spatiotemporal complexity of patchy invasion in a …journal of theoretical biology 238 (2006)...

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Journal of Theoretical Biology 238 (2006) 18–35

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Spatiotemporal complexity of patchy invasion in a predator-preysystem with the Allee effect.

Andrew Morozova, Sergei Petrovskiia,�, Bai-Lian Lib

aShirshov Institute of Oceanology, Russian Academy of Science, Nakhimovsky Prosp. 36, Moscow 117218, Russian FederationbEcological Complexity and Modeling Laboratory, Department of Botany and Plant Sciences, University of California, Riverside, CA 92521-0124, USA

Received 5 January 2005; received in revised form 29 April 2005; accepted 2 May 2005

Available online 6 July 2005

Abstract

Invasion of an exotic species initiated by its local introduction is considered subject to predator–prey interactions and the Allee

effect when the prey growth becomes negative for small values of the prey density. Mathematically, the system dynamics is described

by two nonlinear diffusion–reaction equations in two spatial dimensions. Regimes of invasion are studied by means of extensive

numerical simulations. We show that, in this system, along with well-known scenarios of species spread via propagation of

continuous population fronts, there exists an essentially different invasion regime which we call a patchy invasion. In this regime, the

species spreads over space via irregular motion and interaction of separate population patches without formation of any continuous

front, the population density between the patches being nearly zero. We show that this type of the system dynamics corresponds to

spatiotemporal chaos and calculate the dominant Lyapunov exponent. We then show that, surprisingly, in the regime of patchy

invasion the spatially average prey density appears to be below the survival threshold. We also show that a variation of parameters

can destroy this regime and either restore the usual invasion scenario via propagation of continuous fronts or brings the species to

extinction; thus, the patchy spread can be qualified as the invasion at the edge of extinction. Finally, we discuss the implications of

this phenomenon for invasive species management and control.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Biological invasion; Predator–prey system; Patchy spread; Allee effect; Spatiotemporal chaos

1. Introduction

Understanding of patterns and mechanisms of speciesspatial dispersal is an issue of significant current interestin conservation biology and ecology. It arises frommany ecological applications; in particular, it plays amajor role in connection to biological invasion andepidemic spread (Drake et al., 1989; Hengeveld, 1989;Murray, 1989; Shigesada and Kawasaki, 1997). Avariety of theoretical approaches has been developed

e front matter r 2005 Elsevier Ltd. All rights reserved.

i.2005.05.021

ing author. Current address: 114 Winchester Gardens,

1 2QB, UK. Tel.:+44 121 258 0899.

esses: [email protected] (A. Morozov),

ex.ru (S. Petrovskii), [email protected] (B.-L. Li).

and a considerable progress has been made during thelast decade. However, many aspects related to speciesdispersal have never been properly addressed yet.

Regarding the spread of exotic species, a problem ofhigh practical importance is how to create an effectiveprogram of management and control of invasive species(Andow et al., 1990; Sakai et al., 2001; Fagan et al.,2002). Such a program must include both good under-standing of the mechanisms underlying species spreadand an optimal monitoring strategy. In its turn, arelevant strategy is likely to be different for differentspecies (e.g. depending on whether a given species isdetectable with satellite-based remote sensing imagerydata) and should be also based on the knowledge of thepattern of spread. For instance, in many cases invasion

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ARTICLE IN PRESSA. Morozov et al. / Journal of Theoretical Biology 238 (2006) 18–35 19

of exotic species takes place via propagation of apopulation front separating the areas where givenspecies is absent, i.e. in front of the front, from theareas where it is present in considerable densities, i.e. inthe wake of the front. In this case, under somewhatidealized assumption that the invasion is going iso-tropically so that its rate does not depend on thedirection of spread, the advance of invading species canbe monitored using a relatively small number of on-siteobservers or sampling stations situated along a certainline coming out of the place of original speciesintroduction.

However, reality is often much more complicated. Asa result of environmental heterogeneity the rates ofinvasion can be significantly different in differentdirections (Shigesada and Kawasaki, 1997). Also, dueto the impact of both environmental and biologicalfactors, the pattern of spread can be more complicatedthan simple population front. There is growing evidence(Davis et al., 1998; Kolb et al., 2004; Swope et al., 2004)that, in some cases, invasion of exotic species takes placethrough dynamics of separate population patches notpreceded by propagation of a continuous populationfront, see also, Shigesada and Kawasaki (1997), Lewis(2000); and Lewis and Pacala (2000) and the referencestherein. Below we will call that pattern of spread apatchy invasion. Obviously, in the case of patchyinvasion an adequate monitoring strategy should bemore complicated and likely include many moreobservers.

Thus, distinguishing between the situations whenspecies spread takes place via propagation of acontinuous population front and when it happens viapatchy invasion, as well as identification of factorsenhancing or hampering patchy invasion are problemsof significant practical and theoretical importance. Theorigin of patchy invasion is often seen either inenvironmental heterogeneity (cf. Murray, 1989) or inthe impact of stochastic factors (Lewis, 2000; Lewisand Pacala, 2000). Indeed, the whole dynamicsof ecological communities appears as a result ofinterplay between numerous deterministic and stochas-tic factors. However, the importance of stochasticityshould not be overestimated. Sometimes stochasticmodels and deterministic models lead to qualitativelysimilar spatiotemporal patterns, although reachedthrough different mechanisms (cf. Kawasaki et al.,1997; Mimura et al., 2000). For a rather general modelof marine ecosystem, Malchow et al. (2002) showed thatthere exists a critical level of noise so that forundercritical noise the system is more driven bydeterministic factors. Even in the supercritical case,when noise can change the system dynamics consider-ably, it was shown that intrinsic spatial scales of thesystem are still controlled by deterministic mechanisms(Malchow et al., 2004).

Recently, it was shown by Petrovskii et al. (2002a,b)that patchy invasion can arise in a fully deterministicpredator–prey system as a result of the Allee effect, i.e.of a threshold phenomenon when the population growthrate becomes negative for low population density (Allee,1938; Dennis, 1989; Courchamp et al., 1999). Determi-nistic patchy invasion was shown to correspond to theinvasion at the edge of extinction (Petrovskii andVenturino, 2004; Petrovskii et al., 2005c) so that a smallfinite variation of the system parameters either restoresusual population front propagation scenario or bringsthe species to extinction. Moreover, it was shown thatthe system dimensionality is a crucial point and thepatchy invasion in two spatial dimensions correspondsto species extinction in the corresponding 1D system(Petrovskii and Venturino, 2004; Petrovskii et al.,2005c).

The above papers, however, left many questions open.In this paper, we make a detailed study of thedeterministic patchy invasion in a predator–prey systemwhere the prey growth is damped by the Allee effect. Inparticular, the following issues are addressed: (i) what isthe succession of invasion regimes in response tovariation of an ecologically meaningful controllingparameter and how the invasion speed depends on thetype of spread, (ii) what is the degree of spatiotemporalcomplexity corresponding to the regime of patchyspread and (iii) what are the ecological implications ofthe patchy invasion. We show that, although theinvasion speed is much lower in the regime of patchyspread than it is in the usual regime(s) of continuousfront propagation, the patchy spread provides ascenario of species invasion below the survival thresh-old. Also, we show that deterministic patchy invasioncorresponds to spatiotemporal chaos and estimate thevalue of the dominant Lyapunov exponent.

2. Mathematical model

We consider 2D dynamics of a predator–prey systemdescribed by two partial differential equations ofdiffusion–reaction type (Nisbet and Gurney, 1982;Murray, 1989; Holmes et al., 1994; Shigesada andKawasaki, 1997; Medvinsky et al., 2002):

qP

qT¼ D1

q2P

qX 2þ

q2P

qY 2

� �þ F ðPÞ � f ðPÞZ, (1)

qZ

qT¼ D2

q2Z

qX 2þ

q2Z

qY 2

� �þ kf ðPÞZ �MZ, (2)

Here P ¼ PðX ;Y ;TÞ and Z ¼ ZðX ;Y ;TÞ are densitiesof prey and predator, respectively, at moment T andposition ðX ;Y Þ. The function F ðPÞ stands for theintrinsic prey growth, f ðPÞ is the predator trophic

ARTICLE IN PRESSA. Morozov et al. / Journal of Theoretical Biology 238 (2006) 18–3520

response, k is the food utilization coefficient, D1 and D2

are diffusion coefficients and MZ describes predatormortality. We assume that the dynamics takes place in ahomogeneous environment, which means that none ofthe parameters exhibit any dependence on space. In themost part of this paper, we consider D1 ¼ D2; the caseof unequal diffusivities will be briefly addressed in thelast section.

We assume that the predator trophic response is ofHolling type II and choose the following parameteriza-tion (Holling, 1965; Murray, 1989):

f ðPÞ ¼ AP

Pþ B, (3)

where B is the half-saturation density and A is themaximum predation rate. Considering Eqs. (2) and (3)together, it is readily seen that the product Ak has themeaning of predator maximum growth rate.

In the case of strong Allee effect, a convenientparameterization for the prey growth is as follows (cf.Lewis and Kareiva, 1993; Owen and Lewis, 2001):

F ðPÞ ¼ mP P� P0ð Þ K � Pð Þ, (4)

where m is a coefficient, K is the prey carrying capacity,and P0 is the prey survival threshold (we assume0oP0oK), so that at low densities 0oPoP0 its growthbecomes negative. The value of P0 is a characteristic ofthe strength of the Allee effect: the less P0 is, the lessprominent is the Allee effect.

Introducing, for convenience, dimensionless variables

u ¼ P=K ; v ¼ Z=ðkKÞ; t ¼ aT ; x ¼ X ða=DÞ1=2,

y ¼ Y ða=DÞ1=2,

where D ¼ D1 ¼ D2 and a ¼ AkK=B, from Eqs. (1)–(4)we arrive at the following equations:

@u

@t¼@2u

@x2þ@2u

@y2þ guðu� bÞð1� uÞ �

uv

1þ au, (5)

@v

@t¼@2v

@x2þ@2v

@y2þ

uv

1þ au� dv, (6)

where a ¼ K=B, b ¼ P0=K , g ¼ K2mB=a and d ¼M=a

are dimensionless parameters. Note that, since themodel (5)–(6) does not take into account possibleexistence of alternative food sources, predator cannotsurvive without prey and extinction of prey leads toextinction of both species.

Biological invasion usually starts with a localintroduction of exotic species; thus, relevant initialconditions for system (5)–(6) should be described byfunctions of compact support when the density of one orboth species at the initial moment of time is non-zeroonly inside a certain domain. The shape of the domainand the profiles of the population densities can bedifferent in different cases. In this paper, however, weare primarily concerned with the large-time dynamics of

the system when small details of the initial speciesdistribution are not likely to play an important role(provided that the initial population size is large enoughto ensure invasion success, see the beginning of the nextsection). For that reason, we consider the initialconditions in the form of elliptic patches:

uðx; y; 0Þ ¼ u0 forx� x1

D11

� �2

þy� y1

D12

� �2

p1; otherwise uðx; y; 0Þ ¼ 0, ð7Þ

vðx; y; 0Þ ¼ v0 forx� x2

D21

� �2

þy� y2

D22

� �2

p1 otherwise vðx; y; 0Þ ¼ 0, ð8Þ

where D11, D12, D21, D22, x1, y1, x2, y2, u0, v0 areparameters with obvious meaning.

In the corresponding 1D case, the system (1)–(2)without Allee effect is known to exhibit a variety oftravelling population waves (Kolmogorov et al., 1937;Fisher, 1937; Aronson and Weinberger, 1978; Volpert etal., 1994; Berezovskaya and Karev, 1999), some of themare linked to spatiotemporal pattern formation, cf.Sherratt et al. (1995), Petrovskii and Malchow (2000),Sherratt (2001) and Malchow and Petrovskii (2002). Theimpact of the Allee effect can modify the systemdynamics significantly resulting in new exotic regimessuch as standing chaotic patches and travelling popula-tion pulses (Morozov, 2003; Morozov et al., 2004;Petrovskii et al., 2005a). In two spatial dimensions, thesystem dynamics appears to be even richer. Along withregimes giving an immediate generalization of theregimes observed in 1D case, there appear morecomplicated patterns that cannot exist in 1D case. Inthe next section, the 2D dynamics of the predator–preysystem with the Allee effect will be considered in muchdetail with a special attention to the patchy invasion andits relation to other regimes of the system dynamics.

3. Results of computer simulations

Eqs. (5)–(6) with initial conditions (7)–(8) were solvednumerically by finite-difference method. Computerexperiments were run in a square numerical domainL�L where L ¼ 300, cf. Figs. 1–4. In most cases, weused the explicit scheme. In order to avoid numericalartifacts we checked the sensitivity of the results to thechoice of the time and space steps and their values havebeen chosen sufficiently small. Also, some of the resultswere reproduced by means of using more advancedalternate directions scheme. Both numerical schemes arestandard hence we do not describe them here; detailsand particulars can be found in Thomas (1995).

Before proceeding to the simulation results, a certainremark is necessary regarding the choice of the initial

ARTICLE IN PRESS

Fig. 1. Snapshots of the prey density (in dimensionless units) obtained

at different times for parameters a ¼ 0:1, b ¼ 0:22, g ¼ 3, d ¼ 0:63.White and black/gray colors correspond to species absence and species

presence at high density, respectively. Behind the expanding circular

front, the species is distributed homogeneously. Predator density

exhibits similar properties.

Fig. 2. Snapshots of the prey density obtained at different times for

d ¼ 0:51, other parameters as in Fig. 1. White and black/gray colors

correspond to species absence and species presence at high density,

respectively. A prominent patchy species spatial distribution appears in

the wake of the expanding front. Predator density exhibits similar

properties.

A. Morozov et al. / Journal of Theoretical Biology 238 (2006) 18–35 21

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Fig. 3. Snapshots of the prey density obtained at different times for

d ¼ 0:43, other parameters as in Fig. 1. White and black/gray colors

correspond to species absence and species presence in high density,

respectively. Note that the species is absent in the wake of the

expanding front. Predator density exhibits similar properties.

A. Morozov et al. / Journal of Theoretical Biology 238 (2006) 18–3522

conditions. It is well known that, in case populationgrowth is affected by the strong Allee effect, not everyspecies introduction leads to successful invasion (Lewisand Kareiva, 1993; Petrovskii et al., 2005a). There existsa minimum viable population size so that invasionsuccess can only be guaranteed when the initiallyinvaded area and the initial population density of thealien species are not too small. Correspondingly, since inthis paper we are primarily concerned with the dynamicsof species spread, in our numerical simulations para-meters u0, v0 and Dij in Eqs. (7)–(8) have always beenchosen sufficiently large.

It must be mentioned that, since the system understudy depends on a relatively large number of parameters(eight for the original system (1)–(4) and four for itsdimensionless version (5)–(6), its detailed numericalinvestigation in the whole parameter space is virtuallyimpossible. Instead, we choose one controlling parameterand consider the changes in the invasion scenario subjectto its variation. In this paper, we choose predatormortality d as the controlling factor, and keep all otherparameters fixed. The choice of d as the controlling factoris justified by the results of field observations showingthat intensity of predation affects the rate of preyinvasion (Fagan and Bishop, 2000). Also, this choice isreasonable from the point of prospective invasive speciesmanagement because, in a real ecosystem, the rate ofpredator mortality can be relatively easy controlled bymeans of additional harvesting.

Results of our computer simulations show that, in thepredator–prey system with the Allee effect for prey,there are several qualitatively different patterns ofinvasion, see also, Petrovskii et al. (2002a,b), Morozov(2003), Morozov et al. (2004) and Petrovskii et al.(2005a). Typically, for large values of d the systemexhibits propagation of population fronts with station-ary spatially homogeneous species distribution behind,see Fig. 1 showing snapshots of the prey density (thepredator density has similar features) obtained forparameters a ¼ 0:1, b ¼ 0:22, g ¼ 3, d ¼ 0:63 and theinitial conditions (7)–(8) with D2

11 ¼ 12:5, D212 ¼ 12:5,

D221 ¼ 5, D2

22 ¼ 10, x1 ¼ 153:5, y1 ¼ 145, x2 ¼ 150,y2 ¼ 150, u0 ¼ 1 and v0 ¼ 0:2. For somewhat smallerd, this regime changes to travelling fronts withspatiotemporal oscillations in the wake, cf. Fig. 2obtained for a ¼ 0:1, b ¼ 0:22, g ¼ 3, d ¼ 0:51 and thesame initial conditions. (Note that these two patternsare also typical for a predator–prey system withoutAllee effect, see Sherratt et al. (1995) and Petrovskii andMalchow (2000).) In both of these cases, the populationsare absent in front of the front and they are present inconsiderable densities behind the front. Apparently,both patterns correspond to successful invasion.

A further decrease in d changes it to travellingpopulation pulses (in 1D case), or rings (in 2D case)travelling/expanding from the place of the species

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Fig. 4. Snapshots of the prey densities obtained at different times for the regime of patchy invasion, d ¼ 0:42, other parameters as in Fig. 1. White

and black/gray colors correspond to species absence and species presence in high density, respectively. The species spreads without forming a

continuous travelling front. Predator density exhibits similar properties.


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0 100 200 300 400 500 6000

5

10

15x 10 4

t

1

23

t0 200 400 600 800 1000

0

4

8

12

16x 10

3

1

2

3

(a)

(b)

Pop

ulat

ion

Size

of

Pre

yP

opul

atio

n Si

ze o

f P

rey

Fig. 5. Temporal variation of the prey population size for different

invasion scenarios: (a) regimes of species invasion through propaga-

tion of continuous travelling fronts, curve 1 for d ¼ 0:51, curve 2 for

d ¼ 0:46, curve 3 for d ¼ 0:43; (b) patchy invasion regime, curve 1 for

d ¼ 0:42, curve 2 for d ¼ 0:417, curve 3 for d ¼ 0:414, respectively.Other parameters are same as in Fig. 1. Note different order of

magnitude in (a) and (b).


introduction, see Fig. 3 obtained for a ¼ 0:1, b ¼ 0:22,g ¼ 3, d ¼ 0:43 and the same initial conditions. In thiscase, the species are absent both in front of the front andbehind the front. This type of the system dynamics issomewhat paradoxical from ecological point of viewbecause successful species spread ( ¼ propagation ofpopulation front of invasive species) yet leads toinvasion failure ( ¼ no species behind the front).

In all above cases, the species spread takes place viapropagation of a continuous travelling front. However,the impact of the strong Allee effect in the system (1)–(2)can yield a completely different pattern of populationspread, in particular, the regime of patchy invasion(Petrovskii et al., 2002a,b). An example of patchyinvasion is shown in Fig. 4 (obtained for parametersa ¼ 0:1, b ¼ 0:22, g ¼ 3, d ¼ 0:42 and the initial condi-tions the same as in Figs. 1–3). At the early stage of thesystem dynamics, the species spread is characterized byformation of a continuous round front (see Fig. 4a)similar to what is observed for other regimes, cf. Fig. 2a.At later time, however, the front breaks to pieces, cf.Figs. 4b and 4c, and at further stages of invasion thepopulation spreads over space via irregular motion ofseparate patches. No continuous front arises again, cf.Figs. 4d–4f. The patches move, merge, disappear orproduce new patches, etc. After the population patchesinvade over the whole domain, the spatiotemporaldynamics of the system does not change and the spatialdistribution of species at any time is qualitatively similarto the one shown in Fig. 4f.

We want to emphasize that, although the parameterrange where the patchy invasion can be observed isrelatively narrow compare to the range where invasiontakes place through propagation of continuous travel-ling fronts, the parameter set chosen for Fig. 4 is not atall unique and the patchy invasion can be observed forother parameter values as well (cf. Petrovskii et al.,2002a; Petrovskii and Venturino, 2004; Petrovskii et al.,2005c).

The invasion of introduced species is naturallyfollowed by a gradual increase in the population sizesU and V:

UðtÞ ¼

Z L

0

Z L

0

uðx; y; tÞdydx,

V ðtÞ ¼

Z L

0

Z L

0

vðx; y; tÞdydx. ð9Þ

Fig. 5 shows U and V calculated at different time forparameter values corresponding to different invasionregimes. Fig. 5a performs U(t) and V(t) obtained forinvasion through propagation of continuous travellingfronts, curve 1 for d ¼ 0:51, curve 2 for d ¼ 0:46, curve 3for d ¼ 0:43. Here curve 1 corresponds to travellingfronts with oscillations in the wake, cf. Fig. 2, andcurves 2 and 3 correspond to pulse/rings propagation,

cf. Fig. 3. Fig. 5b shows U(t) and V(t) obtained forpatchy invasion, curve 1 for d ¼ 0:42, curve 2 ford ¼ 0:417, curve 3 for d ¼ 0:414. Thus, not only thespatial distribution of the population density exhibitsprominent spatial irregularity (cf. Fig. 4) but also thepopulation sizes experience irregular temporal fluctua-tions. This irregularity will come into the focus of ourstudy in Section 3.2.

3.1. Calculation of the invasion speed

When managing invasion of exotic species, one of themost important issues is to estimate the rate of theirspread and to identify the main factors the rate can beaffected by. For single-species models of populationdynamics, this problem is well studied, cf. Mikhailov(1990) and Volpert et al. (1994). However, the problembecomes much more difficult when the spread ofinvading species is complicated by interspecies interac-tions, e.g. through predator–prey relations. An analy-tical solution of this problem is hardly possible (but see

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0 50 100 150 200 250 300 350 400 450 500 5500

50

100

150

I II

III

t

R1(t

),R

2(t

)

0

0.7

0.6

0.5

0.4

0.3

0.2

0.1

1C

,2

C

IIIII I

0.42 0.44 0.46 0.48 0.50

δ

(a)

(b)

Fig. 6. (a) The radii R1 (solid) and R2 (dashed) of invaded area vs time

obtained for three different types of invasion regimes (see Figs. 2–4) by

using two different methods, cf. Eqs. (11b) and (11a), respectively.

Curves I are obtained for parameters of Fig. 2 (patterns on the wake),

curve II for parameters of Fig. 3 (travelling pulses), curve III for

parameters of Fig. 4 (patchy invasion). Other parameters are the same

as in Fig. 1; (b) speed of invasion calculated for different d based on

Eq. (12) with R ¼ R1 (dots) and R ¼ R2 (asterisks). Parameter

domains I and II correspond to propagation of continuous travelling

fronts with irregular spatiotemporal patterns and species extinction in

the wake, respectively; domain III corresponds to patchy invasion.


Petrovskii et al., 2005b) and a thorough numericalinvestigation is very difficult as well because of highdimensionality of the corresponding parameter space.

Let us note, however, that in a real ecosystem the rateof spread depends on numerous factors that unlikely canbe all taken into account by a single mathematicalmodel. From this perspective, it seems to be moreimportant to understand the structural dependence ofthe invasion speed on the type of spread (e.g. patchy ornot patchy) rather than to find out its dependence on allmodel parameters.

For the invasion regimes showing continuous travel-ling fronts, estimation of the invasion speed does notbring much difficulty. In an idealized case of purelyhomogeneous environment when the shape of invadedarea is very close to a circle, it can be readily obtainedfrom the rate of increase of the radius R of the area. Fora more realistic case of inhomogeneous environmentwhen the shape of the invaded area can be ratherdistorted, a special averaging procedure was developedwhich is still essentially based on the existence of thecontinuous travelling front separating invaded and non-invaded areas, see Shigesada and Kawasaki (1997) fordetails and further references.

In case of patchy invasion, however, these approachesdo not immediately apply because it is hardly possible totrace an exact border between invaded and non-invadedareas. In order to overcome this difficulty, we havedeveloped two different methods to estimate theinvasion speed that are not based on existence ofcontinuous travelling front.

The idea of our approach is as follows. First, wedefine the center (xC,yC) of the invaded area by means ofthe following relations:

xCðtÞ ¼1

UðtÞ

Z L

0

Z L

0

xuðx; y; tÞdydx,

yCðtÞ ¼1

UðtÞ

Z L

0

Z L

0

yuðx; y; tÞdydx, ð10Þ

where U(t) is the population size of prey given byEq. (9). Definition (10) is thus similar to the definition ofthe center of mass in mechanics. The radius R of thedomain can be now defined as the maximum distancebetween the center (xC,yC) and all the points in spacewhere the prey density is higher then a certain prescribedvalue u*:

RðtÞ ¼ maxOðtÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix� xCðtÞð Þ

2þ y� yCðtÞ� �2q

, (11a)

where OðtÞ ¼ fðx; yÞjuðx; y; tÞ4u�g and the value of u*should be reasonably small.

According to Eqs. (9)–(10), in general, the position ofthe center can change in the course of the systemdynamics. On the other hand, it is intuitively clear thatthe center of the invaded area should be related to theplace of original species introduction, i.e. the initial

conditions. Applying definition (9)–(10) to the initialcondition (7), we immediately obtain that xCð0Þ ¼ x1,yCð0Þ ¼ y1. Thus, an alternative definition of the arearadius can be as follows:

RðtÞ ¼ maxOðtÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix� x1ð Þ

2þ y� y1

� �2q. (11b)

We denote by R1ðtÞ the radius of the domain given byEq. (11b) and by R2ðtÞ the radius given by Eq. (11a).

Fig. 6a shows R1ðtÞ (solid) and R2ðtÞ (dashed)calculated for u� ¼ 0:025 for the three different regimesshown in Figs. 2–4, i.e. for travelling fronts withirregular oscillations in the wake (I), travelling pulses(II) and patchy spread (III). Note that, contrary tointuitive expectations, definition (11b) of the domainradius (cf. the solid curve) appears to be moreappropriate for speed estimation because it produces acurve very close to a straight line, at least, for sufficiently


large t. Oscillations in the dashed curve indicate that theposition of the dynamical center of the invaded domaingiven by Eqs. (9)–(10) fluctuates with time, likely due tolocal oscillatory population dynamics.

The average slopes of the curves give an estimate ofthe invasion speed. Let us note, however, that, generallyspeaking, in a 2D case with axial symmetry the rate ofinvasion depends on the radius of the growing domain,i.e. C ¼ CðRÞ, and the stationary value can only bereached in the large-time limit when R tends to infinity(Mikhailov, 1990). Indeed, it is easily seen that theprocess has different stages, especially, in the regimes IIand III. Thus, to estimate the invasion speed we use thefollowing equation:

C ¼R00 � R0

t00 � t0, (12)

where R0 should be chosen sufficiently large. Accountingfor the fact that the radius of the domain can grow non-monotonously, we define t0 ¼ min t0ijt

0i : Rðt0i ¼ÞR

0� �

;t00 ¼ min t00i jt

00i : Rðt00i Þ ¼ R00

� �; i.e., t0 and t00are the

moments when the radius of the growing domain forthe first time reaches prescribed values R0 and R00,respectively.

To estimate the speed of invasion by means of Eq.(12), we chose R0 ¼ 100 and R00 ¼ 150; for R(t)o100,the impact of transients and dependence of the speed onR is still strong and for R004150 the species nearlyreaches the domain border.

In order to reveal the ecological implications of thepatchy invasion, we compare the invasion speed fordifferent regimes using predator mortality d as acontrolling parameter, cf. the beginning of Section 3.Fig. 6b shows the speed C1ðdÞ (dots) and C2ðdÞ(asterisks) calculated for R1ðtÞ, R2ðtÞ, respectively (otherparameters are the same as in Figs. 1–4). The symbols I,II and III, show which regime of invasion takes place forgiven range of d. For do0:412, extinction of bothspecies takes place for any initial conditions; ford40:61, regime I (travelling fronts with oscillations inthe wake) changes to propagation of fronts withhomogeneous species distribution in the wake.

Thus, the two methods to estimate the invasion speedgive very close values, C1 � C2 � C, where C is the truevalue of the speed. Our numerical data show that C

exhibits a clear tendency to decrease with a decrease inpredator mortality d. This result is in a good agreementwith data of field observations (Fagan and Bishop,2000) showing that the speed of invasion is the less thehigher is the predation.

Fig. 6b shows that the transition between the patternsof species spread with continuous travelling fronts, i.e.types I and II, does not lead to any significant change inC. On the contrary, the transition from domain II todomain III, i.e. from a pattern of spread via propagationof continuous population front to the patchy invasion, is

characterized by a remarkable drop in the invasionspeed. This dramatic decrease in C can be interpretedheuristically as a transition to a different mechanism ofspecies spread which will be discussed in Section 4.

It should be mentioned that, for d 2 ½0:412; 0:427�, thechoice of initial conditions is important. Our numericalsimulations show that the same parameters in Eqs.(7)–(8) that lead to patchy invasion for a certain d fromthis range can lead to species extinction for a slightlydifferent value d0. However, for this new value d0

another parameter set in Eqs. (7)–(8) can be found thatlead again to the patchy spread. This singular depen-dence of the system dynamics on the initial conditionsbrings a certain difficulty for a regular study of thepatchy invasion, e.g. when investigating invasion speeddependence on given controlling parameter. One possi-ble solution can be reached by tuning the initialconditions for each new d. Our simulations show thatparameter changes in Eqs. (7)–(8) do not affect much thevalue of invasion speed as far as the type of spread isretained. Another approach to the choice of appropriateinitial conditions, which we actually used in our study, isbased on the observation that, in case the patchy spreadhas been initiated, small variation of d normally doesnot destroy it. Thus, it is convenient to use an embryo ofthe patchy distribution obtained for a particular d at aparticular moment of time as the initial condition in thesimulations for other values of d. Specifically, for thatpurpose we used the pattern shown in Fig. 4c.

3.2. Chaotic properties of the patchy dynamics

Irregular temporal fluctuations of the population sizesshown in Fig. 5b seem to indicate that the systemdynamics is chaotic. It should be noted that thecorresponding homogeneous system, i.e. Eqs. (1)–(2)without diffusion terms, cannot exhibit behavior morecomplex than periodic. Thus, chaos in the system (1)–(2)is an essential result of the system dynamics in space andshould be qualified as spatiotemporal (Pascual, 1993;Sherratt et al., 1995; Petrovskii and Malchow, 1999,2001; Sherratt, 2001; Petrovskii et al., 2003).

An apparent irregularity of the system behavior,however, does not necessarily correspond to chaoticdynamics and a more careful analysis is needed. Thatcan be done in different ways. For instance, one of thebasic properties of deterministic chaos is its sensitivityto the initial conditions so that the distance betweenthe solutions corresponding to perturbed and unper-turbed initial conditions grows with time (Nayfeh andBalachandran, 1995).

In order to check whether sensitivity to the initialconditions takes place for the regime of patchy invasion,we varied parameters in Eqs. (7)–(8). Fig. 7 shows theprey density vs. time for the unperturbed system (solidline) and the perturbed system (dashed line) obtained at

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0.2

0.4

0.6

0.8

1

t600

Pre

y D

ensi

ty

Fig. 7. Temporal oscillations of the prey density at a fixed point,

x ¼ 155, y ¼ 145 subject to a small perturbation of the initial

conditions, solid curve for the undisturbed system, dashed curve for

the disturbed system.

1250 1350 1450 1550 1650t

-16

-14

-12

-10

-8

-6

-4

-2

0

2

1750

ln(h

u),

ln(h

v)

Straight line approximation:PreyPredator

Fig. 8. Discrepancies huðtÞ and hvðtÞ between two initially closed

system trajectories, cf. Eqs. (13)–(16), in the regime of patchy invasion.

Parameters are the same as in Fig. 4. The slope of the straight line gives

the value of the dominant Lyapunov exponent, lmax ¼ 0:035� 0:001.


a fixed point in space, x ¼ 155, y ¼ 145, for the sameparameter values as in Fig. 4. The perturbed system wasobtained from the unperturbed one by multiplying u0 byfactor 1.00001. It is readily seen that up to t ¼ 550 thedifference between the two solutions is very small;however, for t4550 the difference grows rapidly and thediscrepancy soon becomes on the order of the solutionsthemselves. Qualitatively similar results were alsoobtained for perturbation of other parameters and forother positions in space.

A more rigorous definition of deterministic chaos,however, requires the discrepancy between the undis-turbed and disturbed systems not just to be increasingbut to be increasing exponentially. There is a number ofmethods to calculate the dominant Lyapunov exponentlmax (cf. Wolf et al., 1985; Kantz and Schreiber, 1997).In this paper, we have estimated the value of lmax basedon its definition.

In order to take into consideration both dynamicalvariables, i.e. the prey and predator densities, and alsoto take into account the spatial aspect, we analyse thebehavior of the following two values:

huðtÞ ¼ uðx; y; tÞ � u1ðx; y; tÞ�� ,

hvðtÞ ¼ vðx; y; tÞ � v1ðx; y; tÞ�� , ð13Þ

where

wk k ¼

Z L

0

Z L

0

w2ðx; yÞdxdy

� �1=2

. (14)

Strictly speaking, the analysis of the system sensitivity tophase perturbations only can be applied to stationary (inthe statistical sense) time series. Meanwhile, until thespecies invade the whole domain, the time series areapparently transient, cf. Fig. 5. Thus, to estimate thedominant Lyapunov exponent, the system dynamics wasstudied for the post-invasion stage, i.e. for t4tinv wheretinv is the time that takes the species to spread over thewhole area. For the parameters of Fig. 4, tinv�1000.

The procedure is as follows. At certain momentt04tinv, the following small perturbation is applied tothe system:

u1ðx; y; t0Þ ¼ uðx; y; t0Þ 1þ e cos2pðxþ yÞ

L0

�

�e cos2pðx� yÞ

L0

�ð15Þ

v1ðx; y; t0Þ ¼ vðx; y; t0Þ. (16)

The dynamical variables of perturbed and unperturbedsystems are then used to construct the discrepancies hu

and hv.Fig. 8 shows lnðhuÞ and lnðhvÞ vs. time calculated for

e ¼ 10�9, t0 ¼ 1250, L0 ¼ 100 and for the sameparameters as in Fig. 4. The straight lines are obtainedby the least square approximation. It is easily seenthat the distances hu and hv between initially closesolutions u,v and u1,v1 grow exponentially with time(up to small fluctuations). The slope of the straight lines(which appears to be the same for both lines) givesan estimate for the dominant Lyapunov exponent,lmax ¼ 0:035� 0:001.

Another evidence of chaotic nature of the patchydynamics can be obtained from the behavior of relevantpower spectra (Nayfeh and Balachandran, 1995). Weapply this method to the time series of the populationdensity obtained in a fixed point x ¼ 160, y ¼ 155, seeFig. 9a (parameters are the same as in Fig. 4). As well asthe population sizes, the local dynamics exhibitsapparent irregularity giving an indication of determi-nistic chaos. This irregularity perhaps can be seen evenbetter in the phase plane of the local populationdensities, cf. Fig. 9b obtained for a somewhat longerperiod of time compare to Fig. 9a. Indeed, the powerspectrum of the prey density time series, see Fig. 9c, hasproperties typical for chaotic dynamics. In particular, itshows the rate of decay slower than exponential (the

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2

3000 3500 4000 4500 50000

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t

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y an

d P

reda

tor

dens

ity

Pre

dato

r de

nsit

y

Pow

er

1/Time

(a)

(b)

(c)

Fig. 9. Dynamics of the population densities calculated in a fixed

point, x ¼ 160, y ¼ 155 for the regime of patchy invasion (parameters

are the same as in Fig. 4): (a) density of prey (solid) and predator

(dashed) vs. time and (b) local phase plane of the system and (c) power

spectrum of the oscillating prey density (semilogarithmic).

0 40 80 120 160 200 240

s

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

K (

s)

Fig. 10. Cross-correlation function K(s), see Eq. (18), calculated for

the regime of patchy invasion. Parameters are the same as in Fig. 4.


exponential decay would correspond to a straight lineenvelope). The power spectrum of predator densitypossesses similar properties. We want to emphasize thatthe results presented in Fig. 9 are not point-specific; ournumerical results show that the same type of behavior isobserved for any other position in space.

The results of the above analysis demonstrate that thedynamics of the system (5)–(6) in the regime of patchyspread (cf. Fig. 4) is chaotic. It should be mentionedthat, although solid evidence is obtained only for thepost-invasion stage when the time series becomestationary, the sensitivity to the initial conditions thatclearly manifest itself already during the invasion stage,see Fig. 7, makes it possible to conclude that thedynamics is chaotic at earlier stages as well.

The above results concern the temporal variations ofthe species density. Meanwhile, the spatial aspect is veryimportant as well, cf. the beginning of this section. Inorder to address this issue and to reveal possible

correlation between different patches in space, wecalculate the spatial cross-correlation function. Correla-tion between temporal variations of prey species densityat two different sites in space as a function of the inter-site distance is given, in a rather general case is given by(Nisbet and Gurney, 1982; Nayfeh and Balachandran,1995)

Kðx; y; x0; y0Þ ¼ limT!1

1

T

1

susu0

�

Z t0þT

t0

uðx0; y0; tÞ � u0ð Þ uðx; y; tÞ � uð Þdt,

ð17Þ

where (x,y) and ðx0; y0Þ give the position of the sites, u

and u0 are the corresponding mean prey densities, su andsu0 are the standard deviations.

In Eq. (17), formally, the value of K depends on theposition of the sites. Assuming, based on our numericalsimulations, that for sufficiently large time the systembecomes statistically homogeneous, the cross-correla-tion function Kðx0; y0; x; yÞ is a function only of thedistance s between the two points. Then, from Eq. (17)we obtain:

KðsÞ ¼ limT!1

1

T�1

s2

Z t0þT

t0

uðx0; y0; tÞ � u0ð Þ uðx; y; tÞ � uð Þdt,

(18)

where su0 ¼ su ¼ s, u0 ¼ u and s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix� x0ð Þ

2þ y� y0ð Þ

2q

: Fig. 10 shows function K(s)

calculated according (18) with t0 ¼ 2000 and T ¼ 10000along the straight line x ¼ 150, y ¼ 25þ s. Parametersare the same as in Fig. 4. The behavior of the cross-correlation function is typical for chaotic dynamics


(Nayfeh and Balachandran, 1995). The first zero of K(s)gives the correlation length of the system: starting fromthat distance the local temporal fluctuations can beregarded as independent. For parameters of Fig. 4,Lcorr � 18: Thus, the whole area L�L appears to be

dynamically split into N�ðL=LcorrÞ2 sub-domains with

virtually independent temporal behavior.Note that Eqs. (17)–(18) can be written in terms of

predator density as well. For the sake of brevity we donot present corresponding results here because they donot lead to any new insights: the properties of thepredator-based cross-correlation function are verysimilar to those shown in Fig. 10.

Since the dynamics of the system (5)–(6) is essentiallyspatiotemporal, it also makes sense to look at itsproperties in terms of the global dynamical variablesthat would explicitly take into account the spatialdimensions. The simplest variables of that kind seemto be the population sizes U and V or the spatially

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0

2

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0.10 0.12 0.14 0.18 0.20 0.220.07

0.08

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0.05

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0.15

0.2

0.25

0.3

t

tu

0.16

tv

tv

tu

,

Pow

er

(a)

(b)

(c)

Fig. 11. Dynamics of the spatially average species densities ou4t and

ov4t for the regime of patchy invasion (parameters are the same as in

Fig. 4): (a) spatially average density of preyou4t (solid) and predator

ov4t (dashed) vs. time and (b) phase plane of the average species

densities and (c) The power spectrum of the oscillating spatially

average prey density (semilogarithmic).

averaged densities, uh it ¼ L�2U ; vh it ¼ L�2V : Timeseries of these values are shown in Fig. 11a. As it couldbe expected, the dynamical splitting of the system to anumber of quasi-independent oscillators results in amuch smaller amplitude of the oscillations of thespatially averaged values than it was for the localdensities, cf. Figs. 9a and 11a. (A similar type of space-dependence was observed earlier for another system, seeBascompte and Sole (1994).) As well as the localvariables, the oscillations of the spatially averageddensities exhibit prominent irregularity, see Fig. 11b.The properties of the corresponding power spectra, seeFig. 11c, indicates that this irregularity arises as a resultof chaotic dynamics.

In conclusion of this section, we want to mention that,although our attention has been mostly focused on thepatchy invasion, it is not the only regime in the system(5)–(6) that exhibit chaotic dynamics. In particular,analysis of our numerical data indicates that irregularoscillations arising behind the continuous travellingfront, cf. Fig. 2, also correspond to spatiotemporalchaos. That regime of the system dynamics, however,has already been observed and studied in much detailfor a predator–prey system without Allee effect(Pascual, 1993; Sherratt et al., 1995; Petrovskii andMalchow, 1999, 2000, 2001; Petrovskii et al., 2003), andthus will not be considered in this paper.

4. Discussion

In this paper, we have considered regimes ofbiological invasion in a 2D predator–prey system wherethe prey growth is damped by the Allee effect. It wasshown in our earlier work (Petrovskii et al., 2002a,b)that, as a result of the strong Allee effect, along withalready well-known regimes of spread such as invasionof species via propagation of travelling population frontwith either spatially homogeneous or patchy speciesdistribution in the wake, this system exhibits acompletely new invasion scenario when the speciesspread is not preceded by propagation of any contin-uous front. Instead, the populations spread over spacevia irregular motion of separate patches of highpopulation density, the population density betweenpatches being nearly zero, cf. Fig. 4. We want toemphasize that this patchiness is not induced by anyenvironmental heterogeneity and is self-organized. Notethat the patterns shown in Fig. 4 were observed in thecase of equal species diffusivity. Thus, the patternformation is not the result of classical Turing mechan-ism that requires sufficiently different diffusion coeffi-cients (Segel and Jackson, 1972) and has a differentorigin. We also want to mention that our model is fullydeterministic and thus the patchy spread cannot be


attributed to the impact of stochastic factors (cf. Lewis,2000; Lewis and Pacala, 2000).

The above-mentioned papers by Petrovskii et al.(2002a,b) were more concerned with reporting the newphenomenon rather than with its comprehensive in-vestigation. In this paper, the phenomenon of determi-nistic patchy invasion was considered in much detail. Inparticular, two important issues were addressed: (i) howthe type of spread, i.e. with or without continuoustravelling front, affects the speed of invasion and (ii)what is the degree of corresponding spatiotemporalcomplexity of the system dynamics.

Concerning the first issue, we showed by means ofextensive numerical experiments that, choosing thepredator mortality d as a controlling parameter, theregime of patchy invasion corresponds to a remarkabledrop in the invasion speed, cf. Fig. 6b. A sufficientlylarge decrease in d brings the invasion speed to zero.This fact is in a very good agreement with other recentresults (Petrovskii and Venturino, 2004; Petrovskii et al.,2005c) showing that the patchy spread gives the scenarioof species invasion at the edge of extinction: a reason-ably small variation of the system parameters eitherrestore the usual invasion regime via propagation of acontinuous travelling front or bring the species toextinction. Moreover, results of our computer experi-ments, see Petrovskii and Venturino (2004) and Pet-rovskii et al. (2005c) for details, show that thedimensionality of space plays a crucial role: for theparameters when the system exhibits the patchy spreadin two spatial dimensions, the species go extinct in thecorresponding 1D system regardless the choice of theinitial conditions.

A heuristic description of invasion failure in 1D caseis as follows. At early stages of species spread, thetravelling front of the prey density propagates intoempty space being followed by a travelling front ofpredator density, the dynamics being similar to the so-called predator–prey pursuit scenario (Murray, 1989).The speed of prey wave appears to be lower than that ofthe predator wave; as a result, prey is caught up bypredator. After some oscillations, predator decreases theprey density below the survival threshold b at everylocation in space, and extinction of both species takesplace.

In the 2D system, however, the patch border iscurvilinear; thus, prey can escape through the lateralsides and create separate patches. Each new patch thenstarts growing through formation and expansion ofcircular population front (cf. Fig. 4a) until prey iscaught by predator, and this scenario occurs again andagain resulting in a prominent patchy structure.

The above description also helps to understand whythe rate of species invasion in the regime of patchyspread appears to be smaller than it is in the regimes ofcontinuous front propagation. First, the travelling front

of prey is periodically stopped by predator. Second, inevery case formation of new patches takes a certaintime. Finally, the growth of the newly formed patchesnormally leads to species spread not in the radialdirection but at different angles.

Note that the effect of patch border curvature ispositive for prey and negative for predator. Indeed, thelarger is curvature (i.e. the smaller is the patch radius)the higher is the rate of population density decrease dueto its out-flux through the patch border. A decrease inthe population densities inside given patch diminishesthe impact of predation on the prey growth and alsodiminishes the growth rate of predator, cf. Eqs. (5)–(6)where the term describing interspecies interaction hasdifferent sign in the equation for prey and in theequation for predator. Thus, prey has more chances tosurvive in small patches where the border curvature islarge than in large patches where the border shape isclose to a straight line.

Regarding the complexity of the system dynamics inthe regime of patchy invasion, we showed that it ischaotic both for local densities and for spatiallyaveraged values, cf. Figs. 9 and 11. Results of ournumerical experiments (omitting details for the sake ofbrevity) indicate that the correlation dimension of thestrange attractor is relatively high, Dcorr46. Forcomparison, in a similar predator-prey system but withthe logistic prey growth the correlation dimension Dcorr

was found to be between 3 and 4 (Pascual, 1993). Thus,it can be inferred that the impact of the Allee effectsignificantly increases system complexity.

4.1. Statistical properties of the patchy dynamics

As it was mentioned above, spatiotemporal chaos inthe predator–prey system with the Allee effect has beenobserved for some other regimes as well (Morozov et al.,2004; Petrovskii et al., 2005a), e.g. in the case of patternformation in the wake of a travelling population front,see Fig. 2. (Note that the patterns in the wake looksqualitatively similar to the patterns arising as a result ofpatchy invasion, cf. the middle of Figs. 2c and 4c.) Thus,chaoticity of the system dynamics alone cannot be usedto distinguish between the two regimes. The questionthat remains open is whether the absence of acontinuous travelling front is the only difference, albeitimportant, between the two regimes of irregularspatiotemporal pattern formation, cf. Figs. 2 and 4.

In order to address this issue, we have to make adeeper insight into the properties of the correspondingdynamics. High complexity of the system behavior(which is quantified by the high correlation dimensionof the attractor) indicates that it is, in some sense, ratherclose to stochastic dynamics. Indeed, it can be shownthat, on the time-scale larger than the correlation timetcorr of the system, its dynamics can be reproduced by

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16

Dis

trib

utio

n fu

ncti

onD

istr

ibut

ion

func

tion

Average Prey Density

Local Prey Density

(a)

(b)

Fig. 12. Probability distribution functions of the system states

calculated for the patchy invasion, parameters are the same as in

Fig. 4: (a) probability distribution function of the spatially average

prey densityou4t. Solid line shows normal distribution with the same

mean and standard deviation and (b) Probability distribution function

of the local prey density at a fixed point ðx ¼ 160; y ¼ 155Þ.


means of purely stochastic process (A.Y.Morozov,unpublished manuscript). These observations seem tomake possible to describe the system dynamics in termsof probability of its different states.

In order to apply this approach, we first estimate thecorrelation time of the system by means of calculatingthe autocorrelation functions and finding their first zero.Then we generate the time series of local populationdensities (calculated in a fixed point x ¼ 160, y ¼ 155)and of their spatially averaged values, the time lagbetween any two consequent terms in these series beingchosen equal to tcorr. Assuming that any two measure-ments in these series are independent, we then restorethe probability distribution functions (PDF) of systemstates.

Since the time series obtained for prey and predatorexhibit similar properties, below we show only theresults obtained for prey dynamics. Fig. 12a shows thePDF of the average prey density obtained for the patchyinvasion regime for the same parameter set as in Figs. 4and 9 (for these parameters, tcorr ¼ 35). One can easilysee that the shape of the PDF is very close to theGaussian law. Indeed, the envelope shown by the thicksolid line gives the normal distribution where the valuesof the mean mu and the standard deviation su areobtained from the same time series; here mu ¼ 0:168 andsu ¼ 0:012.

The result that the dynamics of the spatially averagedvalues is very well described by the normal distributionis easily understood if we recall that, in the patchyregime, the system is virtually reduced to an ensemble ofquasi-independent oscillators, cf. the lines below Eq.(18). Moreover, it can be expected that the PDFs ofspatially averaged values constructed for the irregularpatterns preceded by continuous front propagation, cf.Fig. 2, will be very close to the normal distribution aswell because, as it was mentioned above, the system inthis regime also exhibits spatiotemporal chaos. Fig. 13ashows the average prey density PDF obtained for theparameters of Fig. 2, thick solid curve giving the normaldistribution.

Other useful information can be obtained from thePDFs of the local densities. Figs. 12b and 13b shows thePDFs of the prey density calculated for the patchyspread (Fig. 4) and for patterns in the wake of thepopulation front (Fig. 2), respectively. Note that, inboth cases, the value of the survival threshold is thesame, b ¼ 0:22. It is readily seen that, in the case ofpattern formation preceded by front propagation, highvalues of the prey density are more probable than lowvalues. On the contrary, in the case of the patchy spread,low values of prey density are more probable than highones. A brief inspection of Fig. 12b immediately leads toa somewhat counter-intuitive result that, in the regimeof patchy spread, the probability to detect the preydensity below the survival threshold is higher than the

probability to find it above the threshold. Moreover, asit is immediately inferred from Figs. 12a and 13a, thespatially averaged prey density appears to be well aboveand well below the survival threshold for the regimes ofinvasion with and without continuous population front,respectively.

4.2. Impact of environmental heterogeneity

All the above results have been obtained under anidealized assumption of environmental homogeneity. Inreality, however, species spread often takes place inheterogeneous environment or even fragmented land-scape. An important question thus arises whether theself-organized patchy invasion can be observed when themodel parameters become space-dependent. Appar-ently, it is a rather diversified issue because the actualoutcome of the interplay between intrinsic populationdynamics and external forcing depends on the nature,geometrical shape and the magnitude of environmentalfluctuations, cf. Sherratt et al. (2003). Hence, in thispaper we make only an early insight into the matter in

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2.0

2.5

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trib

utio

n fu

ncti

onD

istr

ibut

ion

func

tion

(a)

(b)

Average Prey Density

Local Prey Density

Fig. 13. Probability distribution functions of the system states

calculated for the regime of invasion via propagation of continuous

population front with spatiotemporal patterns in the wake, parameters

are the same as in Fig. 2: (a) probability distribution function of the

spatially average prey density ou4t. Solid line shows normal

distribution with the same mean and standard deviation and (b)

Probability distribution function of the local prey density at a fixed

point ðx ¼ 160; y ¼ 155Þ.


order to demonstrate robustness of the patchy invasion;a detailed study of this problem will be done elsewhere.

In order to keep it consistent with the earlier analysis,we assume that it is the predator mortality d thatdepends on position in space while other parametersremain constant. Specifically, we consider heterogeneityof the following type:

dðx; yÞ ¼ d0 þ D sin2p xþ yð Þ

L1

� D sin

2p x� yð Þ

L1

,

(19)

where d0 and D are parameters with the meaning of theaverage mortality and the amplitude of spatial varia-tions, respectively, and L1 quantifies the scale of thevariations. Function dðx; yÞ is shown in Fig. 14a ford0 ¼ 0:42, D ¼ 0:02, L1 ¼ 125:67.

We consider species invasion in heterogeneous envir-onment (19) for the same value of parameters a, b, g andthe same initial condition as in Figs. 1–4. Snapshots ofthe prey density at different moments of time are shownin Figs. 14b–14e. In spite of the presence of areas with

rather severe conditions for prey, cf. the white spotswhere do0:41 (which would correspond to speciesextinction in spatially homogeneous case), invasionappears to be successful and species spread takes placevia essentially the same scenario of formation andmotion of separated patches. Qualitatively, the dy-namics looks very similar to the patchy spread inhomogeneous environment, cf. Fig. 4, typical size andshape of the spatial structures being close to thoseobtained earlier. Note that the characteristic width ofemerging patches is several times smaller than the periodof spatial variations in d, which indicates that the patchyspread is still controlled by the intrinsic mechanismrather than by environmental heterogeneity.

4.3. Concluding remarks

Apart from environmental homogeneity, anotherassumption that has been used throughout this paperis that diffusivity is the same for prey and for predator.It should be emphasized, however, that existence ofpatchy spread is not restricted to this case. In ournumerical simulations, we checked whether the mainresults remain qualitatively the same when D2/D1 is notequal to unity. Computer simulations show that, at leastin the range 0:7oD2=D1o1:5, the patchy spread withthe properties described above can be observed as well,although parameters a, b, g, d should be chosen slightlydifferent.

An inspection of the spatial patterns observed duringthe patchy invasion shows that there is an intrinsicspatial scale, which corresponds to the typical size of thepatches, cf. Figs. 4 and 10. For the given parameters, itappears to be, in dimensionless units, somewherebetween 10 and 20. A question of interest is to whatspatial scale and/or to what invasive species it maycorrespond in real population communities. Accordingto the definition of dimensionless variables, see the linesabove Eqs. (5)–(6), the relation between the dimensionalR and dimensionless r spatial scales is given as R ¼

rðD=AkaÞ1=2; where a ¼ K=B. Note that, in our simula-tions, parameter a is always fixed at a hypothetical valueof 0.1, which means that the effect of saturation inpredator response is insignificant. As for D and Ak (herewe recall that Ak has the meaning of predator maximumgrowth rate), they can be different for different species.As an example, we consider the vole–weasels interac-tion, cf. Sherratt et al. (2002), with D ¼ 0:2 km2 year�1

and Ak ¼ 2:7 year�1 as typical values. We then obtainthat R lies between 8.5 and 17 km, which seems to beecologically reasonable. Note that an increase/decreasein D or Ak as much as 10 times corresponds to onlyabout three times increase/decrease in R; thus, anestimate for R to be between a few kms and a fewdozens kms is likely to remain valid for many otherterrestrial species.

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Fig. 14. Patchy invasion in heterogeneous environment: (a) spatial dependence of d as given by Eq. (19) for parameters d0 ¼ 0:42, D ¼ 0:02,L1 ¼ 125:67 and (b)–(e) snapshots of the prey density obtained at different times. White and black/gray colors correspond to prey absence and prey

presence at high density, respectively. Other parameters are the same as in Fig. 4.



From the point of prospective application of ourresults to invasive species management and control, thesingular dependence of invasion success on predatormortality is an important result. High predator mortal-ity corresponds to successful invasion (cf. Figs. 1, 2 andalso 6b): predator is too weak to stop the invasion ofprey. Gradual decrease in d (which corresponds to anincrease in predation) finally leads to invasion failurewhen propagation of continuous travelling fronts witheither homogeneous or patchy species distributionbehind changes to propagation of travelling populationpulses where population density in the wake is zero, cf.Fig. 3. However, a further decrease in d restoressuccessful invasion in the form of the patchy spread.This result seems to have important implications formanaging invasive species and the problem of biologicalcontrol. Indeed, current approaches to invasive speciescontrol are often based on explicit or implicit assump-tion that the higher is the press on invasive species, themore probable is its invasion failure. Our results,however, indicate that, instead of this monotonedependence, there can exist an optimal magnitude ofcontrolling efforts.

Acknowledgements

This research was partially supported by the USNational Science Foundation under Grants DEB-0409984 and DEB-0080529, by Russian Foundationfor Basic Research under Grants 03-04-48018 and 04-04-49649, and by the University of California Agricul-tural Experiment Station.

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