldquoInstabilities in spatiallyextended predator-prey systems spatio-temporal patterns inthe neighborhood of Turing-Hopf bifurcationsrdquo Journal ofTheoretical Biology vol 245 no 2 pp 220ndash229 2007

[32] WWang Q-X Liu and Z Jin ldquoSpatiotemporal complexity of aratio-dependent predator-prey systemrdquo Physical Review E vol75 no 5 Article ID 051913 p 9 2007

[33] WWang L Zhang HWang and Z Li ldquoPattern formation of apredator-prey systemwith Ivlev-type functional responserdquo Eco-logical Modelling vol 221 no 2 pp 131ndash140 2010

[34] R K Upadhyay W Wang and N K Thakur ldquoSpatiotemporaldynamics in a spatial plankton systemrdquoMathematicalModellingof Natural Phenomena vol 5 no 5 pp 102ndash122 2010

[35] JWang J Shi and JWei ldquoDynamics and pattern formation in adiffusive predator-prey system with strong Allee effect in preyrdquo

Journal of Differential Equations vol 251 no 4-5 pp 1276ndash13042011

[36] M Banerjee and S Petrovskii ldquoSelf-organised spatial patternsand chaos in a ratiodependent predatorndashprey systemrdquoTheoret-ical Ecology vol 4 no 1 pp 37ndash53 2011

[37] M Banerjee ldquoSpatial pattern formation in ratio-dependentmodel higher-order stability analysisrdquo Mathematical Medicineand Biology vol 28 no 2 pp 111ndash128 2011

[38] M Banerjee and S Banerjee ldquoTuring instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner modelrdquoMathematical Biosciences vol 236 no 1 pp 64ndash76 2012

[39] S Fasani and S Rinaldi ldquoFactors promoting or inhibitingTuring instability in spatially extended preyndashpredator systemsrdquoEcological Modelling vol 222 no 18 pp 3449ndash3452 2011

[40] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[41] W C Allee Animal Aggregations A Study in General SociologyAMS Press 1978

[42] F Courchamp J Berec and J GascoigneAllee Effects in Ecologyand Conservation Oxford University Press New York NYUSA 2008

[43] BDennis ldquoAllee effects population growth critical density andthe chance of extinctionrdquoNatural Resource Modeling vol 3 no4 pp 481ndash538 1989

[44] MAMcCarthy ldquoTheAllee effect findingmates and theoreticalmodelsrdquo Ecological Modelling vol 103 no 1 pp 99ndash102 1997

[45] GWang X G Liang and F ZWang ldquoThe competitive dynam-ics of populations subject to an Allee effectrdquo Ecological Mod-elling vol 124 no 2 pp 183ndash192 1999

[46] MA Burgman S FersonH RAkcakaya et alRiskAssessmentin Conservation Biology Chapman amp Hall London UK 1993

[47] M Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993

[48] P A Stephens andW J Sutherland ldquoConsequences of the Alleeeffect for behaviour ecology and conservationrdquo Trends in Ecol-ogy and Evolution vol 14 no 10 pp 401ndash405 1999

[49] P A Stephens W J Sutherland and R P Freckleton ldquoWhat isthe Allee effectrdquo Oikos vol 87 no 1 pp 185ndash190 1999

[50] T H Keitt M A Lewis and R D Holt ldquoAllee effects invasionpinning and Speciesrsquo bordersrdquoAmericanNaturalist vol 157 no2 pp 203ndash216 2002

[51] S R Zhou Y F Liu and G Wang ldquoThe stability of predatorndashprey systems subject to the Allee effectsrdquoTheoretical PopulationBiology vol 67 no 1 pp 23ndash31 2005

[52] S Petrovskii A Morozov and B-L Li ldquoRegimes of biologicalinvasion in a predator-prey system with the Allee effectrdquo Bul-letin of Mathematical Biology vol 67 no 3 pp 637ndash661 2005

[53] J Shi and R Shivaji ldquoPersistence in reaction diffusion modelswith weak Allee effectrdquo Journal of Mathematical Biology vol 52no 6 pp 807ndash829 2006

[54] A Morozov S Petrovskii and B-L Li ldquoSpatiotemporal com-plexity of patchy invasion in a predator-prey system with theAllee effectrdquo Journal of Theoretical Biology vol 238 no 1 pp18ndash35 2006

[55] L Roques A Roques H Berestycki and A KretzschmarldquoA population facing climate change joint influences of Alleeeffects and environmental boundary geometryrdquoPopulation Eco-logy vol 50 no 2 pp 215ndash225 2008

10 Abstract and Applied Analysis

[56] C Celik andODuman ldquoAllee effect in a discrete-time predator-prey systemrdquo Chaos Solitons and Fractals vol 40 no 4 pp1956ndash1962 2009

[57] J Zu andMMimura ldquoThe impact of Allee effect on a predator-prey system with Holling type II functional responserdquo AppliedMathematics and Computation vol 217 no 7 pp 3542ndash35562010

[58] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[59] E Gonzalez-Olivares H Meneses-Alcay B Gonzalez-Yanez JMena-Lorca A Rojas-Palma and R Ramos-Jiliberto ldquoMultiplestability and uniqueness of the limit cycle in a Gause-typepredator-prey model considering the Allee effect on preyrdquoNon-linear Analysis Real World Applicationsl vol 12 no 6 pp 2931ndash2942 2011

[60] P Aguirre E Gonzalez-Olivares and E Saez ldquoTwo limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo Nonlinear Analysis Real World Applications vol 10 no3 pp 1401ndash1416 2009

[61] PAguirre EGonzalez-Olivares andE Saez ldquoThree limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo SIAM Journal on Applied Mathematics vol 69 no 5 pp1244ndash1262 2009

[62] Y Cai W Wang and J Wang ldquoDynamics of a diffusivepredator-prey model with additive Allee effectrdquo InternationalJournal of Biomathematics vol 5 no 2 Article ID 1250023 2012

[63] M E Solomon ldquoThenatural control of animal populationsrdquoTheJournal of Animal Ecology vol 18 no 1 pp 1ndash35 1949

[64] A J Lotka Elements of Physical Biology Williams and Wilkins1925

[65] C Jost Comparing PredatorndashPrey Models Qualitatively andQuantitatively with Ecological Time-Series Data InstitutNational Agronomique Paris-Grignon 1998

[66] H Malchow S V Petrovskii and E Venturino SpatiotemporalPatterns in Ecology and Epidemiology Chapman amp Hall BocaRaton Fla USA 2008

[67] M R Garvie ldquoFinite-difference schemes for reaction-diffusionequations modeling predator-prey interactions in MATLABrdquoBulletin of Mathematical Biology vol 69 no 3 pp 931ndash9562007

[68] A Munteanu and R Sole ldquoPattern formation in noisy self-rep-licating spotsrdquo International Journal of Bifurcation and Chaosvol 16 no 12 pp 3679ndash3683 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Page 3: Allee-Effect-Induced Instability in a Reaction-Diffusion ...emis.maths.adelaide.edu.au/.../Volume2013/487810.pdf · 2 AbstractandAppliedAnalysis interactionmodelwithlogisticgrowthrateofthepreyinthe

Abstract and Applied Analysis 3

2 The Model System

Inmodel (1) the product119891(119906)119892(V) gives the rate atwhich preyis consumed The prey consumed per predator 119891(119906)119892(V)Vwas termed as the functional response by Solomon [63]These functions can be defined in differentways In this paperfollowing Lotka [64] we adopt

119891 (119906) = 119888119906 (4)

which is a linear functional response without saturationwhere 119888 gt 0 denotes the capture rate [65] And followingHar-rison [28 29] we set

119892 (V) =V

119898V + 1 (5)

where 119898 gt 0 represents a reduction in the predation rate athigh predator densities due tomutual interference among thepredators while searching for food

The proportionality constant 120590 is the rate of prey con-sumption And the function 119911(V) is given by

119911 (V) = 120574V + 119897V2 120574 gt 0 119897 ge 0 (6)

where 120574 denotes the natural death rate of the predator and119897 gt 0 can be used tomodel predator intraspecific competitionthat is not the direct competition for food such as some typeof territoriality [28] In this paper we will discuss the case119897 = 0 which is used in a much more traditional case

Based on the previous discussions we can establish thefollowing predation model of two partial differential equa-tions with additive Allee effect on prey

120597119906

120597119905= 119906 (120572 minus 120573119906 minus

119902

119906 + 119887) minus

119888119906V

119898V + 1+ 1198891Δ119906

120597V

120597119905= V (minus120574 +

119904119906

119898V + 1) + 1198892ΔV

(7)

with the positive initial conditions

119906 (119909 119910 0) gt 0 V (119909 119910 0) gt 0

(119909 119910) isin Ω = (0 119871) times (0 119871)

(8)

and the zero-flux boundary conditions

120597119906

120597120592=120597V

120597120592= 0 (119909 119910) isin 120597Ω (9)

where 119904 denotes conversion rate and Ω is a bounded opendomain in R2

+with boundary 120597Ω 120592 is the outward unit

normal vector on 120597Ω and zero-flux conditions reflect thesituationwhere the population cannotmove across the boun-dary of the domain

Themain purpose of this paper is to focus on the impactsof diffusion orandAllee effect on themodel system about thepositive equilibrium especially for the instability and patternformation

3 Stability Analysis

31The Case without Allee Effect Wefirst consider the stabil-ity of the positive equilibria of model (7) without Allee effectthat is 119902 = 0 and the model is given by

120597119906

120597119905= 119906 (120572 minus 120573119906) minus

119888119906V

119898V + 1+ 1198891Δ119906

120597V

120597119905= V (minus120574 +

119904119906

119898V + 1) + 1198892ΔV

(10)

Easy to know that model (10) has a unique positive equi-librium 119864lowast = (119906lowast Vlowast) with 119904120572 gt 120573120574 where

119906lowast=119898119904120572 minus 119888119904 + radic1199042(119898120572 minus 119888)

2+ 4119888119898119904120573120574

2119898119904120573

Vlowast=

s119906lowast minus 120574119898120574

(11)

which is locally asymptotically stable Next we will discussthe effect of diffusion on 119864lowast

Set 1198801= 119906 minus 119906

lowast 1198811= V minus Vlowast and the linearized system

(10) around 119864lowast = (119906lowast Vlowast) is as follows

1205971198801

120597119905= 1198891Δ1198801minus 120573119906lowast1198801minus

119888119906lowast

(119898Vlowast + 1)21198802

1205971198802

120597119905= 1198892Δ1198802+

119904Vlowast

119898Vlowast + 11198801minus119898119904119906lowastVlowast

(119898Vlowast + 1)21198802

1205971198801

120597]

10038161003816100381610038161003816100381610038161003816120597Ω=1205971198802

120597]

10038161003816100381610038161003816100381610038161003816120597Ω= 0

(12)

FollowingMalchow et al [66] we know that any solutionof system (12) can be expanded into a Fourier series as follows

1198801(r 119905) =

infin

sum

119899119898=0

119906119899119898(r 119905) =

infin

sum

119899119898=0

120572119899119898(119905) sin kr

1198802(r 119905) =

infin

sum

119899119898=0

V119899119898(r 119905) =

infin

sum

119899119898=0

120573119899119898(119905) sin kr

(13)

where r = (119909 119910) and 0 lt 119909 lt 119871 0 lt 119910 lt 119871 k = (119896119899 119896119898) and

119896119899= 119899120587119871 119896

119898= 119898120587119871 are the corresponding wavenumbers

Having substituted 119906119899119898

and V119899119898

into (12) we obtain

119889120572119899119898

119889119905= (minus120573119906

lowastminus 11988911198962) 120572119899119898+ minus

119888119906lowast

(119898Vlowast + 1)2120573119899119898

119889120573119899119898

119889119905=

119904Vlowast

119898Vlowast + 1120572119899119898+ (minus

119898119904119906lowastVlowast

(119898Vlowast + 1)2minus 11988921198962)120573119899119898

(14)

where 1198962 = 1198962119899+ 1198962

119898

A general solution of (14) has the form 1198621exp(120582

1119905) +

1198622exp(120582

2119905) where the constants 119862

1and 119862

2are determined

4 Abstract and Applied Analysis

by the initial conditions (8) and the exponents 1205821 1205822are the

eigenvalues of the following matrix

119869119864lowast = (

minus120573119906lowastminus 11988911198962

minus119888119906lowast

(119898Vlowast + 1)2

119904Vlowast

119898Vlowast + 1minus119898119904119906lowastVlowast

(119898Vlowast + 1)2minus 11988921198962

) (15)

Correspondingly 120582119894(119894 = 1 2) arises as the solution of fol-

lowing equation

1205822

119894minus tr (119869

119864lowast) 120582119894+ det (119869

119864lowast) = 0 (16)

where the trace and determinant of 119869119864lowast are respectively

tr (119869119864lowast) = minus (119889

1+ 1198892) 1198962minus 120573119906lowastminus119898119904119906lowastVlowast

(119898Vlowast + 1)2

det (119869119864lowast) = 119889

111988921198964+ (1198892120573119906lowast+1198891119898119904119906lowastVlowast

(119898Vlowast + 1)2)1198962

+119887119898120573119906

lowast2Vlowast

(119898Vlowast + 1)2+

119887119888119906lowastVlowast

(119898Vlowast + 1)3

(17)

It is easy to know that tr(119869119864lowast) lt 0 and det(119869

119864lowast) gt 0

Hence the positive equilibrium119864lowast ofmodel (10) is uniformlyasymptotically stable

Obviously there is no effect on the stability of the positiveequilibrium whether model (10) with diffusion or not Thatis to say there is nonexistence of diffusion-driven instabilityin model (10) which is the special case of model (7) withoutAllee effect

32 The Case with Allee Effect

321 Allee-Diffusion-Driven Instability In this subsectionwe restrict ourselves to the stability analysis of spatial model(7) which is in the presence of Allee effect on prey

For the sake of learning the effect of Allee effect on thepositive equilibrium of model (7) we first give a definitioncalled Allee-diffusion-driven instability as follows

Definition 1 If a positive equilibrium is uniformly asymptot-ically stable in the reaction-diffusion model without Allee-effect (eg model (10)) but unstable with respect to solutionsof the reaction-diffusion model with Allee effect (eg model(7)) then this instability is called Allee-diffusion-driveninstability

Next we will only investigate the stability of the positiveequilibrium of model (7) For simplicity we take the weakAllee effect case (0 lt 119902 lt 119887120572) as an example and the uniquepositive equilibrium is named 119864

119908= (119906119908 V119908) = (119906

119908 (119904119906119908minus

120574)119898120574) We first give the stability of 119864119908in the case without

diffusion as follows that is 1198891= 1198892= 0 in model (7)

d119906d119905= 119906 (120572 minus 120573119906 minus

119902

119906 + 119887) minus

119888119906V

119898V + 1≜ 119891 (119906 V)

dVd119905= V (minus120574 +

119904119906

119898V + 1) ≜ 119892 (119906 V)

(18)

The Jacobianmatrix of (18) evaluated in the positive equi-librium 119864

119908takes the form

119869119864119908

= (

minus120573119906119908+

119902119906119908

(119906119908+ 119887)2

minus1198881205742

1199042119906119908

119904119906119908minus 120574

119898119906119908

(120574 minus 119904119906119908) 120574

119904119906119908

) (19)

Suppose that (119906119908+ 119887)2(119888120574 + 119898119904120573119906

2

119908) minus 119898119902119904119906

2

119908gt 0 and set

119902[119906119908]= (120573119906

119908minus(120574 minus 119904119906

119908) 120574

119904119906119908

)(119906119908+ 119887)2

119906119908

(20)

By some computational analysis we obtain tr(119869119864119908

) lt 0det(119869119864119908

) gt 0 Hence 119864119908= (119906119908 (119904119906119908minus 120574)119898120574) is locally

asymptotically stableAnd the Jacobian matrix of model (7) at 119864

119908= (119906119908 V119908) is

given by

119864119908

= (

(minus120573 +119902

(119906119908+ 119887)2)119906119908minus 11988911198962

minus1198881205742

1199042119906119908

119904119906119908minus 120574

119898119906119908

minus120574 (119904119906119908minus 120574)

119904119906119908

minus 11988921198962

)

(21)

and the characteristic equation of 119864119908

at 119864119908is

1205822minus tr (

119864119908

) 120582 + det (119864119908

) = 0 (22)

where

tr (119864119908

) = tr (119869119864119908

) minus (1198891+ 1198892) 1198962

det (119864119908

) = det (119869119864119908

) + 119889111988921198964

+ (1198891120574 (119904119906119908minus 120574)

119904119906119908

+(120573 minus119902

(119906119908+ 119887)2)1198892119906119908)1198962

(23)

And the instability sets in when at least tr(119864119908

) gt 0 ordet(119864119908

) lt 0 is violatedSince tr(119869

119864119908

) lt 0

tr (119864119908

) = tr (119869119864119908

) minus (1198891+ 1198892) 1198962lt 0 (24)

is always true Hence only violation of det(119864119908

) lt 0 gives riseto Allee-diffusion-driven instability which leads to

1198891120574 (119904119906119908minus 120574)

119904119906119908

+ (120573 minus119902

(119906119908+ 119887)2)1198892119906119908≜ Θ lt 0 (25)

otherwise det(119864119908

) gt 0 for all 119896 if det(119869119864119908

) gt 0

Abstract and Applied Analysis 5

03305

033

03295

0329

03285

0328

03275

(a)

0336

0334

0332

033

0328

0326

0324

0322

032

(b)

045

04

035

03

025

02

015

01

(c)

045

04

035

03

025

02

015

01

(d)

Figure 1 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 1751198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

Notice that det(119864119908

) achieves its minimum

min119896

det (119864119908

) =411988911198892det (119869

119864119908

) minus Θ2

411988911198892

(26)

at the critical value 119896lowast2 gt 0 where

119896lowast2

= minusΘ

211988911198892

(27)

And Θ lt 0 is equivalent to

(1198891120574 (119904119906119908minus 120574)

11988921199041199062119908

+ 120573) (119906119908+ 119887)2lt 119902 lt 119887120572 (28)

where min119896det(119864119908

) lt 0 is equivalent to 411988911198892det(119869119864119908

) minus

Θ2lt 0 which is equivalent to

119902 gt (119906119908+ 119887)2

(120573 +1198891120574 (119904119906119908minus 120574)

11988921199041199062119908

+

2radic11988911198892det (119869

119864119908

)

1198892119906119908

)

(29)

And from det(119864119908

) = 0 we can determine 1198961and 1198962as

1198962

1=

minusΘ + radicΘ2 minus 411988911198891det (119869

119864119908

)

211988911198892

1198962

2=

minusΘ minus radicΘ2 minus 411988911198891det (119869

119864119908

)

211988911198892

(30)

In conclusion if 11989621lt 1198962lt 1198962

2 then det(

119864119908

) lt 0 and thepositive equilibrium 119864

119908of model (7) is unstableThatrsquos to say

6 Abstract and Applied Analysis

02835

0283

02825

0282

02815

0281

02805

(a)

032

031

03

029

028

027

026

025

024

023

022

(b)

05

045

04

035

03

025

02

015

01

005

(c)

05

045

04

035

03

025

02

015

01

005

(d)

Figure 2 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 21198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

Allee-diffusion-driven instability occurs and model (7) mayexhibit Turing pattern formation

322 Pattern Formation In this subsection in two-dimen-sional space we perform extensive numerical simulations ofthe spatially extended model (7) in the case with weak Alleeeffect and show qualitative results All of the numerical simu-lations employ the zero-flux boundary conditions (9) with asystem size of 200times 200 Other parameters are fixed as 120572 = 1120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 0351198891= 0015 and 119889

2= 1

The numerical integration of model (7) is performed byusing an explicit Euler method for the time integration [67]with a time step size Δ119905 = 1100 and the standard five-pointapproximation [68] for the 2119863 Laplacian with the zero-fluxboundary conditionsThe initial conditions are always a smallamplitude random perturbation around the positive constant

steady state solution119864119908 After the initial period during which

the perturbation spreads the model goes into either a time-dependent state or an essentially steady state solution (time-independent state)

In the numerical simulations different types of dynamicscan be observed and it is found that the distributions ofpredator and prey are always of the same type Consequentlywe can restrict our analysis of pattern formation to onedistribution We only show the distribution of prey 119906 as aninstance

In Figure 1 with 119904 = 175 there is a pattern consisting ofblue hexagons (minimum density of 119906) in a red (maximumdensity of 119906) background that is isolated zones with lowpopulation densities We call this pattern as ldquoholesrdquo

When increasing 119904 to 119904 = 2 the model dynamics exhibitsa transition from stripes-holes growth to stripes replicationthat is holes decay and the stripes pattern emerges (cfFigure 2)

Abstract and Applied Analysis 7

0177

01765

0176

01755

0175

01745

0174

01735

(a)

03

025

02

015

01

(b)

045

04

035

03

025

02

015

01

005

(c)

045

04

035

03

025

02

015

01

005

(d)

Figure 3 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 301198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

When 119904 increasing to 119904 = 30 the later random pertur-bations make these stripes decay end with the time-indepen-dent regular spots (cf Figure 3) which is isolated zones withhigh prey densities

In Figure 4 we show patterns of time-independentstripes-holes and stripes-spots mixture obtained with model(7) These two patterns are similar to each other With 119904 =19 (cf Figure 4(a)) the stripes-holes mixture pattern is atrelatively low prey densities while 119904 = 245 (cf Figure 4(b))at high prey densities

From Figures 1 to 4 one can see that on increasing thecontrol parameter 119904 the pattern sequence ldquoholes rarr stripes-holes mixture rarr stripes rarr stripes-spots mixture rarr spotsrdquois observed

From the viewpoint of population dynamics ldquospotsrdquo pat-tern (cf Figure 3) shows that the prey population is driven bypredator to a very low level in those regions The final resultis the formation of patches of high prey density surroundedby areas of low prey densities [30] That is to say under the

control of these parameters the prey is predominant in thedomain In contrast ldquoholesrdquo pattern (cf Figure 1) indicatesthat the predator is predominant in the domain

4 Conclusions and Remarks

In summary in this paper we have investigated the spa-tiotemporal dynamics of a predator-prey model that involvesAllee effect on prey analytically and numerically

For model (7) in the case without Allee effect there isno effect on the stability of the positive equilibrium whetherwith diffusion or not That is to say there is nonexistence ofdiffusion-driven instability in the model without Allee effectMore precisely the distribution of species converge to a spa-tially homogeneous steady state which varies in time

And in the case with Allee effect the positive equilibriummay be unstableThis instability is induced byAllee effect anddiffusion together so we give a new definition called ldquoAllee-diffusion-driven instabilityrdquo and present the analysis of this

8 Abstract and Applied Analysis

05

045

04

035

03

025

02

015

01

005

(a)

05

045

04

035

03

025

02

015

01

005

(b)

Figure 4 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 1198891= 0015

and 1198892= 1 (a) 119904 = 19 (b) 119904 = 245

instability of the model in details To the best of our knowl-edge this is the first reported case Furthermore via numer-ical simulations it is found that the model dynamics exhibitsboth Allee effect and diffusion controlled pattern formationgrowth to holes stripes-holes mixtures stripes stripes-spotsmixtures and spots replicationThat is to say the distributionof species is aggregation This indicates that the pattern for-mation of the model with Allee effect is not simple but richand complex

In fact for a predator-prey system Okubo and Levin [21]noted Allee effect on the functional response and a density-dependent death rate of the predator is necessary to generatespatial patterns And in this paper we show that a predator-prey system with Allee effect on prey can generate complexTuring spatial patterns which may be a supplementary to[21]

It is needed to note that in this paper we investigate thedynamics of localized patterns inmodel (7) Such patterns arecharacterized by a highly spatially heterogeneous solutionsand are far from the spatially uniform state These patternsoccur in two-component systems when the ratio of the twodiffusion coefficients are very large In the numerical simula-tions we take the diffusivity ratio as 10015 ≫ 1 and so weare close to the regime of localized patterns And the existenceof these spatial patterns can be rigorously proved usingtools from nonlinear functional analysis such as Liapunov-Schmidt reduction and fixed-point theorems [13 14] this isdesirable in future studies

Acknowledgments

The authors would like to thank the anonymous refereefor very helpful suggestions and comments which led toimprovements of our original paper This research was sup-ported by Natural Science Foundation of Zhejiang Province(LY12A01014 and LQ12A01009) and the National BasicResearch Program of China (2012 CB 426510)

References

[1] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society B vol 237 pp 37ndash72 1952

[2] S Levin ldquoThe problem of pattern and scale in ecologyrdquo Ecologyvol 73 no 6 pp 1943ndash1967 1992

[3] Z A Wang and T Hillen ldquoClassical solutions and pattern for-mation for a volume filling chemotaxis modelrdquo Chaos vol 17no 3 Article ID 037108 2007

[4] N F Britton Essential Mathematical Biology Springer 2003[5] M C Cross and P H Hohenberg ldquoPattern formation outside of

equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp 851ndash1112 1993

[6] K J Lee W D McCormick Q Ouyang and H L SwinneyldquoPattern formation by interacting chemical frontsrdquo Science vol261 no 5118 pp 192ndash194 1993

[7] A B Medvinsky S V Petrovskii I A Tikhonova H Malchowand B-L Li ldquoSpatiotemporal complexity of plankton and fishdynamicsrdquo SIAM Review vol 44 no 3 pp 311ndash370 2002

[8] W-M Ni andM Tang ldquoTuring patterns in the Lengyel-Epsteinsystem for the CIMA reactionrdquo Transactions of the AmericanMathematical Society vol 357 no 10 pp 3953ndash3969 2005

[9] R B Hoyle Pattern Formation An Introduction to MethodsCambridge University Press Cambridge UK 2006

[10] M JWard and JWei ldquoThe existence and stability of asymmetricspike patterns for the Schnakenberg modelrdquo Studies in AppliedMathematics vol 109 no 3 pp 229ndash264 2002

[11] M J Ward ldquoAsymptotic methods for reaction-diffusion sys-tems past and presentrdquo Bulletin of Mathematical Biology vol68 no 5 pp 1151ndash1167 2006

[12] J Wei ldquoPattern formations in two-dimensional Gray-Scottmodel existence of single-spot solutions and their stabilityrdquoPhysica D vol 148 no 1-2 pp 20ndash48 2001

[13] J Wei and M Winter ldquoStationary multiple spots for reaction-diffusion systemsrdquo Journal of Mathematical Biology vol 57 no1 pp 53ndash89 2008

[14] W Chen and M J Ward ldquoThe stability and dynamics of local-ized spot patterns in the two-dimensional Gray-Scott modelrdquo

Abstract and Applied Analysis 9

SIAM Journal on Applied Dynamical Systems vol 10 no 2 pp582ndash666 2011

[15] R McKay and T Kolokolnikov ldquoStability transitions anddynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimensionrdquo Discrete and Con-tinuous Dynamical Systems B vol 17 no 1 pp 191ndash220 2012

[16] L Segel and J Jackson ldquoDissipative structure an explanationand an ecological examplerdquo Journal of Theoretical Biology vol37 no 3 pp 545ndash559 1972

[17] A Gierer and H Meinhardt ldquoA theory of biological patternformationrdquo Biological Cybernetics vol 12 no 1 pp 30ndash39 1972

[18] S Levin and L Segel ldquoHypothesis for origin of planktonic pat-chinessrdquo Nature vol 259 no 5545 p 659 1976

[19] S A Levin and L A Segel ldquoPattern generation in space andaspectrdquo SIAM Review A vol 27 no 1 pp 45ndash67 1985

[20] J D MurrayMathematical Biology Springer 2003[21] A Okubo and S A Levin Diffusion and Ecological Problems

Modern Perspectives Springer NewYorkNYUSA 2nd edition2001

[22] C Neuhauser ldquoMathematical challenges in spatial ecologyrdquoNotices of the AmericanMathematical Society vol 48 no 11 pp1304ndash1314 2001

[23] R S Cantrell and C Cosner Spatial Ecology via Reaction-Diffusion Equations JohnWiley amp SonsWest Sussex UK 2003

[24] T K Callahan and E Knobloch ldquoPattern formation in three-dimensional reaction-diffusion systemsrdquo Physica D vol 132 no3 pp 339ndash362 1999

[25] A Bhattacharyay ldquoSpirals and targets in reaction-diffusionsystemsrdquo Physical Review E vol 64 no 1 Article ID 0161132001

[26] T Leppanen M Karttunen K Kaski and R Barrio ldquoDimen-sionality effects inTuring pattern formationrdquo International Jour-nal of Modern Physics B vol 17 no 29 pp 5541ndash5553 2003

[27] J He ldquoAsymptotic methods for solitary solutions and compac-tonsrdquoAbstract andAppliedAnalysis vol 2012 Article ID 916793130 pages 2012

[28] G W Harrison ldquoMultiple stable equilibria in a predator-preysystemrdquo Bulletin of Mathematical Biology vol 48 no 2 pp 137ndash148 1986

[29] G W Harrison ldquoComparing predator-prey models to Luckin-billrsquos experiment with didinium and parameciumrdquo Ecology vol76 no 2 pp 357ndash374 1995

[30] D Alonso F Bartumeus and J Catalan ldquoMutual interferencebetween predators can give rise to Turing spatial patternsrdquo Ecol-ogy vol 83 no 1 pp 28ndash34 2002

[31] M Baurmann T Gross and U Feudel ldquoInstabilities in spatiallyextended predator-prey systems spatio-temporal patterns inthe neighborhood of Turing-Hopf bifurcationsrdquo Journal ofTheoretical Biology vol 245 no 2 pp 220ndash229 2007

[32] WWang Q-X Liu and Z Jin ldquoSpatiotemporal complexity of aratio-dependent predator-prey systemrdquo Physical Review E vol75 no 5 Article ID 051913 p 9 2007

[33] WWang L Zhang HWang and Z Li ldquoPattern formation of apredator-prey systemwith Ivlev-type functional responserdquo Eco-logical Modelling vol 221 no 2 pp 131ndash140 2010

[34] R K Upadhyay W Wang and N K Thakur ldquoSpatiotemporaldynamics in a spatial plankton systemrdquoMathematicalModellingof Natural Phenomena vol 5 no 5 pp 102ndash122 2010

[35] JWang J Shi and JWei ldquoDynamics and pattern formation in adiffusive predator-prey system with strong Allee effect in preyrdquo

Journal of Differential Equations vol 251 no 4-5 pp 1276ndash13042011

[36] M Banerjee and S Petrovskii ldquoSelf-organised spatial patternsand chaos in a ratiodependent predatorndashprey systemrdquoTheoret-ical Ecology vol 4 no 1 pp 37ndash53 2011

[37] M Banerjee ldquoSpatial pattern formation in ratio-dependentmodel higher-order stability analysisrdquo Mathematical Medicineand Biology vol 28 no 2 pp 111ndash128 2011

[38] M Banerjee and S Banerjee ldquoTuring instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner modelrdquoMathematical Biosciences vol 236 no 1 pp 64ndash76 2012

[39] S Fasani and S Rinaldi ldquoFactors promoting or inhibitingTuring instability in spatially extended preyndashpredator systemsrdquoEcological Modelling vol 222 no 18 pp 3449ndash3452 2011

[40] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[41] W C Allee Animal Aggregations A Study in General SociologyAMS Press 1978

[42] F Courchamp J Berec and J GascoigneAllee Effects in Ecologyand Conservation Oxford University Press New York NYUSA 2008

[43] BDennis ldquoAllee effects population growth critical density andthe chance of extinctionrdquoNatural Resource Modeling vol 3 no4 pp 481ndash538 1989

[44] MAMcCarthy ldquoTheAllee effect findingmates and theoreticalmodelsrdquo Ecological Modelling vol 103 no 1 pp 99ndash102 1997

[45] GWang X G Liang and F ZWang ldquoThe competitive dynam-ics of populations subject to an Allee effectrdquo Ecological Mod-elling vol 124 no 2 pp 183ndash192 1999

[46] MA Burgman S FersonH RAkcakaya et alRiskAssessmentin Conservation Biology Chapman amp Hall London UK 1993

[47] M Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993

[48] P A Stephens andW J Sutherland ldquoConsequences of the Alleeeffect for behaviour ecology and conservationrdquo Trends in Ecol-ogy and Evolution vol 14 no 10 pp 401ndash405 1999

[49] P A Stephens W J Sutherland and R P Freckleton ldquoWhat isthe Allee effectrdquo Oikos vol 87 no 1 pp 185ndash190 1999

[50] T H Keitt M A Lewis and R D Holt ldquoAllee effects invasionpinning and Speciesrsquo bordersrdquoAmericanNaturalist vol 157 no2 pp 203ndash216 2002

[51] S R Zhou Y F Liu and G Wang ldquoThe stability of predatorndashprey systems subject to the Allee effectsrdquoTheoretical PopulationBiology vol 67 no 1 pp 23ndash31 2005

[52] S Petrovskii A Morozov and B-L Li ldquoRegimes of biologicalinvasion in a predator-prey system with the Allee effectrdquo Bul-letin of Mathematical Biology vol 67 no 3 pp 637ndash661 2005

[53] J Shi and R Shivaji ldquoPersistence in reaction diffusion modelswith weak Allee effectrdquo Journal of Mathematical Biology vol 52no 6 pp 807ndash829 2006

[54] A Morozov S Petrovskii and B-L Li ldquoSpatiotemporal com-plexity of patchy invasion in a predator-prey system with theAllee effectrdquo Journal of Theoretical Biology vol 238 no 1 pp18ndash35 2006

[55] L Roques A Roques H Berestycki and A KretzschmarldquoA population facing climate change joint influences of Alleeeffects and environmental boundary geometryrdquoPopulation Eco-logy vol 50 no 2 pp 215ndash225 2008

10 Abstract and Applied Analysis

[56] C Celik andODuman ldquoAllee effect in a discrete-time predator-prey systemrdquo Chaos Solitons and Fractals vol 40 no 4 pp1956ndash1962 2009

[57] J Zu andMMimura ldquoThe impact of Allee effect on a predator-prey system with Holling type II functional responserdquo AppliedMathematics and Computation vol 217 no 7 pp 3542ndash35562010

[58] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[59] E Gonzalez-Olivares H Meneses-Alcay B Gonzalez-Yanez JMena-Lorca A Rojas-Palma and R Ramos-Jiliberto ldquoMultiplestability and uniqueness of the limit cycle in a Gause-typepredator-prey model considering the Allee effect on preyrdquoNon-linear Analysis Real World Applicationsl vol 12 no 6 pp 2931ndash2942 2011

[60] P Aguirre E Gonzalez-Olivares and E Saez ldquoTwo limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo Nonlinear Analysis Real World Applications vol 10 no3 pp 1401ndash1416 2009

[61] PAguirre EGonzalez-Olivares andE Saez ldquoThree limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo SIAM Journal on Applied Mathematics vol 69 no 5 pp1244ndash1262 2009

[62] Y Cai W Wang and J Wang ldquoDynamics of a diffusivepredator-prey model with additive Allee effectrdquo InternationalJournal of Biomathematics vol 5 no 2 Article ID 1250023 2012

[63] M E Solomon ldquoThenatural control of animal populationsrdquoTheJournal of Animal Ecology vol 18 no 1 pp 1ndash35 1949

[64] A J Lotka Elements of Physical Biology Williams and Wilkins1925

[65] C Jost Comparing PredatorndashPrey Models Qualitatively andQuantitatively with Ecological Time-Series Data InstitutNational Agronomique Paris-Grignon 1998

[66] H Malchow S V Petrovskii and E Venturino SpatiotemporalPatterns in Ecology and Epidemiology Chapman amp Hall BocaRaton Fla USA 2008

[67] M R Garvie ldquoFinite-difference schemes for reaction-diffusionequations modeling predator-prey interactions in MATLABrdquoBulletin of Mathematical Biology vol 69 no 3 pp 931ndash9562007

[68] A Munteanu and R Sole ldquoPattern formation in noisy self-rep-licating spotsrdquo International Journal of Bifurcation and Chaosvol 16 no 12 pp 3679ndash3683 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of

Page 4: Allee-Effect-Induced Instability in a Reaction-Diffusion ...emis.maths.adelaide.edu.au/.../Volume2013/487810.pdf · 2 AbstractandAppliedAnalysis interactionmodelwithlogisticgrowthrateofthepreyinthe

4 Abstract and Applied Analysis

by the initial conditions (8) and the exponents 1205821 1205822are the

eigenvalues of the following matrix

119869119864lowast = (

minus120573119906lowastminus 11988911198962

minus119888119906lowast

(119898Vlowast + 1)2

119904Vlowast

119898Vlowast + 1minus119898119904119906lowastVlowast

(119898Vlowast + 1)2minus 11988921198962

) (15)

Correspondingly 120582119894(119894 = 1 2) arises as the solution of fol-

lowing equation

1205822

119894minus tr (119869

119864lowast) 120582119894+ det (119869

119864lowast) = 0 (16)

where the trace and determinant of 119869119864lowast are respectively

tr (119869119864lowast) = minus (119889

1+ 1198892) 1198962minus 120573119906lowastminus119898119904119906lowastVlowast

(119898Vlowast + 1)2

det (119869119864lowast) = 119889

111988921198964+ (1198892120573119906lowast+1198891119898119904119906lowastVlowast

(119898Vlowast + 1)2)1198962

+119887119898120573119906

lowast2Vlowast

(119898Vlowast + 1)2+

119887119888119906lowastVlowast

(119898Vlowast + 1)3

(17)

It is easy to know that tr(119869119864lowast) lt 0 and det(119869

119864lowast) gt 0

Hence the positive equilibrium119864lowast ofmodel (10) is uniformlyasymptotically stable

Obviously there is no effect on the stability of the positiveequilibrium whether model (10) with diffusion or not Thatis to say there is nonexistence of diffusion-driven instabilityin model (10) which is the special case of model (7) withoutAllee effect

32 The Case with Allee Effect

321 Allee-Diffusion-Driven Instability In this subsectionwe restrict ourselves to the stability analysis of spatial model(7) which is in the presence of Allee effect on prey

For the sake of learning the effect of Allee effect on thepositive equilibrium of model (7) we first give a definitioncalled Allee-diffusion-driven instability as follows

Definition 1 If a positive equilibrium is uniformly asymptot-ically stable in the reaction-diffusion model without Allee-effect (eg model (10)) but unstable with respect to solutionsof the reaction-diffusion model with Allee effect (eg model(7)) then this instability is called Allee-diffusion-driveninstability

Next we will only investigate the stability of the positiveequilibrium of model (7) For simplicity we take the weakAllee effect case (0 lt 119902 lt 119887120572) as an example and the uniquepositive equilibrium is named 119864

119908= (119906119908 V119908) = (119906

119908 (119904119906119908minus

120574)119898120574) We first give the stability of 119864119908in the case without

diffusion as follows that is 1198891= 1198892= 0 in model (7)

d119906d119905= 119906 (120572 minus 120573119906 minus

119902

119906 + 119887) minus

119888119906V

119898V + 1≜ 119891 (119906 V)

dVd119905= V (minus120574 +

119904119906

119898V + 1) ≜ 119892 (119906 V)

(18)

The Jacobianmatrix of (18) evaluated in the positive equi-librium 119864

119908takes the form

119869119864119908

= (

minus120573119906119908+

119902119906119908

(119906119908+ 119887)2

minus1198881205742

1199042119906119908

119904119906119908minus 120574

119898119906119908

(120574 minus 119904119906119908) 120574

119904119906119908

) (19)

Suppose that (119906119908+ 119887)2(119888120574 + 119898119904120573119906

2

119908) minus 119898119902119904119906

2

119908gt 0 and set

119902[119906119908]= (120573119906

119908minus(120574 minus 119904119906

119908) 120574

119904119906119908

)(119906119908+ 119887)2

119906119908

(20)

By some computational analysis we obtain tr(119869119864119908

) lt 0det(119869119864119908

) gt 0 Hence 119864119908= (119906119908 (119904119906119908minus 120574)119898120574) is locally

asymptotically stableAnd the Jacobian matrix of model (7) at 119864

119908= (119906119908 V119908) is

given by

119864119908

= (

(minus120573 +119902

(119906119908+ 119887)2)119906119908minus 11988911198962

minus1198881205742

1199042119906119908

119904119906119908minus 120574

119898119906119908

minus120574 (119904119906119908minus 120574)

119904119906119908

minus 11988921198962

)

(21)

and the characteristic equation of 119864119908

at 119864119908is

1205822minus tr (

119864119908

) 120582 + det (119864119908

) = 0 (22)

where

tr (119864119908

) = tr (119869119864119908

) minus (1198891+ 1198892) 1198962

det (119864119908

) = det (119869119864119908

) + 119889111988921198964

+ (1198891120574 (119904119906119908minus 120574)

119904119906119908

+(120573 minus119902

(119906119908+ 119887)2)1198892119906119908)1198962

(23)

And the instability sets in when at least tr(119864119908

) gt 0 ordet(119864119908

) lt 0 is violatedSince tr(119869

119864119908

) lt 0

tr (119864119908

) = tr (119869119864119908

) minus (1198891+ 1198892) 1198962lt 0 (24)

is always true Hence only violation of det(119864119908

) lt 0 gives riseto Allee-diffusion-driven instability which leads to

1198891120574 (119904119906119908minus 120574)

119904119906119908

+ (120573 minus119902

(119906119908+ 119887)2)1198892119906119908≜ Θ lt 0 (25)

otherwise det(119864119908

) gt 0 for all 119896 if det(119869119864119908

) gt 0

Abstract and Applied Analysis 5

03305

033

03295

0329

03285

0328

03275

(a)

0336

0334

0332

033

0328

0326

0324

0322

032

(b)

045

04

035

03

025

02

015

01

(c)

045

04

035

03

025

02

015

01

(d)

Figure 1 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 1751198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

Notice that det(119864119908

) achieves its minimum

min119896

det (119864119908

) =411988911198892det (119869

119864119908

) minus Θ2

411988911198892

(26)

at the critical value 119896lowast2 gt 0 where

119896lowast2

= minusΘ

211988911198892

(27)

And Θ lt 0 is equivalent to

(1198891120574 (119904119906119908minus 120574)

11988921199041199062119908

+ 120573) (119906119908+ 119887)2lt 119902 lt 119887120572 (28)

where min119896det(119864119908

) lt 0 is equivalent to 411988911198892det(119869119864119908

) minus

Θ2lt 0 which is equivalent to

119902 gt (119906119908+ 119887)2

(120573 +1198891120574 (119904119906119908minus 120574)

11988921199041199062119908

+

2radic11988911198892det (119869

119864119908

)

1198892119906119908

)

(29)

And from det(119864119908

) = 0 we can determine 1198961and 1198962as

1198962

1=

minusΘ + radicΘ2 minus 411988911198891det (119869

119864119908

)

211988911198892

1198962

2=

minusΘ minus radicΘ2 minus 411988911198891det (119869

119864119908

)

211988911198892

(30)

In conclusion if 11989621lt 1198962lt 1198962

2 then det(

119864119908

) lt 0 and thepositive equilibrium 119864

119908of model (7) is unstableThatrsquos to say

6 Abstract and Applied Analysis

02835

0283

02825

0282

02815

0281

02805

(a)

032

031

03

029

028

027

026

025

024

023

022

(b)

05

045

04

035

03

025

02

015

01

005

(c)

05

045

04

035

03

025

02

015

01

005

(d)

Figure 2 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 21198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

Allee-diffusion-driven instability occurs and model (7) mayexhibit Turing pattern formation

322 Pattern Formation In this subsection in two-dimen-sional space we perform extensive numerical simulations ofthe spatially extended model (7) in the case with weak Alleeeffect and show qualitative results All of the numerical simu-lations employ the zero-flux boundary conditions (9) with asystem size of 200times 200 Other parameters are fixed as 120572 = 1120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 0351198891= 0015 and 119889

2= 1

The numerical integration of model (7) is performed byusing an explicit Euler method for the time integration [67]with a time step size Δ119905 = 1100 and the standard five-pointapproximation [68] for the 2119863 Laplacian with the zero-fluxboundary conditionsThe initial conditions are always a smallamplitude random perturbation around the positive constant

steady state solution119864119908 After the initial period during which

the perturbation spreads the model goes into either a time-dependent state or an essentially steady state solution (time-independent state)

In the numerical simulations different types of dynamicscan be observed and it is found that the distributions ofpredator and prey are always of the same type Consequentlywe can restrict our analysis of pattern formation to onedistribution We only show the distribution of prey 119906 as aninstance

In Figure 1 with 119904 = 175 there is a pattern consisting ofblue hexagons (minimum density of 119906) in a red (maximumdensity of 119906) background that is isolated zones with lowpopulation densities We call this pattern as ldquoholesrdquo

When increasing 119904 to 119904 = 2 the model dynamics exhibitsa transition from stripes-holes growth to stripes replicationthat is holes decay and the stripes pattern emerges (cfFigure 2)

Abstract and Applied Analysis 7

0177

01765

0176

01755

0175

01745

0174

01735

(a)

03

025

02

015

01

(b)

045

04

035

03

025

02

015

01

005

(c)

045

04

035

03

025

02

015

01

005

(d)

Figure 3 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 301198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

When 119904 increasing to 119904 = 30 the later random pertur-bations make these stripes decay end with the time-indepen-dent regular spots (cf Figure 3) which is isolated zones withhigh prey densities

In Figure 4 we show patterns of time-independentstripes-holes and stripes-spots mixture obtained with model(7) These two patterns are similar to each other With 119904 =19 (cf Figure 4(a)) the stripes-holes mixture pattern is atrelatively low prey densities while 119904 = 245 (cf Figure 4(b))at high prey densities

From Figures 1 to 4 one can see that on increasing thecontrol parameter 119904 the pattern sequence ldquoholes rarr stripes-holes mixture rarr stripes rarr stripes-spots mixture rarr spotsrdquois observed

From the viewpoint of population dynamics ldquospotsrdquo pat-tern (cf Figure 3) shows that the prey population is driven bypredator to a very low level in those regions The final resultis the formation of patches of high prey density surroundedby areas of low prey densities [30] That is to say under the

control of these parameters the prey is predominant in thedomain In contrast ldquoholesrdquo pattern (cf Figure 1) indicatesthat the predator is predominant in the domain

4 Conclusions and Remarks

In summary in this paper we have investigated the spa-tiotemporal dynamics of a predator-prey model that involvesAllee effect on prey analytically and numerically

For model (7) in the case without Allee effect there isno effect on the stability of the positive equilibrium whetherwith diffusion or not That is to say there is nonexistence ofdiffusion-driven instability in the model without Allee effectMore precisely the distribution of species converge to a spa-tially homogeneous steady state which varies in time

And in the case with Allee effect the positive equilibriummay be unstableThis instability is induced byAllee effect anddiffusion together so we give a new definition called ldquoAllee-diffusion-driven instabilityrdquo and present the analysis of this

8 Abstract and Applied Analysis

05

045

04

035

03

025

02

015

01

005

(a)

05

045

04

035

03

025

02

015

01

005

(b)

Figure 4 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 1198891= 0015

and 1198892= 1 (a) 119904 = 19 (b) 119904 = 245

instability of the model in details To the best of our knowl-edge this is the first reported case Furthermore via numer-ical simulations it is found that the model dynamics exhibitsboth Allee effect and diffusion controlled pattern formationgrowth to holes stripes-holes mixtures stripes stripes-spotsmixtures and spots replicationThat is to say the distributionof species is aggregation This indicates that the pattern for-mation of the model with Allee effect is not simple but richand complex

In fact for a predator-prey system Okubo and Levin [21]noted Allee effect on the functional response and a density-dependent death rate of the predator is necessary to generatespatial patterns And in this paper we show that a predator-prey system with Allee effect on prey can generate complexTuring spatial patterns which may be a supplementary to[21]

It is needed to note that in this paper we investigate thedynamics of localized patterns inmodel (7) Such patterns arecharacterized by a highly spatially heterogeneous solutionsand are far from the spatially uniform state These patternsoccur in two-component systems when the ratio of the twodiffusion coefficients are very large In the numerical simula-tions we take the diffusivity ratio as 10015 ≫ 1 and so weare close to the regime of localized patterns And the existenceof these spatial patterns can be rigorously proved usingtools from nonlinear functional analysis such as Liapunov-Schmidt reduction and fixed-point theorems [13 14] this isdesirable in future studies

Acknowledgments

The authors would like to thank the anonymous refereefor very helpful suggestions and comments which led toimprovements of our original paper This research was sup-ported by Natural Science Foundation of Zhejiang Province(LY12A01014 and LQ12A01009) and the National BasicResearch Program of China (2012 CB 426510)

References

[1] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society B vol 237 pp 37ndash72 1952

[2] S Levin ldquoThe problem of pattern and scale in ecologyrdquo Ecologyvol 73 no 6 pp 1943ndash1967 1992

[3] Z A Wang and T Hillen ldquoClassical solutions and pattern for-mation for a volume filling chemotaxis modelrdquo Chaos vol 17no 3 Article ID 037108 2007

[4] N F Britton Essential Mathematical Biology Springer 2003[5] M C Cross and P H Hohenberg ldquoPattern formation outside of

equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp 851ndash1112 1993

[6] K J Lee W D McCormick Q Ouyang and H L SwinneyldquoPattern formation by interacting chemical frontsrdquo Science vol261 no 5118 pp 192ndash194 1993

[7] A B Medvinsky S V Petrovskii I A Tikhonova H Malchowand B-L Li ldquoSpatiotemporal complexity of plankton and fishdynamicsrdquo SIAM Review vol 44 no 3 pp 311ndash370 2002

[8] W-M Ni andM Tang ldquoTuring patterns in the Lengyel-Epsteinsystem for the CIMA reactionrdquo Transactions of the AmericanMathematical Society vol 357 no 10 pp 3953ndash3969 2005

[9] R B Hoyle Pattern Formation An Introduction to MethodsCambridge University Press Cambridge UK 2006

[10] M JWard and JWei ldquoThe existence and stability of asymmetricspike patterns for the Schnakenberg modelrdquo Studies in AppliedMathematics vol 109 no 3 pp 229ndash264 2002

[11] M J Ward ldquoAsymptotic methods for reaction-diffusion sys-tems past and presentrdquo Bulletin of Mathematical Biology vol68 no 5 pp 1151ndash1167 2006

[12] J Wei ldquoPattern formations in two-dimensional Gray-Scottmodel existence of single-spot solutions and their stabilityrdquoPhysica D vol 148 no 1-2 pp 20ndash48 2001

[13] J Wei and M Winter ldquoStationary multiple spots for reaction-diffusion systemsrdquo Journal of Mathematical Biology vol 57 no1 pp 53ndash89 2008

[14] W Chen and M J Ward ldquoThe stability and dynamics of local-ized spot patterns in the two-dimensional Gray-Scott modelrdquo

Abstract and Applied Analysis 9

SIAM Journal on Applied Dynamical Systems vol 10 no 2 pp582ndash666 2011

[15] R McKay and T Kolokolnikov ldquoStability transitions anddynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimensionrdquo Discrete and Con-tinuous Dynamical Systems B vol 17 no 1 pp 191ndash220 2012

[16] L Segel and J Jackson ldquoDissipative structure an explanationand an ecological examplerdquo Journal of Theoretical Biology vol37 no 3 pp 545ndash559 1972

[17] A Gierer and H Meinhardt ldquoA theory of biological patternformationrdquo Biological Cybernetics vol 12 no 1 pp 30ndash39 1972

[18] S Levin and L Segel ldquoHypothesis for origin of planktonic pat-chinessrdquo Nature vol 259 no 5545 p 659 1976

[19] S A Levin and L A Segel ldquoPattern generation in space andaspectrdquo SIAM Review A vol 27 no 1 pp 45ndash67 1985

[20] J D MurrayMathematical Biology Springer 2003[21] A Okubo and S A Levin Diffusion and Ecological Problems

Modern Perspectives Springer NewYorkNYUSA 2nd edition2001

[22] C Neuhauser ldquoMathematical challenges in spatial ecologyrdquoNotices of the AmericanMathematical Society vol 48 no 11 pp1304ndash1314 2001

[23] R S Cantrell and C Cosner Spatial Ecology via Reaction-Diffusion Equations JohnWiley amp SonsWest Sussex UK 2003

[24] T K Callahan and E Knobloch ldquoPattern formation in three-dimensional reaction-diffusion systemsrdquo Physica D vol 132 no3 pp 339ndash362 1999

[25] A Bhattacharyay ldquoSpirals and targets in reaction-diffusionsystemsrdquo Physical Review E vol 64 no 1 Article ID 0161132001

[26] T Leppanen M Karttunen K Kaski and R Barrio ldquoDimen-sionality effects inTuring pattern formationrdquo International Jour-nal of Modern Physics B vol 17 no 29 pp 5541ndash5553 2003

[27] J He ldquoAsymptotic methods for solitary solutions and compac-tonsrdquoAbstract andAppliedAnalysis vol 2012 Article ID 916793130 pages 2012

[28] G W Harrison ldquoMultiple stable equilibria in a predator-preysystemrdquo Bulletin of Mathematical Biology vol 48 no 2 pp 137ndash148 1986

[29] G W Harrison ldquoComparing predator-prey models to Luckin-billrsquos experiment with didinium and parameciumrdquo Ecology vol76 no 2 pp 357ndash374 1995

[30] D Alonso F Bartumeus and J Catalan ldquoMutual interferencebetween predators can give rise to Turing spatial patternsrdquo Ecol-ogy vol 83 no 1 pp 28ndash34 2002

[31] M Baurmann T Gross and U Feudel ldquoInstabilities in spatiallyextended predator-prey systems spatio-temporal patterns inthe neighborhood of Turing-Hopf bifurcationsrdquo Journal ofTheoretical Biology vol 245 no 2 pp 220ndash229 2007

[32] WWang Q-X Liu and Z Jin ldquoSpatiotemporal complexity of aratio-dependent predator-prey systemrdquo Physical Review E vol75 no 5 Article ID 051913 p 9 2007

[33] WWang L Zhang HWang and Z Li ldquoPattern formation of apredator-prey systemwith Ivlev-type functional responserdquo Eco-logical Modelling vol 221 no 2 pp 131ndash140 2010

[34] R K Upadhyay W Wang and N K Thakur ldquoSpatiotemporaldynamics in a spatial plankton systemrdquoMathematicalModellingof Natural Phenomena vol 5 no 5 pp 102ndash122 2010

[35] JWang J Shi and JWei ldquoDynamics and pattern formation in adiffusive predator-prey system with strong Allee effect in preyrdquo

Journal of Differential Equations vol 251 no 4-5 pp 1276ndash13042011

[36] M Banerjee and S Petrovskii ldquoSelf-organised spatial patternsand chaos in a ratiodependent predatorndashprey systemrdquoTheoret-ical Ecology vol 4 no 1 pp 37ndash53 2011

[37] M Banerjee ldquoSpatial pattern formation in ratio-dependentmodel higher-order stability analysisrdquo Mathematical Medicineand Biology vol 28 no 2 pp 111ndash128 2011

[38] M Banerjee and S Banerjee ldquoTuring instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner modelrdquoMathematical Biosciences vol 236 no 1 pp 64ndash76 2012

[39] S Fasani and S Rinaldi ldquoFactors promoting or inhibitingTuring instability in spatially extended preyndashpredator systemsrdquoEcological Modelling vol 222 no 18 pp 3449ndash3452 2011

[40] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[41] W C Allee Animal Aggregations A Study in General SociologyAMS Press 1978

[42] F Courchamp J Berec and J GascoigneAllee Effects in Ecologyand Conservation Oxford University Press New York NYUSA 2008

[43] BDennis ldquoAllee effects population growth critical density andthe chance of extinctionrdquoNatural Resource Modeling vol 3 no4 pp 481ndash538 1989

[44] MAMcCarthy ldquoTheAllee effect findingmates and theoreticalmodelsrdquo Ecological Modelling vol 103 no 1 pp 99ndash102 1997

[45] GWang X G Liang and F ZWang ldquoThe competitive dynam-ics of populations subject to an Allee effectrdquo Ecological Mod-elling vol 124 no 2 pp 183ndash192 1999

[46] MA Burgman S FersonH RAkcakaya et alRiskAssessmentin Conservation Biology Chapman amp Hall London UK 1993

[47] M Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993

[48] P A Stephens andW J Sutherland ldquoConsequences of the Alleeeffect for behaviour ecology and conservationrdquo Trends in Ecol-ogy and Evolution vol 14 no 10 pp 401ndash405 1999

[49] P A Stephens W J Sutherland and R P Freckleton ldquoWhat isthe Allee effectrdquo Oikos vol 87 no 1 pp 185ndash190 1999

[50] T H Keitt M A Lewis and R D Holt ldquoAllee effects invasionpinning and Speciesrsquo bordersrdquoAmericanNaturalist vol 157 no2 pp 203ndash216 2002

[51] S R Zhou Y F Liu and G Wang ldquoThe stability of predatorndashprey systems subject to the Allee effectsrdquoTheoretical PopulationBiology vol 67 no 1 pp 23ndash31 2005

[52] S Petrovskii A Morozov and B-L Li ldquoRegimes of biologicalinvasion in a predator-prey system with the Allee effectrdquo Bul-letin of Mathematical Biology vol 67 no 3 pp 637ndash661 2005

[53] J Shi and R Shivaji ldquoPersistence in reaction diffusion modelswith weak Allee effectrdquo Journal of Mathematical Biology vol 52no 6 pp 807ndash829 2006

[54] A Morozov S Petrovskii and B-L Li ldquoSpatiotemporal com-plexity of patchy invasion in a predator-prey system with theAllee effectrdquo Journal of Theoretical Biology vol 238 no 1 pp18ndash35 2006

[55] L Roques A Roques H Berestycki and A KretzschmarldquoA population facing climate change joint influences of Alleeeffects and environmental boundary geometryrdquoPopulation Eco-logy vol 50 no 2 pp 215ndash225 2008

10 Abstract and Applied Analysis

[56] C Celik andODuman ldquoAllee effect in a discrete-time predator-prey systemrdquo Chaos Solitons and Fractals vol 40 no 4 pp1956ndash1962 2009

[57] J Zu andMMimura ldquoThe impact of Allee effect on a predator-prey system with Holling type II functional responserdquo AppliedMathematics and Computation vol 217 no 7 pp 3542ndash35562010

[58] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[59] E Gonzalez-Olivares H Meneses-Alcay B Gonzalez-Yanez JMena-Lorca A Rojas-Palma and R Ramos-Jiliberto ldquoMultiplestability and uniqueness of the limit cycle in a Gause-typepredator-prey model considering the Allee effect on preyrdquoNon-linear Analysis Real World Applicationsl vol 12 no 6 pp 2931ndash2942 2011

[60] P Aguirre E Gonzalez-Olivares and E Saez ldquoTwo limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo Nonlinear Analysis Real World Applications vol 10 no3 pp 1401ndash1416 2009

[61] PAguirre EGonzalez-Olivares andE Saez ldquoThree limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo SIAM Journal on Applied Mathematics vol 69 no 5 pp1244ndash1262 2009

[62] Y Cai W Wang and J Wang ldquoDynamics of a diffusivepredator-prey model with additive Allee effectrdquo InternationalJournal of Biomathematics vol 5 no 2 Article ID 1250023 2012

[63] M E Solomon ldquoThenatural control of animal populationsrdquoTheJournal of Animal Ecology vol 18 no 1 pp 1ndash35 1949

[64] A J Lotka Elements of Physical Biology Williams and Wilkins1925

[65] C Jost Comparing PredatorndashPrey Models Qualitatively andQuantitatively with Ecological Time-Series Data InstitutNational Agronomique Paris-Grignon 1998

[66] H Malchow S V Petrovskii and E Venturino SpatiotemporalPatterns in Ecology and Epidemiology Chapman amp Hall BocaRaton Fla USA 2008

[67] M R Garvie ldquoFinite-difference schemes for reaction-diffusionequations modeling predator-prey interactions in MATLABrdquoBulletin of Mathematical Biology vol 69 no 3 pp 931ndash9562007

[68] A Munteanu and R Sole ldquoPattern formation in noisy self-rep-licating spotsrdquo International Journal of Bifurcation and Chaosvol 16 no 12 pp 3679ndash3683 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Allee-Effect-Induced Instability in a Reaction-Diffusion ...emis.maths.adelaide.edu.au/.../Volume2013/487810.pdf · 2 AbstractandAppliedAnalysis interactionmodelwithlogisticgrowthrateofthepreyinthe

Abstract and Applied Analysis 5

03305

033

03295

0329

03285

0328

03275

(a)

0336

0334

0332

033

0328

0326

0324

0322

032

(b)

045

04

035

03

025

02

015

01

(c)

045

04

035

03

025

02

015

01

(d)

Figure 1 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 1751198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

Notice that det(119864119908

) achieves its minimum

min119896

det (119864119908

) =411988911198892det (119869

119864119908

) minus Θ2

411988911198892

(26)

at the critical value 119896lowast2 gt 0 where

119896lowast2

= minusΘ

211988911198892

(27)

And Θ lt 0 is equivalent to

(1198891120574 (119904119906119908minus 120574)

11988921199041199062119908

+ 120573) (119906119908+ 119887)2lt 119902 lt 119887120572 (28)

where min119896det(119864119908

) lt 0 is equivalent to 411988911198892det(119869119864119908

) minus

Θ2lt 0 which is equivalent to

119902 gt (119906119908+ 119887)2

(120573 +1198891120574 (119904119906119908minus 120574)

11988921199041199062119908

+

2radic11988911198892det (119869

119864119908

)

1198892119906119908

)

(29)

And from det(119864119908

) = 0 we can determine 1198961and 1198962as

1198962

1=

minusΘ + radicΘ2 minus 411988911198891det (119869

119864119908

)

211988911198892

1198962

2=

minusΘ minus radicΘ2 minus 411988911198891det (119869

119864119908

)

211988911198892

(30)

In conclusion if 11989621lt 1198962lt 1198962

2 then det(

119864119908

) lt 0 and thepositive equilibrium 119864

119908of model (7) is unstableThatrsquos to say

6 Abstract and Applied Analysis

02835

0283

02825

0282

02815

0281

02805

(a)

032

031

03

029

028

027

026

025

024

023

022

(b)

05

045

04

035

03

025

02

015

01

005

(c)

05

045

04

035

03

025

02

015

01

005

(d)

Figure 2 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 21198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

Allee-diffusion-driven instability occurs and model (7) mayexhibit Turing pattern formation

322 Pattern Formation In this subsection in two-dimen-sional space we perform extensive numerical simulations ofthe spatially extended model (7) in the case with weak Alleeeffect and show qualitative results All of the numerical simu-lations employ the zero-flux boundary conditions (9) with asystem size of 200times 200 Other parameters are fixed as 120572 = 1120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 0351198891= 0015 and 119889

2= 1

The numerical integration of model (7) is performed byusing an explicit Euler method for the time integration [67]with a time step size Δ119905 = 1100 and the standard five-pointapproximation [68] for the 2119863 Laplacian with the zero-fluxboundary conditionsThe initial conditions are always a smallamplitude random perturbation around the positive constant

steady state solution119864119908 After the initial period during which

the perturbation spreads the model goes into either a time-dependent state or an essentially steady state solution (time-independent state)

In the numerical simulations different types of dynamicscan be observed and it is found that the distributions ofpredator and prey are always of the same type Consequentlywe can restrict our analysis of pattern formation to onedistribution We only show the distribution of prey 119906 as aninstance

In Figure 1 with 119904 = 175 there is a pattern consisting ofblue hexagons (minimum density of 119906) in a red (maximumdensity of 119906) background that is isolated zones with lowpopulation densities We call this pattern as ldquoholesrdquo

When increasing 119904 to 119904 = 2 the model dynamics exhibitsa transition from stripes-holes growth to stripes replicationthat is holes decay and the stripes pattern emerges (cfFigure 2)

Abstract and Applied Analysis 7

0177

01765

0176

01755

0175

01745

0174

01735

(a)

03

025

02

015

01

(b)

045

04

035

03

025

02

015

01

005

(c)

045

04

035

03

025

02

015

01

005

(d)

Figure 3 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 301198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

When 119904 increasing to 119904 = 30 the later random pertur-bations make these stripes decay end with the time-indepen-dent regular spots (cf Figure 3) which is isolated zones withhigh prey densities

In Figure 4 we show patterns of time-independentstripes-holes and stripes-spots mixture obtained with model(7) These two patterns are similar to each other With 119904 =19 (cf Figure 4(a)) the stripes-holes mixture pattern is atrelatively low prey densities while 119904 = 245 (cf Figure 4(b))at high prey densities

From Figures 1 to 4 one can see that on increasing thecontrol parameter 119904 the pattern sequence ldquoholes rarr stripes-holes mixture rarr stripes rarr stripes-spots mixture rarr spotsrdquois observed

From the viewpoint of population dynamics ldquospotsrdquo pat-tern (cf Figure 3) shows that the prey population is driven bypredator to a very low level in those regions The final resultis the formation of patches of high prey density surroundedby areas of low prey densities [30] That is to say under the

control of these parameters the prey is predominant in thedomain In contrast ldquoholesrdquo pattern (cf Figure 1) indicatesthat the predator is predominant in the domain

4 Conclusions and Remarks

In summary in this paper we have investigated the spa-tiotemporal dynamics of a predator-prey model that involvesAllee effect on prey analytically and numerically

For model (7) in the case without Allee effect there isno effect on the stability of the positive equilibrium whetherwith diffusion or not That is to say there is nonexistence ofdiffusion-driven instability in the model without Allee effectMore precisely the distribution of species converge to a spa-tially homogeneous steady state which varies in time

And in the case with Allee effect the positive equilibriummay be unstableThis instability is induced byAllee effect anddiffusion together so we give a new definition called ldquoAllee-diffusion-driven instabilityrdquo and present the analysis of this

8 Abstract and Applied Analysis

05

045

04

035

03

025

02

015

01

005

(a)

05

045

04

035

03

025

02

015

01

005

(b)

Figure 4 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 1198891= 0015

and 1198892= 1 (a) 119904 = 19 (b) 119904 = 245

instability of the model in details To the best of our knowl-edge this is the first reported case Furthermore via numer-ical simulations it is found that the model dynamics exhibitsboth Allee effect and diffusion controlled pattern formationgrowth to holes stripes-holes mixtures stripes stripes-spotsmixtures and spots replicationThat is to say the distributionof species is aggregation This indicates that the pattern for-mation of the model with Allee effect is not simple but richand complex

In fact for a predator-prey system Okubo and Levin [21]noted Allee effect on the functional response and a density-dependent death rate of the predator is necessary to generatespatial patterns And in this paper we show that a predator-prey system with Allee effect on prey can generate complexTuring spatial patterns which may be a supplementary to[21]

It is needed to note that in this paper we investigate thedynamics of localized patterns inmodel (7) Such patterns arecharacterized by a highly spatially heterogeneous solutionsand are far from the spatially uniform state These patternsoccur in two-component systems when the ratio of the twodiffusion coefficients are very large In the numerical simula-tions we take the diffusivity ratio as 10015 ≫ 1 and so weare close to the regime of localized patterns And the existenceof these spatial patterns can be rigorously proved usingtools from nonlinear functional analysis such as Liapunov-Schmidt reduction and fixed-point theorems [13 14] this isdesirable in future studies

Acknowledgments

The authors would like to thank the anonymous refereefor very helpful suggestions and comments which led toimprovements of our original paper This research was sup-ported by Natural Science Foundation of Zhejiang Province(LY12A01014 and LQ12A01009) and the National BasicResearch Program of China (2012 CB 426510)

References

[1] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society B vol 237 pp 37ndash72 1952

[2] S Levin ldquoThe problem of pattern and scale in ecologyrdquo Ecologyvol 73 no 6 pp 1943ndash1967 1992

[3] Z A Wang and T Hillen ldquoClassical solutions and pattern for-mation for a volume filling chemotaxis modelrdquo Chaos vol 17no 3 Article ID 037108 2007

[4] N F Britton Essential Mathematical Biology Springer 2003[5] M C Cross and P H Hohenberg ldquoPattern formation outside of

equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp 851ndash1112 1993

[6] K J Lee W D McCormick Q Ouyang and H L SwinneyldquoPattern formation by interacting chemical frontsrdquo Science vol261 no 5118 pp 192ndash194 1993

[7] A B Medvinsky S V Petrovskii I A Tikhonova H Malchowand B-L Li ldquoSpatiotemporal complexity of plankton and fishdynamicsrdquo SIAM Review vol 44 no 3 pp 311ndash370 2002

[8] W-M Ni andM Tang ldquoTuring patterns in the Lengyel-Epsteinsystem for the CIMA reactionrdquo Transactions of the AmericanMathematical Society vol 357 no 10 pp 3953ndash3969 2005

[9] R B Hoyle Pattern Formation An Introduction to MethodsCambridge University Press Cambridge UK 2006

[10] M JWard and JWei ldquoThe existence and stability of asymmetricspike patterns for the Schnakenberg modelrdquo Studies in AppliedMathematics vol 109 no 3 pp 229ndash264 2002

[11] M J Ward ldquoAsymptotic methods for reaction-diffusion sys-tems past and presentrdquo Bulletin of Mathematical Biology vol68 no 5 pp 1151ndash1167 2006

[12] J Wei ldquoPattern formations in two-dimensional Gray-Scottmodel existence of single-spot solutions and their stabilityrdquoPhysica D vol 148 no 1-2 pp 20ndash48 2001

[13] J Wei and M Winter ldquoStationary multiple spots for reaction-diffusion systemsrdquo Journal of Mathematical Biology vol 57 no1 pp 53ndash89 2008

[14] W Chen and M J Ward ldquoThe stability and dynamics of local-ized spot patterns in the two-dimensional Gray-Scott modelrdquo

Abstract and Applied Analysis 9

SIAM Journal on Applied Dynamical Systems vol 10 no 2 pp582ndash666 2011

[15] R McKay and T Kolokolnikov ldquoStability transitions anddynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimensionrdquo Discrete and Con-tinuous Dynamical Systems B vol 17 no 1 pp 191ndash220 2012

[16] L Segel and J Jackson ldquoDissipative structure an explanationand an ecological examplerdquo Journal of Theoretical Biology vol37 no 3 pp 545ndash559 1972

[17] A Gierer and H Meinhardt ldquoA theory of biological patternformationrdquo Biological Cybernetics vol 12 no 1 pp 30ndash39 1972

[18] S Levin and L Segel ldquoHypothesis for origin of planktonic pat-chinessrdquo Nature vol 259 no 5545 p 659 1976

[19] S A Levin and L A Segel ldquoPattern generation in space andaspectrdquo SIAM Review A vol 27 no 1 pp 45ndash67 1985

[20] J D MurrayMathematical Biology Springer 2003[21] A Okubo and S A Levin Diffusion and Ecological Problems

Modern Perspectives Springer NewYorkNYUSA 2nd edition2001

[22] C Neuhauser ldquoMathematical challenges in spatial ecologyrdquoNotices of the AmericanMathematical Society vol 48 no 11 pp1304ndash1314 2001

[23] R S Cantrell and C Cosner Spatial Ecology via Reaction-Diffusion Equations JohnWiley amp SonsWest Sussex UK 2003

[24] T K Callahan and E Knobloch ldquoPattern formation in three-dimensional reaction-diffusion systemsrdquo Physica D vol 132 no3 pp 339ndash362 1999

[25] A Bhattacharyay ldquoSpirals and targets in reaction-diffusionsystemsrdquo Physical Review E vol 64 no 1 Article ID 0161132001

[26] T Leppanen M Karttunen K Kaski and R Barrio ldquoDimen-sionality effects inTuring pattern formationrdquo International Jour-nal of Modern Physics B vol 17 no 29 pp 5541ndash5553 2003

[27] J He ldquoAsymptotic methods for solitary solutions and compac-tonsrdquoAbstract andAppliedAnalysis vol 2012 Article ID 916793130 pages 2012

[28] G W Harrison ldquoMultiple stable equilibria in a predator-preysystemrdquo Bulletin of Mathematical Biology vol 48 no 2 pp 137ndash148 1986

[29] G W Harrison ldquoComparing predator-prey models to Luckin-billrsquos experiment with didinium and parameciumrdquo Ecology vol76 no 2 pp 357ndash374 1995

[30] D Alonso F Bartumeus and J Catalan ldquoMutual interferencebetween predators can give rise to Turing spatial patternsrdquo Ecol-ogy vol 83 no 1 pp 28ndash34 2002

[31] M Baurmann T Gross and U Feudel ldquoInstabilities in spatiallyextended predator-prey systems spatio-temporal patterns inthe neighborhood of Turing-Hopf bifurcationsrdquo Journal ofTheoretical Biology vol 245 no 2 pp 220ndash229 2007

[32] WWang Q-X Liu and Z Jin ldquoSpatiotemporal complexity of aratio-dependent predator-prey systemrdquo Physical Review E vol75 no 5 Article ID 051913 p 9 2007

[33] WWang L Zhang HWang and Z Li ldquoPattern formation of apredator-prey systemwith Ivlev-type functional responserdquo Eco-logical Modelling vol 221 no 2 pp 131ndash140 2010

[34] R K Upadhyay W Wang and N K Thakur ldquoSpatiotemporaldynamics in a spatial plankton systemrdquoMathematicalModellingof Natural Phenomena vol 5 no 5 pp 102ndash122 2010

[35] JWang J Shi and JWei ldquoDynamics and pattern formation in adiffusive predator-prey system with strong Allee effect in preyrdquo

Journal of Differential Equations vol 251 no 4-5 pp 1276ndash13042011

[36] M Banerjee and S Petrovskii ldquoSelf-organised spatial patternsand chaos in a ratiodependent predatorndashprey systemrdquoTheoret-ical Ecology vol 4 no 1 pp 37ndash53 2011

[37] M Banerjee ldquoSpatial pattern formation in ratio-dependentmodel higher-order stability analysisrdquo Mathematical Medicineand Biology vol 28 no 2 pp 111ndash128 2011

[38] M Banerjee and S Banerjee ldquoTuring instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner modelrdquoMathematical Biosciences vol 236 no 1 pp 64ndash76 2012

[39] S Fasani and S Rinaldi ldquoFactors promoting or inhibitingTuring instability in spatially extended preyndashpredator systemsrdquoEcological Modelling vol 222 no 18 pp 3449ndash3452 2011

[40] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[41] W C Allee Animal Aggregations A Study in General SociologyAMS Press 1978

[42] F Courchamp J Berec and J GascoigneAllee Effects in Ecologyand Conservation Oxford University Press New York NYUSA 2008

[43] BDennis ldquoAllee effects population growth critical density andthe chance of extinctionrdquoNatural Resource Modeling vol 3 no4 pp 481ndash538 1989

[44] MAMcCarthy ldquoTheAllee effect findingmates and theoreticalmodelsrdquo Ecological Modelling vol 103 no 1 pp 99ndash102 1997

[45] GWang X G Liang and F ZWang ldquoThe competitive dynam-ics of populations subject to an Allee effectrdquo Ecological Mod-elling vol 124 no 2 pp 183ndash192 1999

[46] MA Burgman S FersonH RAkcakaya et alRiskAssessmentin Conservation Biology Chapman amp Hall London UK 1993

[47] M Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993

[48] P A Stephens andW J Sutherland ldquoConsequences of the Alleeeffect for behaviour ecology and conservationrdquo Trends in Ecol-ogy and Evolution vol 14 no 10 pp 401ndash405 1999

[49] P A Stephens W J Sutherland and R P Freckleton ldquoWhat isthe Allee effectrdquo Oikos vol 87 no 1 pp 185ndash190 1999

[50] T H Keitt M A Lewis and R D Holt ldquoAllee effects invasionpinning and Speciesrsquo bordersrdquoAmericanNaturalist vol 157 no2 pp 203ndash216 2002

[51] S R Zhou Y F Liu and G Wang ldquoThe stability of predatorndashprey systems subject to the Allee effectsrdquoTheoretical PopulationBiology vol 67 no 1 pp 23ndash31 2005

[52] S Petrovskii A Morozov and B-L Li ldquoRegimes of biologicalinvasion in a predator-prey system with the Allee effectrdquo Bul-letin of Mathematical Biology vol 67 no 3 pp 637ndash661 2005

[53] J Shi and R Shivaji ldquoPersistence in reaction diffusion modelswith weak Allee effectrdquo Journal of Mathematical Biology vol 52no 6 pp 807ndash829 2006

[54] A Morozov S Petrovskii and B-L Li ldquoSpatiotemporal com-plexity of patchy invasion in a predator-prey system with theAllee effectrdquo Journal of Theoretical Biology vol 238 no 1 pp18ndash35 2006

[55] L Roques A Roques H Berestycki and A KretzschmarldquoA population facing climate change joint influences of Alleeeffects and environmental boundary geometryrdquoPopulation Eco-logy vol 50 no 2 pp 215ndash225 2008

10 Abstract and Applied Analysis

[56] C Celik andODuman ldquoAllee effect in a discrete-time predator-prey systemrdquo Chaos Solitons and Fractals vol 40 no 4 pp1956ndash1962 2009

[57] J Zu andMMimura ldquoThe impact of Allee effect on a predator-prey system with Holling type II functional responserdquo AppliedMathematics and Computation vol 217 no 7 pp 3542ndash35562010

[58] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[59] E Gonzalez-Olivares H Meneses-Alcay B Gonzalez-Yanez JMena-Lorca A Rojas-Palma and R Ramos-Jiliberto ldquoMultiplestability and uniqueness of the limit cycle in a Gause-typepredator-prey model considering the Allee effect on preyrdquoNon-linear Analysis Real World Applicationsl vol 12 no 6 pp 2931ndash2942 2011

[60] P Aguirre E Gonzalez-Olivares and E Saez ldquoTwo limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo Nonlinear Analysis Real World Applications vol 10 no3 pp 1401ndash1416 2009

[61] PAguirre EGonzalez-Olivares andE Saez ldquoThree limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo SIAM Journal on Applied Mathematics vol 69 no 5 pp1244ndash1262 2009

[62] Y Cai W Wang and J Wang ldquoDynamics of a diffusivepredator-prey model with additive Allee effectrdquo InternationalJournal of Biomathematics vol 5 no 2 Article ID 1250023 2012

[63] M E Solomon ldquoThenatural control of animal populationsrdquoTheJournal of Animal Ecology vol 18 no 1 pp 1ndash35 1949

[64] A J Lotka Elements of Physical Biology Williams and Wilkins1925

[65] C Jost Comparing PredatorndashPrey Models Qualitatively andQuantitatively with Ecological Time-Series Data InstitutNational Agronomique Paris-Grignon 1998

[66] H Malchow S V Petrovskii and E Venturino SpatiotemporalPatterns in Ecology and Epidemiology Chapman amp Hall BocaRaton Fla USA 2008

[67] M R Garvie ldquoFinite-difference schemes for reaction-diffusionequations modeling predator-prey interactions in MATLABrdquoBulletin of Mathematical Biology vol 69 no 3 pp 931ndash9562007

[68] A Munteanu and R Sole ldquoPattern formation in noisy self-rep-licating spotsrdquo International Journal of Bifurcation and Chaosvol 16 no 12 pp 3679ndash3683 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Allee-Effect-Induced Instability in a Reaction-Diffusion ...emis.maths.adelaide.edu.au/.../Volume2013/487810.pdf · 2 AbstractandAppliedAnalysis interactionmodelwithlogisticgrowthrateofthepreyinthe

6 Abstract and Applied Analysis

02835

0283

02825

0282

02815

0281

02805

(a)

032

031

03

029

028

027

026

025

024

023

022

(b)

05

045

04

035

03

025

02

015

01

005

(c)

05

045

04

035

03

025

02

015

01

005

(d)

Figure 2 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 21198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

Allee-diffusion-driven instability occurs and model (7) mayexhibit Turing pattern formation

322 Pattern Formation In this subsection in two-dimen-sional space we perform extensive numerical simulations ofthe spatially extended model (7) in the case with weak Alleeeffect and show qualitative results All of the numerical simu-lations employ the zero-flux boundary conditions (9) with asystem size of 200times 200 Other parameters are fixed as 120572 = 1120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 0351198891= 0015 and 119889

2= 1

The numerical integration of model (7) is performed byusing an explicit Euler method for the time integration [67]with a time step size Δ119905 = 1100 and the standard five-pointapproximation [68] for the 2119863 Laplacian with the zero-fluxboundary conditionsThe initial conditions are always a smallamplitude random perturbation around the positive constant

steady state solution119864119908 After the initial period during which

the perturbation spreads the model goes into either a time-dependent state or an essentially steady state solution (time-independent state)

In the numerical simulations different types of dynamicscan be observed and it is found that the distributions ofpredator and prey are always of the same type Consequentlywe can restrict our analysis of pattern formation to onedistribution We only show the distribution of prey 119906 as aninstance

In Figure 1 with 119904 = 175 there is a pattern consisting ofblue hexagons (minimum density of 119906) in a red (maximumdensity of 119906) background that is isolated zones with lowpopulation densities We call this pattern as ldquoholesrdquo

When increasing 119904 to 119904 = 2 the model dynamics exhibitsa transition from stripes-holes growth to stripes replicationthat is holes decay and the stripes pattern emerges (cfFigure 2)

Abstract and Applied Analysis 7

0177

01765

0176

01755

0175

01745

0174

01735

(a)

03

025

02

015

01

(b)

045

04

035

03

025

02

015

01

005

(c)

045

04

035

03

025

02

015

01

005

(d)

Figure 3 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 301198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

When 119904 increasing to 119904 = 30 the later random pertur-bations make these stripes decay end with the time-indepen-dent regular spots (cf Figure 3) which is isolated zones withhigh prey densities

In Figure 4 we show patterns of time-independentstripes-holes and stripes-spots mixture obtained with model(7) These two patterns are similar to each other With 119904 =19 (cf Figure 4(a)) the stripes-holes mixture pattern is atrelatively low prey densities while 119904 = 245 (cf Figure 4(b))at high prey densities

From Figures 1 to 4 one can see that on increasing thecontrol parameter 119904 the pattern sequence ldquoholes rarr stripes-holes mixture rarr stripes rarr stripes-spots mixture rarr spotsrdquois observed

From the viewpoint of population dynamics ldquospotsrdquo pat-tern (cf Figure 3) shows that the prey population is driven bypredator to a very low level in those regions The final resultis the formation of patches of high prey density surroundedby areas of low prey densities [30] That is to say under the

control of these parameters the prey is predominant in thedomain In contrast ldquoholesrdquo pattern (cf Figure 1) indicatesthat the predator is predominant in the domain

4 Conclusions and Remarks

In summary in this paper we have investigated the spa-tiotemporal dynamics of a predator-prey model that involvesAllee effect on prey analytically and numerically

For model (7) in the case without Allee effect there isno effect on the stability of the positive equilibrium whetherwith diffusion or not That is to say there is nonexistence ofdiffusion-driven instability in the model without Allee effectMore precisely the distribution of species converge to a spa-tially homogeneous steady state which varies in time

And in the case with Allee effect the positive equilibriummay be unstableThis instability is induced byAllee effect anddiffusion together so we give a new definition called ldquoAllee-diffusion-driven instabilityrdquo and present the analysis of this

8 Abstract and Applied Analysis

05

045

04

035

03

025

02

015

01

005

(a)

05

045

04

035

03

025

02

015

01

005

(b)

Figure 4 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 1198891= 0015

and 1198892= 1 (a) 119904 = 19 (b) 119904 = 245

instability of the model in details To the best of our knowl-edge this is the first reported case Furthermore via numer-ical simulations it is found that the model dynamics exhibitsboth Allee effect and diffusion controlled pattern formationgrowth to holes stripes-holes mixtures stripes stripes-spotsmixtures and spots replicationThat is to say the distributionof species is aggregation This indicates that the pattern for-mation of the model with Allee effect is not simple but richand complex

In fact for a predator-prey system Okubo and Levin [21]noted Allee effect on the functional response and a density-dependent death rate of the predator is necessary to generatespatial patterns And in this paper we show that a predator-prey system with Allee effect on prey can generate complexTuring spatial patterns which may be a supplementary to[21]

It is needed to note that in this paper we investigate thedynamics of localized patterns inmodel (7) Such patterns arecharacterized by a highly spatially heterogeneous solutionsand are far from the spatially uniform state These patternsoccur in two-component systems when the ratio of the twodiffusion coefficients are very large In the numerical simula-tions we take the diffusivity ratio as 10015 ≫ 1 and so weare close to the regime of localized patterns And the existenceof these spatial patterns can be rigorously proved usingtools from nonlinear functional analysis such as Liapunov-Schmidt reduction and fixed-point theorems [13 14] this isdesirable in future studies

Acknowledgments

The authors would like to thank the anonymous refereefor very helpful suggestions and comments which led toimprovements of our original paper This research was sup-ported by Natural Science Foundation of Zhejiang Province(LY12A01014 and LQ12A01009) and the National BasicResearch Program of China (2012 CB 426510)

References

[1] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society B vol 237 pp 37ndash72 1952

[2] S Levin ldquoThe problem of pattern and scale in ecologyrdquo Ecologyvol 73 no 6 pp 1943ndash1967 1992

[3] Z A Wang and T Hillen ldquoClassical solutions and pattern for-mation for a volume filling chemotaxis modelrdquo Chaos vol 17no 3 Article ID 037108 2007

[4] N F Britton Essential Mathematical Biology Springer 2003[5] M C Cross and P H Hohenberg ldquoPattern formation outside of

equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp 851ndash1112 1993

[6] K J Lee W D McCormick Q Ouyang and H L SwinneyldquoPattern formation by interacting chemical frontsrdquo Science vol261 no 5118 pp 192ndash194 1993

[7] A B Medvinsky S V Petrovskii I A Tikhonova H Malchowand B-L Li ldquoSpatiotemporal complexity of plankton and fishdynamicsrdquo SIAM Review vol 44 no 3 pp 311ndash370 2002

[8] W-M Ni andM Tang ldquoTuring patterns in the Lengyel-Epsteinsystem for the CIMA reactionrdquo Transactions of the AmericanMathematical Society vol 357 no 10 pp 3953ndash3969 2005

[9] R B Hoyle Pattern Formation An Introduction to MethodsCambridge University Press Cambridge UK 2006

[10] M JWard and JWei ldquoThe existence and stability of asymmetricspike patterns for the Schnakenberg modelrdquo Studies in AppliedMathematics vol 109 no 3 pp 229ndash264 2002

[11] M J Ward ldquoAsymptotic methods for reaction-diffusion sys-tems past and presentrdquo Bulletin of Mathematical Biology vol68 no 5 pp 1151ndash1167 2006

[12] J Wei ldquoPattern formations in two-dimensional Gray-Scottmodel existence of single-spot solutions and their stabilityrdquoPhysica D vol 148 no 1-2 pp 20ndash48 2001

[13] J Wei and M Winter ldquoStationary multiple spots for reaction-diffusion systemsrdquo Journal of Mathematical Biology vol 57 no1 pp 53ndash89 2008

[14] W Chen and M J Ward ldquoThe stability and dynamics of local-ized spot patterns in the two-dimensional Gray-Scott modelrdquo

Abstract and Applied Analysis 9

SIAM Journal on Applied Dynamical Systems vol 10 no 2 pp582ndash666 2011

[15] R McKay and T Kolokolnikov ldquoStability transitions anddynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimensionrdquo Discrete and Con-tinuous Dynamical Systems B vol 17 no 1 pp 191ndash220 2012

[16] L Segel and J Jackson ldquoDissipative structure an explanationand an ecological examplerdquo Journal of Theoretical Biology vol37 no 3 pp 545ndash559 1972

[17] A Gierer and H Meinhardt ldquoA theory of biological patternformationrdquo Biological Cybernetics vol 12 no 1 pp 30ndash39 1972

[18] S Levin and L Segel ldquoHypothesis for origin of planktonic pat-chinessrdquo Nature vol 259 no 5545 p 659 1976

[19] S A Levin and L A Segel ldquoPattern generation in space andaspectrdquo SIAM Review A vol 27 no 1 pp 45ndash67 1985

[20] J D MurrayMathematical Biology Springer 2003[21] A Okubo and S A Levin Diffusion and Ecological Problems

Modern Perspectives Springer NewYorkNYUSA 2nd edition2001

[22] C Neuhauser ldquoMathematical challenges in spatial ecologyrdquoNotices of the AmericanMathematical Society vol 48 no 11 pp1304ndash1314 2001

[23] R S Cantrell and C Cosner Spatial Ecology via Reaction-Diffusion Equations JohnWiley amp SonsWest Sussex UK 2003

[24] T K Callahan and E Knobloch ldquoPattern formation in three-dimensional reaction-diffusion systemsrdquo Physica D vol 132 no3 pp 339ndash362 1999

[25] A Bhattacharyay ldquoSpirals and targets in reaction-diffusionsystemsrdquo Physical Review E vol 64 no 1 Article ID 0161132001

[26] T Leppanen M Karttunen K Kaski and R Barrio ldquoDimen-sionality effects inTuring pattern formationrdquo International Jour-nal of Modern Physics B vol 17 no 29 pp 5541ndash5553 2003

[27] J He ldquoAsymptotic methods for solitary solutions and compac-tonsrdquoAbstract andAppliedAnalysis vol 2012 Article ID 916793130 pages 2012

[28] G W Harrison ldquoMultiple stable equilibria in a predator-preysystemrdquo Bulletin of Mathematical Biology vol 48 no 2 pp 137ndash148 1986

[29] G W Harrison ldquoComparing predator-prey models to Luckin-billrsquos experiment with didinium and parameciumrdquo Ecology vol76 no 2 pp 357ndash374 1995

[30] D Alonso F Bartumeus and J Catalan ldquoMutual interferencebetween predators can give rise to Turing spatial patternsrdquo Ecol-ogy vol 83 no 1 pp 28ndash34 2002

[31] M Baurmann T Gross and U Feudel ldquoInstabilities in spatiallyextended predator-prey systems spatio-temporal patterns inthe neighborhood of Turing-Hopf bifurcationsrdquo Journal ofTheoretical Biology vol 245 no 2 pp 220ndash229 2007

[32] WWang Q-X Liu and Z Jin ldquoSpatiotemporal complexity of aratio-dependent predator-prey systemrdquo Physical Review E vol75 no 5 Article ID 051913 p 9 2007

[33] WWang L Zhang HWang and Z Li ldquoPattern formation of apredator-prey systemwith Ivlev-type functional responserdquo Eco-logical Modelling vol 221 no 2 pp 131ndash140 2010

[34] R K Upadhyay W Wang and N K Thakur ldquoSpatiotemporaldynamics in a spatial plankton systemrdquoMathematicalModellingof Natural Phenomena vol 5 no 5 pp 102ndash122 2010

[35] JWang J Shi and JWei ldquoDynamics and pattern formation in adiffusive predator-prey system with strong Allee effect in preyrdquo

Journal of Differential Equations vol 251 no 4-5 pp 1276ndash13042011

[36] M Banerjee and S Petrovskii ldquoSelf-organised spatial patternsand chaos in a ratiodependent predatorndashprey systemrdquoTheoret-ical Ecology vol 4 no 1 pp 37ndash53 2011

[37] M Banerjee ldquoSpatial pattern formation in ratio-dependentmodel higher-order stability analysisrdquo Mathematical Medicineand Biology vol 28 no 2 pp 111ndash128 2011

[38] M Banerjee and S Banerjee ldquoTuring instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner modelrdquoMathematical Biosciences vol 236 no 1 pp 64ndash76 2012

[39] S Fasani and S Rinaldi ldquoFactors promoting or inhibitingTuring instability in spatially extended preyndashpredator systemsrdquoEcological Modelling vol 222 no 18 pp 3449ndash3452 2011

[40] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[41] W C Allee Animal Aggregations A Study in General SociologyAMS Press 1978

[42] F Courchamp J Berec and J GascoigneAllee Effects in Ecologyand Conservation Oxford University Press New York NYUSA 2008

[43] BDennis ldquoAllee effects population growth critical density andthe chance of extinctionrdquoNatural Resource Modeling vol 3 no4 pp 481ndash538 1989

[44] MAMcCarthy ldquoTheAllee effect findingmates and theoreticalmodelsrdquo Ecological Modelling vol 103 no 1 pp 99ndash102 1997

[45] GWang X G Liang and F ZWang ldquoThe competitive dynam-ics of populations subject to an Allee effectrdquo Ecological Mod-elling vol 124 no 2 pp 183ndash192 1999

[46] MA Burgman S FersonH RAkcakaya et alRiskAssessmentin Conservation Biology Chapman amp Hall London UK 1993

[47] M Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993

[48] P A Stephens andW J Sutherland ldquoConsequences of the Alleeeffect for behaviour ecology and conservationrdquo Trends in Ecol-ogy and Evolution vol 14 no 10 pp 401ndash405 1999

[49] P A Stephens W J Sutherland and R P Freckleton ldquoWhat isthe Allee effectrdquo Oikos vol 87 no 1 pp 185ndash190 1999

[50] T H Keitt M A Lewis and R D Holt ldquoAllee effects invasionpinning and Speciesrsquo bordersrdquoAmericanNaturalist vol 157 no2 pp 203ndash216 2002

[51] S R Zhou Y F Liu and G Wang ldquoThe stability of predatorndashprey systems subject to the Allee effectsrdquoTheoretical PopulationBiology vol 67 no 1 pp 23ndash31 2005

[52] S Petrovskii A Morozov and B-L Li ldquoRegimes of biologicalinvasion in a predator-prey system with the Allee effectrdquo Bul-letin of Mathematical Biology vol 67 no 3 pp 637ndash661 2005

[53] J Shi and R Shivaji ldquoPersistence in reaction diffusion modelswith weak Allee effectrdquo Journal of Mathematical Biology vol 52no 6 pp 807ndash829 2006

[54] A Morozov S Petrovskii and B-L Li ldquoSpatiotemporal com-plexity of patchy invasion in a predator-prey system with theAllee effectrdquo Journal of Theoretical Biology vol 238 no 1 pp18ndash35 2006

[55] L Roques A Roques H Berestycki and A KretzschmarldquoA population facing climate change joint influences of Alleeeffects and environmental boundary geometryrdquoPopulation Eco-logy vol 50 no 2 pp 215ndash225 2008

10 Abstract and Applied Analysis

[56] C Celik andODuman ldquoAllee effect in a discrete-time predator-prey systemrdquo Chaos Solitons and Fractals vol 40 no 4 pp1956ndash1962 2009

[57] J Zu andMMimura ldquoThe impact of Allee effect on a predator-prey system with Holling type II functional responserdquo AppliedMathematics and Computation vol 217 no 7 pp 3542ndash35562010

[58] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[59] E Gonzalez-Olivares H Meneses-Alcay B Gonzalez-Yanez JMena-Lorca A Rojas-Palma and R Ramos-Jiliberto ldquoMultiplestability and uniqueness of the limit cycle in a Gause-typepredator-prey model considering the Allee effect on preyrdquoNon-linear Analysis Real World Applicationsl vol 12 no 6 pp 2931ndash2942 2011

[60] P Aguirre E Gonzalez-Olivares and E Saez ldquoTwo limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo Nonlinear Analysis Real World Applications vol 10 no3 pp 1401ndash1416 2009

[61] PAguirre EGonzalez-Olivares andE Saez ldquoThree limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo SIAM Journal on Applied Mathematics vol 69 no 5 pp1244ndash1262 2009

[62] Y Cai W Wang and J Wang ldquoDynamics of a diffusivepredator-prey model with additive Allee effectrdquo InternationalJournal of Biomathematics vol 5 no 2 Article ID 1250023 2012

[63] M E Solomon ldquoThenatural control of animal populationsrdquoTheJournal of Animal Ecology vol 18 no 1 pp 1ndash35 1949

[64] A J Lotka Elements of Physical Biology Williams and Wilkins1925

[65] C Jost Comparing PredatorndashPrey Models Qualitatively andQuantitatively with Ecological Time-Series Data InstitutNational Agronomique Paris-Grignon 1998

[66] H Malchow S V Petrovskii and E Venturino SpatiotemporalPatterns in Ecology and Epidemiology Chapman amp Hall BocaRaton Fla USA 2008

[67] M R Garvie ldquoFinite-difference schemes for reaction-diffusionequations modeling predator-prey interactions in MATLABrdquoBulletin of Mathematical Biology vol 69 no 3 pp 931ndash9562007

[68] A Munteanu and R Sole ldquoPattern formation in noisy self-rep-licating spotsrdquo International Journal of Bifurcation and Chaosvol 16 no 12 pp 3679ndash3683 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Allee-Effect-Induced Instability in a Reaction-Diffusion ...emis.maths.adelaide.edu.au/.../Volume2013/487810.pdf · 2 AbstractandAppliedAnalysis interactionmodelwithlogisticgrowthrateofthepreyinthe

Abstract and Applied Analysis 7

0177

01765

0176

01755

0175

01745

0174

01735

(a)

03

025

02

015

01

(b)

045

04

035

03

025

02

015

01

005

(c)

045

04

035

03

025

02

015

01

005

(d)

Figure 3 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 119904 = 301198891= 0015 and 119889

2= 1 Times (a) 0 (b) 50 (c) 250 (d) 2500

When 119904 increasing to 119904 = 30 the later random pertur-bations make these stripes decay end with the time-indepen-dent regular spots (cf Figure 3) which is isolated zones withhigh prey densities

In Figure 4 we show patterns of time-independentstripes-holes and stripes-spots mixture obtained with model(7) These two patterns are similar to each other With 119904 =19 (cf Figure 4(a)) the stripes-holes mixture pattern is atrelatively low prey densities while 119904 = 245 (cf Figure 4(b))at high prey densities

From Figures 1 to 4 one can see that on increasing thecontrol parameter 119904 the pattern sequence ldquoholes rarr stripes-holes mixture rarr stripes rarr stripes-spots mixture rarr spotsrdquois observed

From the viewpoint of population dynamics ldquospotsrdquo pat-tern (cf Figure 3) shows that the prey population is driven bypredator to a very low level in those regions The final resultis the formation of patches of high prey density surroundedby areas of low prey densities [30] That is to say under the

control of these parameters the prey is predominant in thedomain In contrast ldquoholesrdquo pattern (cf Figure 1) indicatesthat the predator is predominant in the domain

4 Conclusions and Remarks

In summary in this paper we have investigated the spa-tiotemporal dynamics of a predator-prey model that involvesAllee effect on prey analytically and numerically

For model (7) in the case without Allee effect there isno effect on the stability of the positive equilibrium whetherwith diffusion or not That is to say there is nonexistence ofdiffusion-driven instability in the model without Allee effectMore precisely the distribution of species converge to a spa-tially homogeneous steady state which varies in time

And in the case with Allee effect the positive equilibriummay be unstableThis instability is induced byAllee effect anddiffusion together so we give a new definition called ldquoAllee-diffusion-driven instabilityrdquo and present the analysis of this

8 Abstract and Applied Analysis

05

045

04

035

03

025

02

015

01

005

(a)

05

045

04

035

03

025

02

015

01

005

(b)

Figure 4 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 1198891= 0015

and 1198892= 1 (a) 119904 = 19 (b) 119904 = 245

instability of the model in details To the best of our knowl-edge this is the first reported case Furthermore via numer-ical simulations it is found that the model dynamics exhibitsboth Allee effect and diffusion controlled pattern formationgrowth to holes stripes-holes mixtures stripes stripes-spotsmixtures and spots replicationThat is to say the distributionof species is aggregation This indicates that the pattern for-mation of the model with Allee effect is not simple but richand complex

In fact for a predator-prey system Okubo and Levin [21]noted Allee effect on the functional response and a density-dependent death rate of the predator is necessary to generatespatial patterns And in this paper we show that a predator-prey system with Allee effect on prey can generate complexTuring spatial patterns which may be a supplementary to[21]

It is needed to note that in this paper we investigate thedynamics of localized patterns inmodel (7) Such patterns arecharacterized by a highly spatially heterogeneous solutionsand are far from the spatially uniform state These patternsoccur in two-component systems when the ratio of the twodiffusion coefficients are very large In the numerical simula-tions we take the diffusivity ratio as 10015 ≫ 1 and so weare close to the regime of localized patterns And the existenceof these spatial patterns can be rigorously proved usingtools from nonlinear functional analysis such as Liapunov-Schmidt reduction and fixed-point theorems [13 14] this isdesirable in future studies

Acknowledgments

The authors would like to thank the anonymous refereefor very helpful suggestions and comments which led toimprovements of our original paper This research was sup-ported by Natural Science Foundation of Zhejiang Province(LY12A01014 and LQ12A01009) and the National BasicResearch Program of China (2012 CB 426510)

References

[1] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society B vol 237 pp 37ndash72 1952

[2] S Levin ldquoThe problem of pattern and scale in ecologyrdquo Ecologyvol 73 no 6 pp 1943ndash1967 1992

[3] Z A Wang and T Hillen ldquoClassical solutions and pattern for-mation for a volume filling chemotaxis modelrdquo Chaos vol 17no 3 Article ID 037108 2007

[4] N F Britton Essential Mathematical Biology Springer 2003[5] M C Cross and P H Hohenberg ldquoPattern formation outside of

equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp 851ndash1112 1993

[6] K J Lee W D McCormick Q Ouyang and H L SwinneyldquoPattern formation by interacting chemical frontsrdquo Science vol261 no 5118 pp 192ndash194 1993

[7] A B Medvinsky S V Petrovskii I A Tikhonova H Malchowand B-L Li ldquoSpatiotemporal complexity of plankton and fishdynamicsrdquo SIAM Review vol 44 no 3 pp 311ndash370 2002

[8] W-M Ni andM Tang ldquoTuring patterns in the Lengyel-Epsteinsystem for the CIMA reactionrdquo Transactions of the AmericanMathematical Society vol 357 no 10 pp 3953ndash3969 2005

[9] R B Hoyle Pattern Formation An Introduction to MethodsCambridge University Press Cambridge UK 2006

[10] M JWard and JWei ldquoThe existence and stability of asymmetricspike patterns for the Schnakenberg modelrdquo Studies in AppliedMathematics vol 109 no 3 pp 229ndash264 2002

[11] M J Ward ldquoAsymptotic methods for reaction-diffusion sys-tems past and presentrdquo Bulletin of Mathematical Biology vol68 no 5 pp 1151ndash1167 2006

[12] J Wei ldquoPattern formations in two-dimensional Gray-Scottmodel existence of single-spot solutions and their stabilityrdquoPhysica D vol 148 no 1-2 pp 20ndash48 2001

[13] J Wei and M Winter ldquoStationary multiple spots for reaction-diffusion systemsrdquo Journal of Mathematical Biology vol 57 no1 pp 53ndash89 2008

[14] W Chen and M J Ward ldquoThe stability and dynamics of local-ized spot patterns in the two-dimensional Gray-Scott modelrdquo

Abstract and Applied Analysis 9

SIAM Journal on Applied Dynamical Systems vol 10 no 2 pp582ndash666 2011

[15] R McKay and T Kolokolnikov ldquoStability transitions anddynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimensionrdquo Discrete and Con-tinuous Dynamical Systems B vol 17 no 1 pp 191ndash220 2012

[16] L Segel and J Jackson ldquoDissipative structure an explanationand an ecological examplerdquo Journal of Theoretical Biology vol37 no 3 pp 545ndash559 1972

[17] A Gierer and H Meinhardt ldquoA theory of biological patternformationrdquo Biological Cybernetics vol 12 no 1 pp 30ndash39 1972

[18] S Levin and L Segel ldquoHypothesis for origin of planktonic pat-chinessrdquo Nature vol 259 no 5545 p 659 1976

[19] S A Levin and L A Segel ldquoPattern generation in space andaspectrdquo SIAM Review A vol 27 no 1 pp 45ndash67 1985

[20] J D MurrayMathematical Biology Springer 2003[21] A Okubo and S A Levin Diffusion and Ecological Problems

Modern Perspectives Springer NewYorkNYUSA 2nd edition2001

[22] C Neuhauser ldquoMathematical challenges in spatial ecologyrdquoNotices of the AmericanMathematical Society vol 48 no 11 pp1304ndash1314 2001

[23] R S Cantrell and C Cosner Spatial Ecology via Reaction-Diffusion Equations JohnWiley amp SonsWest Sussex UK 2003

[24] T K Callahan and E Knobloch ldquoPattern formation in three-dimensional reaction-diffusion systemsrdquo Physica D vol 132 no3 pp 339ndash362 1999

[25] A Bhattacharyay ldquoSpirals and targets in reaction-diffusionsystemsrdquo Physical Review E vol 64 no 1 Article ID 0161132001

[26] T Leppanen M Karttunen K Kaski and R Barrio ldquoDimen-sionality effects inTuring pattern formationrdquo International Jour-nal of Modern Physics B vol 17 no 29 pp 5541ndash5553 2003

[27] J He ldquoAsymptotic methods for solitary solutions and compac-tonsrdquoAbstract andAppliedAnalysis vol 2012 Article ID 916793130 pages 2012

[28] G W Harrison ldquoMultiple stable equilibria in a predator-preysystemrdquo Bulletin of Mathematical Biology vol 48 no 2 pp 137ndash148 1986

[29] G W Harrison ldquoComparing predator-prey models to Luckin-billrsquos experiment with didinium and parameciumrdquo Ecology vol76 no 2 pp 357ndash374 1995

[30] D Alonso F Bartumeus and J Catalan ldquoMutual interferencebetween predators can give rise to Turing spatial patternsrdquo Ecol-ogy vol 83 no 1 pp 28ndash34 2002

[31] M Baurmann T Gross and U Feudel ldquoInstabilities in spatiallyextended predator-prey systems spatio-temporal patterns inthe neighborhood of Turing-Hopf bifurcationsrdquo Journal ofTheoretical Biology vol 245 no 2 pp 220ndash229 2007

[32] WWang Q-X Liu and Z Jin ldquoSpatiotemporal complexity of aratio-dependent predator-prey systemrdquo Physical Review E vol75 no 5 Article ID 051913 p 9 2007

[33] WWang L Zhang HWang and Z Li ldquoPattern formation of apredator-prey systemwith Ivlev-type functional responserdquo Eco-logical Modelling vol 221 no 2 pp 131ndash140 2010

[34] R K Upadhyay W Wang and N K Thakur ldquoSpatiotemporaldynamics in a spatial plankton systemrdquoMathematicalModellingof Natural Phenomena vol 5 no 5 pp 102ndash122 2010

[35] JWang J Shi and JWei ldquoDynamics and pattern formation in adiffusive predator-prey system with strong Allee effect in preyrdquo

Journal of Differential Equations vol 251 no 4-5 pp 1276ndash13042011

[36] M Banerjee and S Petrovskii ldquoSelf-organised spatial patternsand chaos in a ratiodependent predatorndashprey systemrdquoTheoret-ical Ecology vol 4 no 1 pp 37ndash53 2011

[37] M Banerjee ldquoSpatial pattern formation in ratio-dependentmodel higher-order stability analysisrdquo Mathematical Medicineand Biology vol 28 no 2 pp 111ndash128 2011

[38] M Banerjee and S Banerjee ldquoTuring instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner modelrdquoMathematical Biosciences vol 236 no 1 pp 64ndash76 2012

[39] S Fasani and S Rinaldi ldquoFactors promoting or inhibitingTuring instability in spatially extended preyndashpredator systemsrdquoEcological Modelling vol 222 no 18 pp 3449ndash3452 2011

[40] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[41] W C Allee Animal Aggregations A Study in General SociologyAMS Press 1978

[42] F Courchamp J Berec and J GascoigneAllee Effects in Ecologyand Conservation Oxford University Press New York NYUSA 2008

[43] BDennis ldquoAllee effects population growth critical density andthe chance of extinctionrdquoNatural Resource Modeling vol 3 no4 pp 481ndash538 1989

[44] MAMcCarthy ldquoTheAllee effect findingmates and theoreticalmodelsrdquo Ecological Modelling vol 103 no 1 pp 99ndash102 1997

[45] GWang X G Liang and F ZWang ldquoThe competitive dynam-ics of populations subject to an Allee effectrdquo Ecological Mod-elling vol 124 no 2 pp 183ndash192 1999

[46] MA Burgman S FersonH RAkcakaya et alRiskAssessmentin Conservation Biology Chapman amp Hall London UK 1993

[47] M Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993

[48] P A Stephens andW J Sutherland ldquoConsequences of the Alleeeffect for behaviour ecology and conservationrdquo Trends in Ecol-ogy and Evolution vol 14 no 10 pp 401ndash405 1999

[49] P A Stephens W J Sutherland and R P Freckleton ldquoWhat isthe Allee effectrdquo Oikos vol 87 no 1 pp 185ndash190 1999

[50] T H Keitt M A Lewis and R D Holt ldquoAllee effects invasionpinning and Speciesrsquo bordersrdquoAmericanNaturalist vol 157 no2 pp 203ndash216 2002

[51] S R Zhou Y F Liu and G Wang ldquoThe stability of predatorndashprey systems subject to the Allee effectsrdquoTheoretical PopulationBiology vol 67 no 1 pp 23ndash31 2005

[52] S Petrovskii A Morozov and B-L Li ldquoRegimes of biologicalinvasion in a predator-prey system with the Allee effectrdquo Bul-letin of Mathematical Biology vol 67 no 3 pp 637ndash661 2005

[53] J Shi and R Shivaji ldquoPersistence in reaction diffusion modelswith weak Allee effectrdquo Journal of Mathematical Biology vol 52no 6 pp 807ndash829 2006

[54] A Morozov S Petrovskii and B-L Li ldquoSpatiotemporal com-plexity of patchy invasion in a predator-prey system with theAllee effectrdquo Journal of Theoretical Biology vol 238 no 1 pp18ndash35 2006

[55] L Roques A Roques H Berestycki and A KretzschmarldquoA population facing climate change joint influences of Alleeeffects and environmental boundary geometryrdquoPopulation Eco-logy vol 50 no 2 pp 215ndash225 2008

10 Abstract and Applied Analysis

[56] C Celik andODuman ldquoAllee effect in a discrete-time predator-prey systemrdquo Chaos Solitons and Fractals vol 40 no 4 pp1956ndash1962 2009

[57] J Zu andMMimura ldquoThe impact of Allee effect on a predator-prey system with Holling type II functional responserdquo AppliedMathematics and Computation vol 217 no 7 pp 3542ndash35562010

[58] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[59] E Gonzalez-Olivares H Meneses-Alcay B Gonzalez-Yanez JMena-Lorca A Rojas-Palma and R Ramos-Jiliberto ldquoMultiplestability and uniqueness of the limit cycle in a Gause-typepredator-prey model considering the Allee effect on preyrdquoNon-linear Analysis Real World Applicationsl vol 12 no 6 pp 2931ndash2942 2011

[60] P Aguirre E Gonzalez-Olivares and E Saez ldquoTwo limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo Nonlinear Analysis Real World Applications vol 10 no3 pp 1401ndash1416 2009

[61] PAguirre EGonzalez-Olivares andE Saez ldquoThree limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo SIAM Journal on Applied Mathematics vol 69 no 5 pp1244ndash1262 2009

[62] Y Cai W Wang and J Wang ldquoDynamics of a diffusivepredator-prey model with additive Allee effectrdquo InternationalJournal of Biomathematics vol 5 no 2 Article ID 1250023 2012

[63] M E Solomon ldquoThenatural control of animal populationsrdquoTheJournal of Animal Ecology vol 18 no 1 pp 1ndash35 1949

[64] A J Lotka Elements of Physical Biology Williams and Wilkins1925

[65] C Jost Comparing PredatorndashPrey Models Qualitatively andQuantitatively with Ecological Time-Series Data InstitutNational Agronomique Paris-Grignon 1998

[66] H Malchow S V Petrovskii and E Venturino SpatiotemporalPatterns in Ecology and Epidemiology Chapman amp Hall BocaRaton Fla USA 2008

[67] M R Garvie ldquoFinite-difference schemes for reaction-diffusionequations modeling predator-prey interactions in MATLABrdquoBulletin of Mathematical Biology vol 69 no 3 pp 931ndash9562007

[68] A Munteanu and R Sole ldquoPattern formation in noisy self-rep-licating spotsrdquo International Journal of Bifurcation and Chaosvol 16 no 12 pp 3679ndash3683 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Allee-Effect-Induced Instability in a Reaction-Diffusion ...emis.maths.adelaide.edu.au/.../Volume2013/487810.pdf · 2 AbstractandAppliedAnalysis interactionmodelwithlogisticgrowthrateofthepreyinthe

8 Abstract and Applied Analysis

05

045

04

035

03

025

02

015

01

005

(a)

05

045

04

035

03

025

02

015

01

005

(b)

Figure 4 Typical Turing patterns of 119906 in model (7) with parameters 120572 = 1 120573 = 03 120574 = 03 119887 = 05 119888 = 06 119898 = 06 119902 = 035 1198891= 0015

and 1198892= 1 (a) 119904 = 19 (b) 119904 = 245

instability of the model in details To the best of our knowl-edge this is the first reported case Furthermore via numer-ical simulations it is found that the model dynamics exhibitsboth Allee effect and diffusion controlled pattern formationgrowth to holes stripes-holes mixtures stripes stripes-spotsmixtures and spots replicationThat is to say the distributionof species is aggregation This indicates that the pattern for-mation of the model with Allee effect is not simple but richand complex

In fact for a predator-prey system Okubo and Levin [21]noted Allee effect on the functional response and a density-dependent death rate of the predator is necessary to generatespatial patterns And in this paper we show that a predator-prey system with Allee effect on prey can generate complexTuring spatial patterns which may be a supplementary to[21]

It is needed to note that in this paper we investigate thedynamics of localized patterns inmodel (7) Such patterns arecharacterized by a highly spatially heterogeneous solutionsand are far from the spatially uniform state These patternsoccur in two-component systems when the ratio of the twodiffusion coefficients are very large In the numerical simula-tions we take the diffusivity ratio as 10015 ≫ 1 and so weare close to the regime of localized patterns And the existenceof these spatial patterns can be rigorously proved usingtools from nonlinear functional analysis such as Liapunov-Schmidt reduction and fixed-point theorems [13 14] this isdesirable in future studies

Acknowledgments

The authors would like to thank the anonymous refereefor very helpful suggestions and comments which led toimprovements of our original paper This research was sup-ported by Natural Science Foundation of Zhejiang Province(LY12A01014 and LQ12A01009) and the National BasicResearch Program of China (2012 CB 426510)

References

[1] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society B vol 237 pp 37ndash72 1952

[2] S Levin ldquoThe problem of pattern and scale in ecologyrdquo Ecologyvol 73 no 6 pp 1943ndash1967 1992

[3] Z A Wang and T Hillen ldquoClassical solutions and pattern for-mation for a volume filling chemotaxis modelrdquo Chaos vol 17no 3 Article ID 037108 2007

[4] N F Britton Essential Mathematical Biology Springer 2003[5] M C Cross and P H Hohenberg ldquoPattern formation outside of

equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp 851ndash1112 1993

[6] K J Lee W D McCormick Q Ouyang and H L SwinneyldquoPattern formation by interacting chemical frontsrdquo Science vol261 no 5118 pp 192ndash194 1993

[7] A B Medvinsky S V Petrovskii I A Tikhonova H Malchowand B-L Li ldquoSpatiotemporal complexity of plankton and fishdynamicsrdquo SIAM Review vol 44 no 3 pp 311ndash370 2002

[8] W-M Ni andM Tang ldquoTuring patterns in the Lengyel-Epsteinsystem for the CIMA reactionrdquo Transactions of the AmericanMathematical Society vol 357 no 10 pp 3953ndash3969 2005

[9] R B Hoyle Pattern Formation An Introduction to MethodsCambridge University Press Cambridge UK 2006

[10] M JWard and JWei ldquoThe existence and stability of asymmetricspike patterns for the Schnakenberg modelrdquo Studies in AppliedMathematics vol 109 no 3 pp 229ndash264 2002

[11] M J Ward ldquoAsymptotic methods for reaction-diffusion sys-tems past and presentrdquo Bulletin of Mathematical Biology vol68 no 5 pp 1151ndash1167 2006

[12] J Wei ldquoPattern formations in two-dimensional Gray-Scottmodel existence of single-spot solutions and their stabilityrdquoPhysica D vol 148 no 1-2 pp 20ndash48 2001

[13] J Wei and M Winter ldquoStationary multiple spots for reaction-diffusion systemsrdquo Journal of Mathematical Biology vol 57 no1 pp 53ndash89 2008

[14] W Chen and M J Ward ldquoThe stability and dynamics of local-ized spot patterns in the two-dimensional Gray-Scott modelrdquo

Abstract and Applied Analysis 9

SIAM Journal on Applied Dynamical Systems vol 10 no 2 pp582ndash666 2011

[15] R McKay and T Kolokolnikov ldquoStability transitions anddynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimensionrdquo Discrete and Con-tinuous Dynamical Systems B vol 17 no 1 pp 191ndash220 2012

[16] L Segel and J Jackson ldquoDissipative structure an explanationand an ecological examplerdquo Journal of Theoretical Biology vol37 no 3 pp 545ndash559 1972

[17] A Gierer and H Meinhardt ldquoA theory of biological patternformationrdquo Biological Cybernetics vol 12 no 1 pp 30ndash39 1972

[18] S Levin and L Segel ldquoHypothesis for origin of planktonic pat-chinessrdquo Nature vol 259 no 5545 p 659 1976

[19] S A Levin and L A Segel ldquoPattern generation in space andaspectrdquo SIAM Review A vol 27 no 1 pp 45ndash67 1985

[20] J D MurrayMathematical Biology Springer 2003[21] A Okubo and S A Levin Diffusion and Ecological Problems

Modern Perspectives Springer NewYorkNYUSA 2nd edition2001

[22] C Neuhauser ldquoMathematical challenges in spatial ecologyrdquoNotices of the AmericanMathematical Society vol 48 no 11 pp1304ndash1314 2001

[23] R S Cantrell and C Cosner Spatial Ecology via Reaction-Diffusion Equations JohnWiley amp SonsWest Sussex UK 2003

[24] T K Callahan and E Knobloch ldquoPattern formation in three-dimensional reaction-diffusion systemsrdquo Physica D vol 132 no3 pp 339ndash362 1999

[25] A Bhattacharyay ldquoSpirals and targets in reaction-diffusionsystemsrdquo Physical Review E vol 64 no 1 Article ID 0161132001

[26] T Leppanen M Karttunen K Kaski and R Barrio ldquoDimen-sionality effects inTuring pattern formationrdquo International Jour-nal of Modern Physics B vol 17 no 29 pp 5541ndash5553 2003

[27] J He ldquoAsymptotic methods for solitary solutions and compac-tonsrdquoAbstract andAppliedAnalysis vol 2012 Article ID 916793130 pages 2012

[28] G W Harrison ldquoMultiple stable equilibria in a predator-preysystemrdquo Bulletin of Mathematical Biology vol 48 no 2 pp 137ndash148 1986

[29] G W Harrison ldquoComparing predator-prey models to Luckin-billrsquos experiment with didinium and parameciumrdquo Ecology vol76 no 2 pp 357ndash374 1995

[30] D Alonso F Bartumeus and J Catalan ldquoMutual interferencebetween predators can give rise to Turing spatial patternsrdquo Ecol-ogy vol 83 no 1 pp 28ndash34 2002

[31] M Baurmann T Gross and U Feudel ldquoInstabilities in spatiallyextended predator-prey systems spatio-temporal patterns inthe neighborhood of Turing-Hopf bifurcationsrdquo Journal ofTheoretical Biology vol 245 no 2 pp 220ndash229 2007

[32] WWang Q-X Liu and Z Jin ldquoSpatiotemporal complexity of aratio-dependent predator-prey systemrdquo Physical Review E vol75 no 5 Article ID 051913 p 9 2007

[33] WWang L Zhang HWang and Z Li ldquoPattern formation of apredator-prey systemwith Ivlev-type functional responserdquo Eco-logical Modelling vol 221 no 2 pp 131ndash140 2010

[34] R K Upadhyay W Wang and N K Thakur ldquoSpatiotemporaldynamics in a spatial plankton systemrdquoMathematicalModellingof Natural Phenomena vol 5 no 5 pp 102ndash122 2010

[35] JWang J Shi and JWei ldquoDynamics and pattern formation in adiffusive predator-prey system with strong Allee effect in preyrdquo

Journal of Differential Equations vol 251 no 4-5 pp 1276ndash13042011

[36] M Banerjee and S Petrovskii ldquoSelf-organised spatial patternsand chaos in a ratiodependent predatorndashprey systemrdquoTheoret-ical Ecology vol 4 no 1 pp 37ndash53 2011

[37] M Banerjee ldquoSpatial pattern formation in ratio-dependentmodel higher-order stability analysisrdquo Mathematical Medicineand Biology vol 28 no 2 pp 111ndash128 2011

[38] M Banerjee and S Banerjee ldquoTuring instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner modelrdquoMathematical Biosciences vol 236 no 1 pp 64ndash76 2012

[39] S Fasani and S Rinaldi ldquoFactors promoting or inhibitingTuring instability in spatially extended preyndashpredator systemsrdquoEcological Modelling vol 222 no 18 pp 3449ndash3452 2011

[40] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[41] W C Allee Animal Aggregations A Study in General SociologyAMS Press 1978

[42] F Courchamp J Berec and J GascoigneAllee Effects in Ecologyand Conservation Oxford University Press New York NYUSA 2008

[43] BDennis ldquoAllee effects population growth critical density andthe chance of extinctionrdquoNatural Resource Modeling vol 3 no4 pp 481ndash538 1989

[44] MAMcCarthy ldquoTheAllee effect findingmates and theoreticalmodelsrdquo Ecological Modelling vol 103 no 1 pp 99ndash102 1997

[45] GWang X G Liang and F ZWang ldquoThe competitive dynam-ics of populations subject to an Allee effectrdquo Ecological Mod-elling vol 124 no 2 pp 183ndash192 1999

[46] MA Burgman S FersonH RAkcakaya et alRiskAssessmentin Conservation Biology Chapman amp Hall London UK 1993

[47] M Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993

[48] P A Stephens andW J Sutherland ldquoConsequences of the Alleeeffect for behaviour ecology and conservationrdquo Trends in Ecol-ogy and Evolution vol 14 no 10 pp 401ndash405 1999

[49] P A Stephens W J Sutherland and R P Freckleton ldquoWhat isthe Allee effectrdquo Oikos vol 87 no 1 pp 185ndash190 1999

[50] T H Keitt M A Lewis and R D Holt ldquoAllee effects invasionpinning and Speciesrsquo bordersrdquoAmericanNaturalist vol 157 no2 pp 203ndash216 2002

[51] S R Zhou Y F Liu and G Wang ldquoThe stability of predatorndashprey systems subject to the Allee effectsrdquoTheoretical PopulationBiology vol 67 no 1 pp 23ndash31 2005

[52] S Petrovskii A Morozov and B-L Li ldquoRegimes of biologicalinvasion in a predator-prey system with the Allee effectrdquo Bul-letin of Mathematical Biology vol 67 no 3 pp 637ndash661 2005

[53] J Shi and R Shivaji ldquoPersistence in reaction diffusion modelswith weak Allee effectrdquo Journal of Mathematical Biology vol 52no 6 pp 807ndash829 2006

[54] A Morozov S Petrovskii and B-L Li ldquoSpatiotemporal com-plexity of patchy invasion in a predator-prey system with theAllee effectrdquo Journal of Theoretical Biology vol 238 no 1 pp18ndash35 2006

[55] L Roques A Roques H Berestycki and A KretzschmarldquoA population facing climate change joint influences of Alleeeffects and environmental boundary geometryrdquoPopulation Eco-logy vol 50 no 2 pp 215ndash225 2008

10 Abstract and Applied Analysis

[56] C Celik andODuman ldquoAllee effect in a discrete-time predator-prey systemrdquo Chaos Solitons and Fractals vol 40 no 4 pp1956ndash1962 2009

[57] J Zu andMMimura ldquoThe impact of Allee effect on a predator-prey system with Holling type II functional responserdquo AppliedMathematics and Computation vol 217 no 7 pp 3542ndash35562010

[58] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[59] E Gonzalez-Olivares H Meneses-Alcay B Gonzalez-Yanez JMena-Lorca A Rojas-Palma and R Ramos-Jiliberto ldquoMultiplestability and uniqueness of the limit cycle in a Gause-typepredator-prey model considering the Allee effect on preyrdquoNon-linear Analysis Real World Applicationsl vol 12 no 6 pp 2931ndash2942 2011

[60] P Aguirre E Gonzalez-Olivares and E Saez ldquoTwo limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo Nonlinear Analysis Real World Applications vol 10 no3 pp 1401ndash1416 2009

[61] PAguirre EGonzalez-Olivares andE Saez ldquoThree limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo SIAM Journal on Applied Mathematics vol 69 no 5 pp1244ndash1262 2009

[62] Y Cai W Wang and J Wang ldquoDynamics of a diffusivepredator-prey model with additive Allee effectrdquo InternationalJournal of Biomathematics vol 5 no 2 Article ID 1250023 2012

[63] M E Solomon ldquoThenatural control of animal populationsrdquoTheJournal of Animal Ecology vol 18 no 1 pp 1ndash35 1949

[64] A J Lotka Elements of Physical Biology Williams and Wilkins1925

[65] C Jost Comparing PredatorndashPrey Models Qualitatively andQuantitatively with Ecological Time-Series Data InstitutNational Agronomique Paris-Grignon 1998

[66] H Malchow S V Petrovskii and E Venturino SpatiotemporalPatterns in Ecology and Epidemiology Chapman amp Hall BocaRaton Fla USA 2008

[67] M R Garvie ldquoFinite-difference schemes for reaction-diffusionequations modeling predator-prey interactions in MATLABrdquoBulletin of Mathematical Biology vol 69 no 3 pp 931ndash9562007

[68] A Munteanu and R Sole ldquoPattern formation in noisy self-rep-licating spotsrdquo International Journal of Bifurcation and Chaosvol 16 no 12 pp 3679ndash3683 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Allee-Effect-Induced Instability in a Reaction-Diffusion ...emis.maths.adelaide.edu.au/.../Volume2013/487810.pdf · 2 AbstractandAppliedAnalysis interactionmodelwithlogisticgrowthrateofthepreyinthe

Abstract and Applied Analysis 9

SIAM Journal on Applied Dynamical Systems vol 10 no 2 pp582ndash666 2011

[15] R McKay and T Kolokolnikov ldquoStability transitions anddynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimensionrdquo Discrete and Con-tinuous Dynamical Systems B vol 17 no 1 pp 191ndash220 2012

[16] L Segel and J Jackson ldquoDissipative structure an explanationand an ecological examplerdquo Journal of Theoretical Biology vol37 no 3 pp 545ndash559 1972

[17] A Gierer and H Meinhardt ldquoA theory of biological patternformationrdquo Biological Cybernetics vol 12 no 1 pp 30ndash39 1972

[18] S Levin and L Segel ldquoHypothesis for origin of planktonic pat-chinessrdquo Nature vol 259 no 5545 p 659 1976

[19] S A Levin and L A Segel ldquoPattern generation in space andaspectrdquo SIAM Review A vol 27 no 1 pp 45ndash67 1985

[20] J D MurrayMathematical Biology Springer 2003[21] A Okubo and S A Levin Diffusion and Ecological Problems

Modern Perspectives Springer NewYorkNYUSA 2nd edition2001

[22] C Neuhauser ldquoMathematical challenges in spatial ecologyrdquoNotices of the AmericanMathematical Society vol 48 no 11 pp1304ndash1314 2001

[23] R S Cantrell and C Cosner Spatial Ecology via Reaction-Diffusion Equations JohnWiley amp SonsWest Sussex UK 2003

[24] T K Callahan and E Knobloch ldquoPattern formation in three-dimensional reaction-diffusion systemsrdquo Physica D vol 132 no3 pp 339ndash362 1999

[25] A Bhattacharyay ldquoSpirals and targets in reaction-diffusionsystemsrdquo Physical Review E vol 64 no 1 Article ID 0161132001

[26] T Leppanen M Karttunen K Kaski and R Barrio ldquoDimen-sionality effects inTuring pattern formationrdquo International Jour-nal of Modern Physics B vol 17 no 29 pp 5541ndash5553 2003

[27] J He ldquoAsymptotic methods for solitary solutions and compac-tonsrdquoAbstract andAppliedAnalysis vol 2012 Article ID 916793130 pages 2012

[28] G W Harrison ldquoMultiple stable equilibria in a predator-preysystemrdquo Bulletin of Mathematical Biology vol 48 no 2 pp 137ndash148 1986

[29] G W Harrison ldquoComparing predator-prey models to Luckin-billrsquos experiment with didinium and parameciumrdquo Ecology vol76 no 2 pp 357ndash374 1995

[30] D Alonso F Bartumeus and J Catalan ldquoMutual interferencebetween predators can give rise to Turing spatial patternsrdquo Ecol-ogy vol 83 no 1 pp 28ndash34 2002

[31] M Baurmann T Gross and U Feudel ldquoInstabilities in spatiallyextended predator-prey systems spatio-temporal patterns inthe neighborhood of Turing-Hopf bifurcationsrdquo Journal ofTheoretical Biology vol 245 no 2 pp 220ndash229 2007

[32] WWang Q-X Liu and Z Jin ldquoSpatiotemporal complexity of aratio-dependent predator-prey systemrdquo Physical Review E vol75 no 5 Article ID 051913 p 9 2007

[33] WWang L Zhang HWang and Z Li ldquoPattern formation of apredator-prey systemwith Ivlev-type functional responserdquo Eco-logical Modelling vol 221 no 2 pp 131ndash140 2010

[34] R K Upadhyay W Wang and N K Thakur ldquoSpatiotemporaldynamics in a spatial plankton systemrdquoMathematicalModellingof Natural Phenomena vol 5 no 5 pp 102ndash122 2010

[35] JWang J Shi and JWei ldquoDynamics and pattern formation in adiffusive predator-prey system with strong Allee effect in preyrdquo

Journal of Differential Equations vol 251 no 4-5 pp 1276ndash13042011

[36] M Banerjee and S Petrovskii ldquoSelf-organised spatial patternsand chaos in a ratiodependent predatorndashprey systemrdquoTheoret-ical Ecology vol 4 no 1 pp 37ndash53 2011

[37] M Banerjee ldquoSpatial pattern formation in ratio-dependentmodel higher-order stability analysisrdquo Mathematical Medicineand Biology vol 28 no 2 pp 111ndash128 2011

[38] M Banerjee and S Banerjee ldquoTuring instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner modelrdquoMathematical Biosciences vol 236 no 1 pp 64ndash76 2012

[39] S Fasani and S Rinaldi ldquoFactors promoting or inhibitingTuring instability in spatially extended preyndashpredator systemsrdquoEcological Modelling vol 222 no 18 pp 3449ndash3452 2011

[40] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[41] W C Allee Animal Aggregations A Study in General SociologyAMS Press 1978

[42] F Courchamp J Berec and J GascoigneAllee Effects in Ecologyand Conservation Oxford University Press New York NYUSA 2008

[43] BDennis ldquoAllee effects population growth critical density andthe chance of extinctionrdquoNatural Resource Modeling vol 3 no4 pp 481ndash538 1989

[44] MAMcCarthy ldquoTheAllee effect findingmates and theoreticalmodelsrdquo Ecological Modelling vol 103 no 1 pp 99ndash102 1997

[45] GWang X G Liang and F ZWang ldquoThe competitive dynam-ics of populations subject to an Allee effectrdquo Ecological Mod-elling vol 124 no 2 pp 183ndash192 1999

[46] MA Burgman S FersonH RAkcakaya et alRiskAssessmentin Conservation Biology Chapman amp Hall London UK 1993

[47] M Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993

[48] P A Stephens andW J Sutherland ldquoConsequences of the Alleeeffect for behaviour ecology and conservationrdquo Trends in Ecol-ogy and Evolution vol 14 no 10 pp 401ndash405 1999

[49] P A Stephens W J Sutherland and R P Freckleton ldquoWhat isthe Allee effectrdquo Oikos vol 87 no 1 pp 185ndash190 1999

[50] T H Keitt M A Lewis and R D Holt ldquoAllee effects invasionpinning and Speciesrsquo bordersrdquoAmericanNaturalist vol 157 no2 pp 203ndash216 2002

[51] S R Zhou Y F Liu and G Wang ldquoThe stability of predatorndashprey systems subject to the Allee effectsrdquoTheoretical PopulationBiology vol 67 no 1 pp 23ndash31 2005

[52] S Petrovskii A Morozov and B-L Li ldquoRegimes of biologicalinvasion in a predator-prey system with the Allee effectrdquo Bul-letin of Mathematical Biology vol 67 no 3 pp 637ndash661 2005

[53] J Shi and R Shivaji ldquoPersistence in reaction diffusion modelswith weak Allee effectrdquo Journal of Mathematical Biology vol 52no 6 pp 807ndash829 2006

[54] A Morozov S Petrovskii and B-L Li ldquoSpatiotemporal com-plexity of patchy invasion in a predator-prey system with theAllee effectrdquo Journal of Theoretical Biology vol 238 no 1 pp18ndash35 2006

[55] L Roques A Roques H Berestycki and A KretzschmarldquoA population facing climate change joint influences of Alleeeffects and environmental boundary geometryrdquoPopulation Eco-logy vol 50 no 2 pp 215ndash225 2008

10 Abstract and Applied Analysis

[56] C Celik andODuman ldquoAllee effect in a discrete-time predator-prey systemrdquo Chaos Solitons and Fractals vol 40 no 4 pp1956ndash1962 2009

[57] J Zu andMMimura ldquoThe impact of Allee effect on a predator-prey system with Holling type II functional responserdquo AppliedMathematics and Computation vol 217 no 7 pp 3542ndash35562010

[58] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[59] E Gonzalez-Olivares H Meneses-Alcay B Gonzalez-Yanez JMena-Lorca A Rojas-Palma and R Ramos-Jiliberto ldquoMultiplestability and uniqueness of the limit cycle in a Gause-typepredator-prey model considering the Allee effect on preyrdquoNon-linear Analysis Real World Applicationsl vol 12 no 6 pp 2931ndash2942 2011

[60] P Aguirre E Gonzalez-Olivares and E Saez ldquoTwo limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo Nonlinear Analysis Real World Applications vol 10 no3 pp 1401ndash1416 2009

[61] PAguirre EGonzalez-Olivares andE Saez ldquoThree limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo SIAM Journal on Applied Mathematics vol 69 no 5 pp1244ndash1262 2009

[62] Y Cai W Wang and J Wang ldquoDynamics of a diffusivepredator-prey model with additive Allee effectrdquo InternationalJournal of Biomathematics vol 5 no 2 Article ID 1250023 2012

[63] M E Solomon ldquoThenatural control of animal populationsrdquoTheJournal of Animal Ecology vol 18 no 1 pp 1ndash35 1949

[64] A J Lotka Elements of Physical Biology Williams and Wilkins1925

[65] C Jost Comparing PredatorndashPrey Models Qualitatively andQuantitatively with Ecological Time-Series Data InstitutNational Agronomique Paris-Grignon 1998

[66] H Malchow S V Petrovskii and E Venturino SpatiotemporalPatterns in Ecology and Epidemiology Chapman amp Hall BocaRaton Fla USA 2008

[67] M R Garvie ldquoFinite-difference schemes for reaction-diffusionequations modeling predator-prey interactions in MATLABrdquoBulletin of Mathematical Biology vol 69 no 3 pp 931ndash9562007

[68] A Munteanu and R Sole ldquoPattern formation in noisy self-rep-licating spotsrdquo International Journal of Bifurcation and Chaosvol 16 no 12 pp 3679ndash3683 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Allee-Effect-Induced Instability in a Reaction-Diffusion ...emis.maths.adelaide.edu.au/.../Volume2013/487810.pdf · 2 AbstractandAppliedAnalysis interactionmodelwithlogisticgrowthrateofthepreyinthe

10 Abstract and Applied Analysis

[56] C Celik andODuman ldquoAllee effect in a discrete-time predator-prey systemrdquo Chaos Solitons and Fractals vol 40 no 4 pp1956ndash1962 2009

[57] J Zu andMMimura ldquoThe impact of Allee effect on a predator-prey system with Holling type II functional responserdquo AppliedMathematics and Computation vol 217 no 7 pp 3542ndash35562010

[58] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[59] E Gonzalez-Olivares H Meneses-Alcay B Gonzalez-Yanez JMena-Lorca A Rojas-Palma and R Ramos-Jiliberto ldquoMultiplestability and uniqueness of the limit cycle in a Gause-typepredator-prey model considering the Allee effect on preyrdquoNon-linear Analysis Real World Applicationsl vol 12 no 6 pp 2931ndash2942 2011

[60] P Aguirre E Gonzalez-Olivares and E Saez ldquoTwo limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo Nonlinear Analysis Real World Applications vol 10 no3 pp 1401ndash1416 2009

[61] PAguirre EGonzalez-Olivares andE Saez ldquoThree limit cyclesin a Leslie-Gower predator-prey model with additive Alleeeffectrdquo SIAM Journal on Applied Mathematics vol 69 no 5 pp1244ndash1262 2009

[62] Y Cai W Wang and J Wang ldquoDynamics of a diffusivepredator-prey model with additive Allee effectrdquo InternationalJournal of Biomathematics vol 5 no 2 Article ID 1250023 2012

[63] M E Solomon ldquoThenatural control of animal populationsrdquoTheJournal of Animal Ecology vol 18 no 1 pp 1ndash35 1949

[64] A J Lotka Elements of Physical Biology Williams and Wilkins1925

[65] C Jost Comparing PredatorndashPrey Models Qualitatively andQuantitatively with Ecological Time-Series Data InstitutNational Agronomique Paris-Grignon 1998

[66] H Malchow S V Petrovskii and E Venturino SpatiotemporalPatterns in Ecology and Epidemiology Chapman amp Hall BocaRaton Fla USA 2008

[67] M R Garvie ldquoFinite-difference schemes for reaction-diffusionequations modeling predator-prey interactions in MATLABrdquoBulletin of Mathematical Biology vol 69 no 3 pp 931ndash9562007

[68] A Munteanu and R Sole ldquoPattern formation in noisy self-rep-licating spotsrdquo International Journal of Bifurcation and Chaosvol 16 no 12 pp 3679ndash3683 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Allee-Effect-Induced Instability in a Reaction-Diffusion ...emis.maths.adelaide.edu.au/.../Volume2013/487810.pdf · 2 AbstractandAppliedAnalysis interactionmodelwithlogisticgrowthrateofthepreyinthe

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of