Research ArticleGlobal Stability for a Discrete Space-Time LotkandashVolterraSystem with Feedback Control

Li Xu 12 and Ruiwen Han1

1School of Science Tianjin University of Commerce Tianjin 300134 China2School of Mathematics Tianjin University Tianjin 300072 China

Correspondence should be addressed to Li Xu beifang_xl163com

Received 26 May 2020 Revised 11 July 2020 Accepted 27 July 2020 Published 19 August 2020

Academic Editor Juan Carlos Cortes

Copyright copy 2020 Li Xu and Ruiwen Han -is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this paper a discrete space-time LotkandashVolterra model with the periodic boundary conditions and feedback control isproposed By means of a discrete version of comparison theorem the boundedness of the nonnegative solution of the system isproved By the combination of the Volterra-type and quadratic Lyapunov functions the global asymptomatic stability of theunique positive equilibrium is investigated Finally numerical simulations are presented to verify the effectiveness of themain results

1 Introduction

It is well known that the ecosystem in the real world is oftendistributed by unpredictable forces or interference factorssuch as natural disturbances (floods fires disease outbreaksand droughts) human-caused interference factors (oilspills) and slowly changing long-term stresses (nutrientenrichment) which may result into changes in the biologicalparameters such as survival rates [1ndash3] -e presence of theunpredictable forces or interference factors in an ecologicalsystem raises the following essential and basic question fromthe practical interest in ecology ldquoCan the ecosystem with-stand those unpredictable forces which persist for a finiteperiod of timerdquo -e question has motivated the develop-ment of some control mechanisms for managing pop-ulations to ensure that the interacting species can coexistsuch as impulsive control optimal vibration control in-termittent control and feedback control [4ndash6] As a basicmechanism by which one can recover stability and move thetrajectory towards the desired orbit the introduction of afeedback control variable is one method that can achieve theobjective

For population dynamical systems with feedback con-trols an important and interesting subject is to study the

effects of feedback controls to the persistence permanenceand extinction of species the stability and dynamicalcomplexity of systems [7] -ere are lots of important andinteresting results on stability research for continuous timepopulation dynamical models [8ndash16] A necessary conditionfor sustained concentration oscillations resulting from smallperturbations of the steady state is derived from a closurerule using a variation of the direct Lyapunov method on abiochemical feedback system of the Yates-Pardee type [8]-e authors study the dynamical behavior of a continuousreaction-diffusion waterborne pathogen model such as theexistence of positive solutions and its boundedness theexistence of equilibria local stability uniform persistenceand global stability [9] -e output feedback stabilization ofstochastic feedforward systems with unknown control co-efficients and unknown output function using the time-varying technique and backstepping method is achieved[16]

-e discrete-time models governed by differenceequation are more realistic than the continuous ones whenthe populations have nonoverlapping generations or thepopulation statistics are compiled from given time intervalsand not continuously Moreover discrete-time models canalso provide efficient computational models of continuous

HindawiComplexityVolume 2020 Article ID 2960503 9 pageshttpsdoiorg10115520202960503

models for numerical simulations-erefore it is reasonableto study discrete-time models governed by differenceequations and there has been some work done on the studyof the persistence permanence and global stability forvarious discrete-time nonlinear population systems withfeedback when the effect of spatial factors is not considered[5 7 17ndash19] A weak sufficient condition for the permanenceof a nonautonomous discrete single-species system withdelays and feedback control is given in the article [7] A two-species competitive system with feedback controls is con-sidered in which the global attractivity of a positive periodicsolution is obtained and the existence and uniqueness of theuniformly asymptotically stable almost periodic solution areshown [17 18] In reference [19] some sufficient conditionson the permanence and the global stability of the system of an-species LotkandashVolterra discrete system with delays andfeedback control by constructing the suitable discrete typeLyapunov functionals are obtained

It is a fact that spatial heterogeneity and dispersal playan important role in the dynamics of populations whichhas been the subject of much research both theoreticaland experimental such as the role of dispersal in themaintenance of patchiness or spatial population variationIf the spatial factors are added more dynamics will occur-e diffusion-driven instability may emerge if the steady-state solution is stable to small spatial perturbations inabsence of diffusion but unstable when diffusion ispresent [20] If the diffusion-driven instability should beavoided in some situations and one may wish recoverystability towards the desired orbit but the system pa-rameters are not easy to adjust then some other waysshould be adopted to achieve the stabilization aim [21]-ere also may exist a situation where the equilibrium ofthe dynamical model is not the desirable one (or af-fordable) and a smaller value of the equilibrium is re-quired then altering the model structure so as to makethe population stabilize at a lower value is necessary [22]Feedback control will be an effective one and can alter thepositions of positive equilibrium or obtain its stability Tothe best of our knowledge there is few work that has beendevoted to global properties of the discrete space-timemodels with feedback control -e robustly asymptoticstability and disturbance attenuation level of the filteringerror system for a two-dimensional Roesser models withpolytopic uncertainties are discussed [23] A two-di-mensional FornasinindashMarchesini local state-space systemis also considered in the article [24] However the dif-fusion terms (discrete Laplace operator) are not directlyintroduced into the model -ere is some work on globalstability of discrete diffusion systems [25 26] in which thepositivity boundedness and global stability of theequilibria are established and the discretized models arederived from the corresponding continuous model bynonstandard finite difference but the Laplace operatorhas been dealt with It is a fact that diffusion will producemuch richer dynamical behaviors and complexity how toanalyze stability of the discrete diffusion system withfeedback control by means of suitable Lyapunov functionsis an important problem to solve

Motivated by above discussions the main purpose ofthis paper is to study the global asymptomatic stability ofan one-dimensional spatially discrete reaction diffusionLotkandashVolterra model with the periodic boundary con-ditions and feedback control So the organization of thispaper is as follows In the Section 2 we formulate thediscrete space-time LotkandashVolterra model with feedbackcontrol and present some assumptions and preparationswhich will be essential to our main proofs and thenonnegativity and boundedness of the solution of thesystem are proved by means of comparison theorem-en global asymptotic stability of the unique positiveequilibrium is proved by constructing a combination ofthe nonnegative Volterra-type and quadratic Lyapunovfunctions in Section 3 In Section 4 numerical simulationsare presented to illustrate the feasibility of our main re-sults In the last section brief discussions and conclusionsare given

2 Model and Preliminaries

It is well known that a LotkandashVolterra system can be de-scribed in the form of

xprime(t) x(t) r1 minus a11x(t) minus a12y(t)( 1113857

yprime(t) y(t) r2 + a21x(t) minus a22y(t)( 1113857(1)

which is called the predator-prey model x(t) is the densityof prey species y(t) is the density of predator species thecoefficients a11 and a22 represent the intraspecific interac-tions a12 and a21 represent the interspecific interactions andr1 and r2 are the intrinsic growth rates of the respectivespecies

A corresponding discrete model for the system (1) can bederived from [27]

xn+1 xn exp r1 minus a11xn minus a12yn( 1113857

yn+1 yn exp r2 + a21xn minus a22yn( 11138571113896 (2)

where aij(i j 1 2)gt 0 Let Xn a11xn and Yn a22yn wehave

Xn+1 Xn exp r1 minus Xn minusa12

a22Yn1113888 1113889

Yn+1 Yn exp r2 +a21

a11Xn minus Yn1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

orxn+1 xn exp r1 minus xn minus a12yn( 1113857

yn+1 yn exp r2 + a21xn minus yn( 1113857

⎧⎪⎨

⎪⎩

(3)

It is believed that the diffusion of individuals can play animportant role in determining collective behavior of thepopulation Space factors can be taken into account in allfundamental aspects of ecological organization and we canget a one-dimensional discrete reaction-diffusion model asfollows

2 Complexity

xn+1i x

ni exp r1 minus x

ni minus a12y

ni( 1113857 + D1nabla

2x

ni

yn+1i y

ni exp r2 + a21x

ni minus y

ni( 1113857 + D2nabla

2y

ni

⎧⎨

⎩ (4)

where i isin 1 2 m [1 m] and m n isin Z+ are positiveintegers and D1 and D2 are diffusion parameters

nabla2xni x

ni+1 minus 2x

ni + x

niminus1

nabla2yni y

ni+1 minus 2y

ni + y

niminus1

(5)

-is also indicates the coupling or diffusion from theunits or individuals to the left and the right respectively-efollowing periodic boundary conditions are considered

xn0 x

nm x

n1 x

nm+1

yn0 y

nm y

n1 y

nm+1

⎧⎨

⎩ (6)

Systems (4)ndash(6) can exhibit rich dynamic behaviors anddiffusion-driven instability may emerge [28] As discussed inthe introduction part unpredictable forces or interferencefactors can be introduced into the forms of feedback controlvariables which can contribute significantly to the biologicalsystems by affecting their dynamics and stability Moreoverstructurally modifying existing systems by incorporatingvariables for defining feedback controls is appropriate forconsidering the unpredictable forces or interference factorsin an ecosystem So in the present study we consider thefollowing one-dimensional discrete space-time Lot-kandashVolterra model with periodic boundary conditions andfeedback control

xn+1i x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857 + D1nabla

2x

ni

yn+1i y

ni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni

un+11i 1 minus η1( 1113857u

n1i + e1x

ni

un+12i 1 minus η2( 1113857u

n2i + e2y

ni

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(7)

with the periodic boundary conditions

xn0 x

nm x

n1 x

nm+1

yn0 y

nm y

n1 y

nm+1

⎧⎨

⎩ (8)

where i isin 1 2 m [1 m] and m n isin Z+ is positiveinteger r1 r2 a12a21 η1 η2 e1 e2 are positive constants andD1 and D2 are diffusion parameters

nabla2xni x

ni+1 minus 2x

ni + x

niminus1

nabla2yni y

ni+1 minus 2y

ni + y

niminus1

(9)

To the best of our knowledge no work on globalasymptomatic stability of the positive equilibrium of systems(7) and (8) has been done yet

By simple computation systems (7) and (8) have apositive equilibrium

Elowast

xlowast ylowast ulowast1 ulowast2( 1113857 (10)

where

xlowast

η1r1 η2 + e2d2( 1113857 minus η1η2a12r2

η2 + e2d2( 1113857 η1 + e1d1( 1113857 + a12a21η1η2

ylowast

η2r2 η1 + e1d1( 1113857 + η1η2a21r1

η2 + e2d2( 1113857 η1 + e1d21( 1113857 + a12a21η1η2

ulowast1

e1

η1xlowast

ulowast2

e2

η2ylowast

(11)

If r1(η2 + e2d2)gt η2a12r2 the equilibrium is positiveTo discuss the global asymptomatic stability of the

unique positive equilibrium the following assumptions andpreparations are essential

From the view point of biology we only need to discussthe positive solution of system (7) So it is assumed that theinitial conditions of (8) are of the form

x0i gt 0 u

0i gt 0 i 1 2 m (12)

For our purpose we first introduce the following lemmawhich can be obtained easily by comparison theorem ofdifference equation

Lemma 1 (see [29]) Let x(n) be a nonnegative solution ofinequality

x(n + 1)lex(n)exp α minus βx(n)1113864 1113865 n isin Z (13)

with x(0)gt 0 and α βgt 0 then

limn⟶infin

supx(n)leαβ

(14)

Lemma 2 (see [30]) Any solution x(n) of system

x(n + 1) x(n)(1 minus c) + ω(n) n isin Z (15)

with x(0)gt 0 satisfies

limn⟶infin

supx(n)leup n isin Zω(n)

c (16)

where ω(n) is a nonnegative bounded sequence of realnumbers and 0lt clt 1

Applying the above lemmas we can obtain the followingresult

Theorem 1 -e solution of (7) with initial condition (8) isdefined and remains nonnegative and bounded iferj minus 2Dj ge 0 and ηj lt 1 j 1 2 hold

Proof From the first equation of system (7) we get

xn+1i x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857 + D1nabla

2x

ni

xni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857 minus 2D1( 1113857

+ D1 xni+1 + x

niminus1( 1113857

(17)

Complexity 3

from which it is true that xni ge 0 holds for all n with

x0i gt 0 u0

i gt 0 i 1 2 m if er1 minus 2D1 ge 0 and appropriateparameters a12 d1 are selected

Similarly from the second equation of system (7) we getthat yn

i ge 0 holds for all n with x0i gt 0 u0

i gt 0 i 1 2 mif er2 minus 2D2 ge 0 and appropriate parameters a21 d2 areselected

If ηj le 1 j 1 2 unji ge 0 can also hold by means of the

third and fourth equations of system (7)Next we will show the boundedness of the solutions

1113944

m

i1x

n+1i 1113944

m

i1x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857 + D1nabla

2x

ni1113872 1113873

1113944

m

i1x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857

le 1113944m

i1x

ni exp r1 minus x

ni( 1113857

(18)

From Lemma 1 we can obtain

limn⟶infin

sup1113944m

i1x

ni le 1113944

m

i1r1 mr1 (19)

Similarly we can also obtain

1113944

m

i1y

n+1i 1113944

m

i1y

ni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni1113872 1113873

1113944m

i1y

ni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857

le 1113944m

i1y

ni exp r2 + a21Mx minus y

ni( 1113857

(20)

where Mx supnisinZxni -en

limn⟶infin

sup1113944m

i1y

ni le 1113944

m

i1r2 + a21Mx( 1113857 m r2 + a21Mx( 1113857

(21)

From Lemma 2 by means of (20) and (21) and η1 η2 le 1we can obtain

limn⟶infin

sup un1i le

e1Mx

η1 (22)

limn⟶infin

sup un2i le

e2My

η2 (23)

where My supnisinZyni -e proof is finished

3 Global Stability

In this section we devote ourselves to studying the globalasymptotic stability of the unique positive equilibrium ElowastBy using global Lyapunov function we derive the sufficient

conditions under which the positive equilibrium is globallyasymptotically stable

Denote

H(1) erj minus 2Dj ge 0 j 1 2

H(2)djej

2 1 minus ηj1113872 1113873le 1 ηj le 1 j 1 2

(24)

Assume xni1113864 1113865

nisinZ+

iisin[1m] yni1113864 1113865

nisinZ+

iisin[1m] are positive solutions ofsystems (7) and (8) we can establish the following result

Theorem 2 Assume H(1) and H(2) hold the positiveequilibrium Elowast of systems (7) and (8) is globally asymptot-ically stable

Proof Let

Vn1 1113944

m

i1x

ni minus xlowast

minus xlowastln

xni

xlowast1113888 1113889 (25)

-en we can obtain

ΔVn1 V

n+11 minus V

n1

1113944m

i1x

n+1i minus x

ni minus xlowastln

xn+1i

xni

1113888 1113889

1113944

m

i1x

n+1i minus x

ni minus xlowastx

n+1i minus x

ni

xni

1113888 1113889 + o(1)

1113944m

i1x

n+1i minus x

ni1113872 1113873 1 minus

xlowast

xni

1113888 1113889 + o(1)

1113944m

i11 minus

xlowast

xni

1113888 1113889 xni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857(

+ D1nabla2x

ni minus x

ni 1113873 + o(1)

1113944m

i11 minus

xlowast

xni

1113888 1113889 xni 1 minus x

ni minus xlowast

( 1113857 minus a12 yni minus ylowast

( 1113857((

minus d1 un1i minus u

lowast1( 11138571113857 + D1nabla

2x

ni minus x

ni 1113873

minus D1 1113944

m

i1xlowast x

ni+1

xni

+x

niminus1

xni

minus 21113888 1113889 + o(1) + o ρ1( 1113857

minus 1113944m

i1xi minus x

lowasti( 1113857

2minus a12 1113944

m

i1x

ni minus xlowast

( 1113857 yni minus ylowast

( 1113857

minus d1 1113944

m

i1x

ni minus xlowast

( 1113857 un1i minus u

lowast1( 1113857 minus D1x

lowast1113944

mminus1

i1

middot

xn

i+1xn

i

1113971

minus

xn

iminus1xn

i

1113971

⎛⎝ ⎞⎠

2

minus D1xlowast

xn

m

xn1

1113971

minus

xn1

xnm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ1( 1113857

(26)

where ρ1

(xni minus xlowast)2 + (yn

i minus ylowast)2 + (un1i minus ulowast1 )2

1113969

4 Complexity

Let

Vn2

a12

a211113944

m

i1y

ni minus ylowast

minus ylowastln

yni

ylowast1113888 1113889 (27)

-en we can obtain

ΔVn2 V

n+12 minus V

n2

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowast ln

yn+1i

yni

1113888 1113889

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowasty

n+1i minus y

ni

yni

1113888 1113889 + o(1)

a12

a211113944

m

i1y

n+1i minus y

ni1113872 1113873 1 minus

ylowast

yni

1113888 1113889 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni minus y

ni1113872 1113873 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni 1 minus y

ni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus y

ni + D2nabla

2y

ni1113872 1113873 + o(1) + o ρ2( 1113857

a12

a211113944

m

i1y

ni minus ylowast

( 1113857 minus yni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus D2

a12

a211113944

m

i1ylowast y

ni+1

yni

+y

niminus1

yni

minus 21113888 1113889 + o(1) + o ρ2( 1113857

minusa12

a211113944

m

i1yi minus y

lowasti( 1113857

2+ 1113944

m

i1a12 x

ni minus xlowast

( 1113857 yni minus ylowast

( 1113857 minusa12d2

a211113944

m

i1y

ni minus ylowast

( 1113857 un2i minus u

lowast2( 1113857 minus D2

a12

a211113944

mminus1

i1

middot

yn

i+1yn

i

1113971

minus

yn

iminus1yn

i

1113971

⎛⎝ ⎞⎠

2

minus D2a12

a21

yn

m

yn1

1113971

minus

yn1

ynm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ2( 1113857

(28)

ΔVn2 V

n+12 minus V

n2

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowast ln

yn+1i

yni

1113888 1113889

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowasty

n+1i minus y

ni

yni

1113888 1113889 + o(1)

a12

a211113944

m

i1y

n+1i minus y

ni1113872 1113873 1 minus

ylowast

yni

1113888 1113889 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni minus y

ni1113872 1113873 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni 1 minus y

ni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus y

ni + D2nabla

2y

ni1113872 1113873 + o(1) + o ρ2( 1113857

a12

a211113944

m

i1y

ni minus ylowast

( 1113857 minus yni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus D2

a12

a211113944

m

i1ylowast y

ni+1

yni

+y

niminus1

yni

minus 21113888 1113889 + o(1) + o ρ2( 1113857

minusa12

a211113944

m

i1yi minus y

lowasti( 1113857

2+ a12 1113944

m

i1x

ni minus xlowast

( 1113857 yni minus ylowast

( 1113857 minusa12d2

a211113944

m

i1y

ni minus ylowast

( 1113857 un2i minus u

lowast2( 1113857 minus D2

a12

a211113944

mminus1

i1

yn

i+1yn

i

1113971

minus

yn

iminus1yn

i

1113971

⎛⎝ ⎞⎠

2

minus D2a12

a21

yn

m

yn1

1113971

minus

yn1

ynm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ2( 1113857

(29)

Complexity 5

where ρ2

(xni minus xlowast)2 + (yn

i minus ylowast)2 + (un2i minus ulowast2 )2

1113969

Let

Vn3

d1

2 1 minus η1( 1113857e1u

n1i minus u

lowast1( 1113857

2 (30)

-en we can obtain

ΔVn3 V

n+13 minus V

n3

d1

2 1 minus η1( 1113857e11113944

m

i1u

n+11i minus u

n1i1113872 1113873 u

n+11i + u

n1i minus 2u

lowast11113872 1113873

d1

2 1 minus η1( 1113857e11113944

m

i1minusη1u

n1i + e1x

ni( 1113857 2 minus η1( 1113857u

n1i(

+ e1xni minus 2u

lowast1 1113857

d1

2 1 minus η1( 1113857e11113944

m

i1minusη1 u

n1i minus u

lowast1( 1113857 + e1 x

ni minus xlowast

( 1113857( 1113857

middot 2 minus η1( 1113857 un1i minus u

lowast1( 1113857 + e1 x

ni minus xlowast

( 1113857( 1113857

minusd1η1 2 minus η1( 1113857

2 1 minus η1( 1113857e11113944

m

i1u

n1i minus u

lowast1( 1113857

2

+d1e1

2 1 minus η1( 11138571113944

m

i1x

ni minus xlowast

( 11138572

+ d1 xni minus xlowast

( 1113857 un1i minus u

lowast1( 1113857

(31)

Let

Vn4

d2a12

2 1 minus η2( 1113857e2a21u

n2i minus u

lowast2( 1113857

2 (32)

-en we can obtain

ΔVn4 V

n+14 minus V

n4

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1u

n+12i minus u

n2i1113872 1113873 u

n+12i + u

n2i minus 2u

lowast21113872 1113873

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1minusη2u

n2i + e2y

ni( 1113857 minusη2( 1113857u

n2i(

+ e2yni minus 2u

lowast2 1113857

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1minusη2 u

n2i minus u

lowast2( 1113857(

+ e2 yni minus ylowast

( 11138571113857 2 minus η2( 1113857 un2i minus u

lowast2( 1113857 + e2 y

ni minus ylowast

( 1113857( 1113857

minusd2a12η2 2 minus η2( 1113857

2 1 minus η2( 1113857e2a211113944

m

i1u

n2i minus u

lowast2( 1113857

2

+d2a12e2

2 1 minus η2( 1113857a211113944

m

i1y

ni minus ylowast

( 11138572

+a12d2

a21

middot yni minus ylowast

( 1113857 un2i minus u

lowast2( 1113857

(33)

Let

Vn

Vn1 + V

n2 + V

n3 + V

n4 (34)

-en

ΔVn V

n+1minus V

n

le minus1 +d1e1

2 1 minus η1( 11138571113888 1113889 1113944

m

i1x

ni minus xlowast

( 11138572

+ minusa12

a21+

d2a12e2

2 1 minus η2( 1113857a211113888 1113889 1113944

m

i1y

ni minus ylowast

( 11138572

+d1η1 η1 minus 2( 1113857

2 1 minus η1( 1113857e11113944

m

i1u

n1i minus u

lowast1( 1113857

2

+d2a12η2 η2 minus 2( 1113857

2 1 minus η2( 1113857e2a211113944

m

i1u

n2i minus u

lowast2( 1113857

2

minus D1 1113944

mminus1

i1

xn

i+1xn

i

1113971

minus

xn

iminus1xn

i

1113971

⎛⎝ ⎞⎠

2

minus D1

xn

m

xn1

1113971

minus

xn1

xnm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ1( 1113857

minus D2a12

a211113944

mminus1

i1

yn

i+1yn

i

1113971

minus

yn

iminus1yn

i

1113971

⎛⎝ ⎞⎠

2

minus D2a12

a21

yn

m

yn1

1113971

minus

yn1

ynm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ2( 1113857

(35)

If (d1e12(1 minus η1))le 1 (d2e22(1 minus η2))le 1 (d1η1(η1 minus

2)2(1 minus η1)e1)le 0 and (d2a12η2(η2 minus 2)2(1 minus η2)e2a21)le 0or(d1e12(1 minus η1))le 1 (d2e22(1 minus η2))le 1 η1 lt 1 andη2 lt 1hold ΔVn le 0 -e proof is completed

4 Example and Numerical Simulations

In the following example we will show the feasibility of ourmain results and discuss the effects of feedback controlsTake i 2 in the system we obtain a model with feedbackcontrols as follows

xn+1i x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857 + D1nabla

2x

ni

yn+1i y

ni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni

un+11i 1 minus η1( 1113857u

n1i + e1x

ni

un+12i 1 minus η2( 1113857u

n2i + e2y

ni

i 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)

with the periodic boundary conditions

6 Complexity

xn0 x

n2

xn1 x

n3

yn0 y

n2

yn1 y

n3

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)

To illustrate our purposes the parameter values arechosen as follows (the choice of parameter values is hypo-thetical with appropriate units and not based on data) r1

12 r2 1a12 08a21 03d1 1d2 1D1 03D2 04

e1 07 e2 06η1 05 andη2 06 then there is only aunique positive equilibrium Elowast(xlowast1 xlowast2 ylowast1 ylowast2 ulowast11u

lowast12 ulowast21

ulowast22) (03181 03181 05487 05487 04453 0445305487

05487) It is easy to see that the conditions in -eorem 2 areverified Dynamic behaviors of systems (36) and (37) with theinitial conditions are shown in Figure 1 and three sets ofdifferent initial conditions are listed in Table 1-e simulations

can illustrate the fact that the positive equilibrium is globallyasymptotically stable

To explore clearly the dynamical behavior of systems(36) and (37) we investigate the effect of diffusion pa-rameter d2 by keeping other parameters of the systemfixed Figure 2 exhibits in detail an interesting situation when

050

045

040

035

030

025

0 10 20 30 40n

50 60

xn1

un11

Solu

tions

xn a

nd u

n 1

(a)

050

045

040

035

030

025

0 10 20 30 40n

50 60

xn2

un12

Solu

tions

xn a

nd u

n 1

(b)

0 10 20 30 40n

50 60

Solu

tions

yn an

d un 2

06

05

04

03

02yn1

un21

(c)

0 10 20 30 40n

50 60

Solu

tions

yn an

d un 2

06

05

04

03

02yn2

un22

(d)

Figure 1 Dynamic behaviors of systems (36) and (37) with three sets of different initial conditions when r1 12

r2 1 a12 08 a21 03 e1 07 e2 06 η1 05 η2 06 d1 1 d2 1 D1 03 andD2 04

Table 1 Different initial values for xn1 xn

2 yn1 yn

2 un11 un

12 un21 un

22

x01 x0

2 y01 y0

2 u011 u0

12 u021 u0

22

1 022 035 027 022 040 046 045 0522 033 026 023 028 047 042 048 0433 025 023 021 030 043 041 053 056 050

055

060

045

040

035

0 10 20 30 40n

50 60

xn1

un11

Solu

tions

xn a

nd u

n 1

Figure 2 Dynamic behaviors of xn1 and un

11 for systems (36) and(37) with initial conditions (036 034 036 034 028 031

028 031 059 052 045 054) when d2 08

Complexity 7

d2 08(r1 12 r2 1 a12 08 a21 03 d1 1 D1

03 D2 04 e1 07 e2 06 η1 05 η2 06) in whichthe solutions converge faster than Figure 1 With the increaseof d2 the solutions converge more slowly as depicted inFigure 3 For the sake of convenience only the dynamicalbehavior of xn

1 and un11 with a group of initial values is shown

in Figures 2 and 3 -e simulation results of adjustingfeedback control coefficients d1 e1 and e2 are omitted

5 Conclusions and Discussion

-is paper investigates the global asymptomatic stability ofthe unique positive equilibrium of a discrete diffusion modelwith the periodic boundary conditions and feedback control-e condition to ensure the nonnegativity and boundednessof the solutions of the discrete model is discussed and theglobally asymptotical stability of the positive equilibrium isproved -rough comparing numerical simulations wenotice that when we improve the feedback control coeffi-cients the solutions will converge more slowly It followsthat we can adjust the rate of convergence by choosingsuitable values of feedback control variables Such work mayalso be applied to other discrete diffusion models

It should be noted that there is only a basic conditionobtained to guarantee the existence of positive solutionJudgement sentence xn

i yni un

1i un2i ge 0 should be added into

the simulation programs under the conditions oferj minus 2Dj ge 0 and ηj le 1 j 1 2 Further improvements areneeded

In this study all of the coefficients of the model systemare constant in many situations they can be assumed to benonconstant bounded nonnegative sequences such as pe-riodic positive sequences which can reflect the seasonalfluctuations [19] On the other hand time delays can have agreat influence on species populations It is worthy toconsider the nonautonomous space-time discrete Lot-kandashVolterra system with feedback control

Different control schemes such as switching controlconstraint control and sliding control can be applied to thesystem models Many interesting results can be obtained

Based on the backstepping recursive technique a neuralnetwork-based finite-time control strategy is proposed for aclass of non-strict-feedback nonlinear systems [31] and anevent-triggered robust fuzzy adaptive prescribed perfor-mance finite-time control strategy is proposed for a class ofstrict-feedback nonlinear systems with external disturbances[32] How to apply these control schemes on the discretediffusion models may be worth considering

It is well known that noise disturbance is unavoidable inreal systems and it has an important effect on the stability ofsystems Also noise can be used to stabilize a given unstablesystem or to make a system even more stable when thesystem is already stable which reveals that the stochasticfeedback control can stabilize and destabilize the deter-ministic systems [33 34]-erefore it will be interesting andchallenging to investigate stabilization or destabilization ofnonlinear discrete space-time systems by stochastic feedbackcontrol in our future work

Data Availability

No data were used to support this study

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was supported by the Applied Study Program(grant nos 171006901B 60204 and WH18012)

References

[1] J Xu and Z Teng ldquoPermanence for a nonautonomous dis-crete single-species system with delays and feedback controlrdquoApplied Mathematics Letters vol 23 no 9 pp 949ndash954 2010

[2] O S Board An Ecosystem Services Approach to Assessing theImpacts of the Deepwater Horizon Oil Spill in the Gulf ofMexico National Academies Press Washington DC USA2013

[3] V Tiwari J P Tripathi R K Upadhyay Y-PWu J-SWangand G-Q Sun ldquoPredator-prey interaction system with mu-tually interfering predator role of feedback controlrdquo AppliedMathematical Modelling vol 87 pp 222ndash244 2020

[4] X Li X Yang and T Huang ldquoPersistence of delayed co-operative models impulsive control methodrdquo AppliedMathematics and Computation vol 342 pp 130ndash146 2019

[5] T Luo ldquoStabilization of multi-group models with multipledispersal and stochastic perturbation via feedback controlbased on discrete-time state observationsrdquo Applied Mathe-matics and Computation vol 354 pp 396ndash410 2019

[6] W Qin X Tan M Tosato and X Liu ldquo-reshold controlstrategy for a non-smooth Filippov ecosystem with groupdefenserdquo Applied Mathematics and Computation vol 362Article ID 124532 2019

[7] J Xu Z Teng and H Jiang ldquoPermanence and globalattractivity for discrete nonautonomous two-species lotka-volterra competitive system with delays and feedback con-trolsrdquo Periodica Mathematica Hungarica vol 63 no 1pp 19ndash45 2011

030

028

026

024

022

020

018

0 10 20 30 40 50 60 70 80n

xn1

un11

Solu

tions

xn a

nd u

n 1

Figure 3 Dynamic behaviors of xn1 and un

11 for systems (36) and(37) with initial conditions (016 021 016 021 021 026 021 026 025 022 045 050) when d2 12

8 Complexity

[8] C Walter ldquoStability of controlled biological systemsrdquo Journalof -eoretical Biology vol 23 no 1 pp 23ndash38 1969

[9] P Wang Z Zhao and W Li ldquoGlobal stability analysis fordiscrete-time coupled systems with both time delay andmultiple dispersal and its applicationrdquo Neurocomputingvol 244 pp 42ndash52 2017

[10] Y Kuang ldquoGlobal stability in delay differential systemswithout dominating instantaneous negative feedbacksrdquoJournal of Differential Equations vol 119 no 2 pp 503ndash5321995

[11] Y Yan and E-O N Ekaka-a ldquoStabilizing a mathematicalmodel of population systemrdquo Journal of the Franklin Institutevol 348 no 10 pp 2744ndash2758 2011

[12] Y Muroya ldquoGlobal stability of a delayed nonlinear lotka-volterra system with feedback controls and patch structurerdquoApplied Mathematics and Computation vol 239 pp 60ndash732014

[13] J L Liu and W C Zhao ldquoDynamic analysis of stochasticlotkandashvolterra predator-prey model with discrete delays andfeedback controlrdquo Complexity vol 2019 p 15 Article ID4873290 2019

[14] Q Zhu and H Wang ldquoOutput feedback stabilization ofstochastic feedforward systems with unknown control coef-ficients and unknown output functionrdquo Automatica vol 87pp 166ndash175 2018

[15] Q Zhu ldquoStabilization of stochastic nonlinear delay systemswith exogenous disturbances and the event-triggered feed-back controlrdquo IEEE Transactions on Automatic Controlvol 64 no 9 pp 3764ndash3771 2019

[16] K Kiss and E Gyurkovics ldquoLMI approach to global stabilityanalysis of stochastic delayed lotka-volterra modelsrdquo AppliedMathematics Letters vol 104 pp 106227 1ndash6 2020

[17] X Chen and C Fengde ldquoStable periodic solution of a discreteperiodic lotka-volterra competition system with a feedbackcontrolrdquo Applied Mathematics and Computation vol 181no 2 pp 1446ndash1454 2006

[18] C Niu and X Chen ldquoAlmost periodic sequence solutions of adiscrete lotka-volterra competitive system with feedbackcontrolrdquo Nonlinear Analysis Real World Applications vol 10no 5 pp 3152ndash3161 2009

[19] X Liao S Zhou and Y Chen ldquoPermanence and globalstability in a discrete n-species competition system withfeedback controlsrdquo Nonlinear Analysis Real World Appli-cations vol 9 no 4 pp 1661ndash1671 2008

[20] A M Turing ldquo-e chemical basis of morphogenesisrdquo Philo-Sophical Transactions of the Royal Society B Biological Sci-ences vol 237 no 641 pp 37ndash72 1952

[21] K Gopalsamy and P X Weng ldquoFeedback regulation of lo-gistic growthrdquo International Journal of Mathematics andMathematical Sciences vol 16 no 1 pp 177ndash192 1993

[22] L Xu S S Lou P Q Xu and G Zhang ldquoFeedback controland parameter invasion for a discrete competitive lot-kandashvolterra systemrdquoDiscrete Dynamics in Nature and Societyvol 2018 p 8 Article ID 7473208 2018

[23] X W Li and H J Gao ldquoRobust finite frequency Hinfinfilteringfor uncertain 2-D roesser systemsrdquo Automatica vol 48pp 1163ndash1170 2012

[24] X W Li H J Gao and C H Wang ldquoGeneralized kal-manndashyakubovichndashpopov lemma for 2-D FM LSS modelrdquoIEEE Transaction on Automatic Control vol 57 no 12pp 3090ndash3193 2012

[25] J Zhou Y Yang and T Zhang ldquoGlobal dynamics of a re-action-diffusion waterborne pathogen model with general

incidence raterdquo Journal of Mathematical Analysis and Ap-plications vol 466 no 1 pp 835ndash859 2018

[26] Y Yang and J Zhou ldquoGlobal stability of a discrete virusdynamics model with diffusion and general infection func-tionrdquo International Journal of Computer Mathematics vol 96no 9 pp 1752ndash1762 2019

[27] P Liu and S N Elaydi ldquoDiscrete competitive and cooperativemodels of lotka-volterra typerdquo Journal of ComputationalAnalysis and Applications vol 3 no 1 pp 53ndash73 2001

[28] L L Meng and Y T Han ldquoBifurcation chaos and patternformation for the discrete predator-prey reaction-diffusionmodelrdquo Discrete Dynamics in Nature and Society vol 2019p 9 Article ID 9592878 2019

[29] R M May and G F Oster ldquoBifurcations and dynamiccomplexity in simple ecological modelsrdquo -e AmericanNaturalist vol 110 no 974 pp 573ndash599 1976

[30] Z Zhou and X Zou ldquoStable periodic solutions in a discreteperiodic logistic equationrdquo Applied Mathematics Lettersvol 16 no 2 pp 165ndash171 2003

[31] K K Sun J B Qiu H R Karimi and H J Gao ldquoA novelfinite-time control for nonstrict feedback saturated nonlinearsystems with tracking error constraintrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash12 2020

[32] K K Sun J B Qiu H R Karimi and Y L Fu ldquoEvent-triggered robust fuzzy adaptive finite-time control of non-linear systems with prescribed performancerdquo IEEE Transac-tions on Fuzzy Systems 2020

[33] X Mao ldquoAlmost sure exponential stabilization by discrete-time stochastic feedback controlrdquo IEEE Transactions onAutomatic Control vol 61 no 6 pp 1619ndash1624 2016

[34] Q Zhu and T Huang ldquoStability analysis for a class of sto-chastic delay nonlinear systems driven by G-Brownian mo-tionrdquo Systems amp Control Letters vol 140 Article ID 1046999 pages 2020

Complexity 9

xn+1i x

ni exp r1 minus x

ni minus a12y

ni( 1113857 + D1nabla

2x

ni

yn+1i y

ni exp r2 + a21x

ni minus y

ni( 1113857 + D2nabla

2y

ni

⎧⎨

⎩ (4)

where i isin 1 2 m [1 m] and m n isin Z+ are positiveintegers and D1 and D2 are diffusion parameters

nabla2xni x

ni+1 minus 2x

ni + x

niminus1

nabla2yni y

ni+1 minus 2y

ni + y

niminus1

(5)

-is also indicates the coupling or diffusion from theunits or individuals to the left and the right respectively-efollowing periodic boundary conditions are considered

xn0 x

nm x

n1 x

nm+1

yn0 y

nm y

n1 y

nm+1

⎧⎨

⎩ (6)

Systems (4)ndash(6) can exhibit rich dynamic behaviors anddiffusion-driven instability may emerge [28] As discussed inthe introduction part unpredictable forces or interferencefactors can be introduced into the forms of feedback controlvariables which can contribute significantly to the biologicalsystems by affecting their dynamics and stability Moreoverstructurally modifying existing systems by incorporatingvariables for defining feedback controls is appropriate forconsidering the unpredictable forces or interference factorsin an ecosystem So in the present study we consider thefollowing one-dimensional discrete space-time Lot-kandashVolterra model with periodic boundary conditions andfeedback control

xn+1i x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857 + D1nabla

2x

ni

yn+1i y

ni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni

un+11i 1 minus η1( 1113857u

n1i + e1x

ni

un+12i 1 minus η2( 1113857u

n2i + e2y

ni

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(7)

with the periodic boundary conditions

xn0 x

nm x

n1 x

nm+1

yn0 y

nm y

n1 y

nm+1

⎧⎨

⎩ (8)

where i isin 1 2 m [1 m] and m n isin Z+ is positiveinteger r1 r2 a12a21 η1 η2 e1 e2 are positive constants andD1 and D2 are diffusion parameters

nabla2xni x

ni+1 minus 2x

ni + x

niminus1

nabla2yni y

ni+1 minus 2y

ni + y

niminus1

(9)

To the best of our knowledge no work on globalasymptomatic stability of the positive equilibrium of systems(7) and (8) has been done yet

By simple computation systems (7) and (8) have apositive equilibrium

Elowast

xlowast ylowast ulowast1 ulowast2( 1113857 (10)

where

xlowast

η1r1 η2 + e2d2( 1113857 minus η1η2a12r2

η2 + e2d2( 1113857 η1 + e1d1( 1113857 + a12a21η1η2

ylowast

η2r2 η1 + e1d1( 1113857 + η1η2a21r1

η2 + e2d2( 1113857 η1 + e1d21( 1113857 + a12a21η1η2

ulowast1

e1

η1xlowast

ulowast2

e2

η2ylowast

(11)

If r1(η2 + e2d2)gt η2a12r2 the equilibrium is positiveTo discuss the global asymptomatic stability of the

unique positive equilibrium the following assumptions andpreparations are essential

From the view point of biology we only need to discussthe positive solution of system (7) So it is assumed that theinitial conditions of (8) are of the form

x0i gt 0 u

0i gt 0 i 1 2 m (12)

For our purpose we first introduce the following lemmawhich can be obtained easily by comparison theorem ofdifference equation

Lemma 1 (see [29]) Let x(n) be a nonnegative solution ofinequality

x(n + 1)lex(n)exp α minus βx(n)1113864 1113865 n isin Z (13)

with x(0)gt 0 and α βgt 0 then

limn⟶infin

supx(n)leαβ

(14)

Lemma 2 (see [30]) Any solution x(n) of system

x(n + 1) x(n)(1 minus c) + ω(n) n isin Z (15)

with x(0)gt 0 satisfies

limn⟶infin

supx(n)leup n isin Zω(n)

c (16)

where ω(n) is a nonnegative bounded sequence of realnumbers and 0lt clt 1

Applying the above lemmas we can obtain the followingresult

Theorem 1 -e solution of (7) with initial condition (8) isdefined and remains nonnegative and bounded iferj minus 2Dj ge 0 and ηj lt 1 j 1 2 hold

Proof From the first equation of system (7) we get

xn+1i x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857 + D1nabla

2x

ni

xni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857 minus 2D1( 1113857

+ D1 xni+1 + x

niminus1( 1113857

(17)

Complexity 3

from which it is true that xni ge 0 holds for all n with

x0i gt 0 u0

i gt 0 i 1 2 m if er1 minus 2D1 ge 0 and appropriateparameters a12 d1 are selected

Similarly from the second equation of system (7) we getthat yn

i ge 0 holds for all n with x0i gt 0 u0

i gt 0 i 1 2 mif er2 minus 2D2 ge 0 and appropriate parameters a21 d2 areselected

If ηj le 1 j 1 2 unji ge 0 can also hold by means of the

third and fourth equations of system (7)Next we will show the boundedness of the solutions

1113944

m

i1x

n+1i 1113944

m

i1x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857 + D1nabla

2x

ni1113872 1113873

1113944

m

i1x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857

le 1113944m

i1x

ni exp r1 minus x

ni( 1113857

(18)

From Lemma 1 we can obtain

limn⟶infin

sup1113944m

i1x

ni le 1113944

m

i1r1 mr1 (19)

Similarly we can also obtain

1113944

m

i1y

n+1i 1113944

m

i1y

ni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni1113872 1113873

1113944m

i1y

ni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857

le 1113944m

i1y

ni exp r2 + a21Mx minus y

ni( 1113857

(20)

where Mx supnisinZxni -en

limn⟶infin

sup1113944m

i1y

ni le 1113944

m

i1r2 + a21Mx( 1113857 m r2 + a21Mx( 1113857

(21)

From Lemma 2 by means of (20) and (21) and η1 η2 le 1we can obtain

limn⟶infin

sup un1i le

e1Mx

η1 (22)

limn⟶infin

sup un2i le

e2My

η2 (23)

where My supnisinZyni -e proof is finished

3 Global Stability

In this section we devote ourselves to studying the globalasymptotic stability of the unique positive equilibrium ElowastBy using global Lyapunov function we derive the sufficient

conditions under which the positive equilibrium is globallyasymptotically stable

Denote

H(1) erj minus 2Dj ge 0 j 1 2

H(2)djej

2 1 minus ηj1113872 1113873le 1 ηj le 1 j 1 2

(24)

Assume xni1113864 1113865

nisinZ+

iisin[1m] yni1113864 1113865

nisinZ+

iisin[1m] are positive solutions ofsystems (7) and (8) we can establish the following result

Theorem 2 Assume H(1) and H(2) hold the positiveequilibrium Elowast of systems (7) and (8) is globally asymptot-ically stable

Proof Let

Vn1 1113944

m

i1x

ni minus xlowast

minus xlowastln

xni

xlowast1113888 1113889 (25)

-en we can obtain

ΔVn1 V

n+11 minus V

n1

1113944m

i1x

n+1i minus x

ni minus xlowastln

xn+1i

xni

1113888 1113889

1113944

m

i1x

n+1i minus x

ni minus xlowastx

n+1i minus x

ni

xni

1113888 1113889 + o(1)

1113944m

i1x

n+1i minus x

ni1113872 1113873 1 minus

xlowast

xni

1113888 1113889 + o(1)

1113944m

i11 minus

xlowast

xni

1113888 1113889 xni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857(

+ D1nabla2x

ni minus x

ni 1113873 + o(1)

1113944m

i11 minus

xlowast

xni

1113888 1113889 xni 1 minus x

ni minus xlowast

( 1113857 minus a12 yni minus ylowast

( 1113857((

minus d1 un1i minus u

lowast1( 11138571113857 + D1nabla

2x

ni minus x

ni 1113873

minus D1 1113944

m

i1xlowast x

ni+1

xni

+x

niminus1

xni

minus 21113888 1113889 + o(1) + o ρ1( 1113857

minus 1113944m

i1xi minus x

lowasti( 1113857

2minus a12 1113944

m

i1x

ni minus xlowast

( 1113857 yni minus ylowast

( 1113857

minus d1 1113944

m

i1x

ni minus xlowast

( 1113857 un1i minus u

lowast1( 1113857 minus D1x

lowast1113944

mminus1

i1

middot

xn

i+1xn

i

1113971

minus

xn

iminus1xn

i

1113971

⎛⎝ ⎞⎠

2

minus D1xlowast

xn

m

xn1

1113971

minus

xn1

xnm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ1( 1113857

(26)

where ρ1

(xni minus xlowast)2 + (yn

i minus ylowast)2 + (un1i minus ulowast1 )2

1113969

4 Complexity

Let

Vn2

a12

a211113944

m

i1y

ni minus ylowast

minus ylowastln

yni

ylowast1113888 1113889 (27)

-en we can obtain

ΔVn2 V

n+12 minus V

n2

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowast ln

yn+1i

yni

1113888 1113889

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowasty

n+1i minus y

ni

yni

1113888 1113889 + o(1)

a12

a211113944

m

i1y

n+1i minus y

ni1113872 1113873 1 minus

ylowast

yni

1113888 1113889 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni minus y

ni1113872 1113873 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni 1 minus y

ni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus y

ni + D2nabla

2y

ni1113872 1113873 + o(1) + o ρ2( 1113857

a12

a211113944

m

i1y

ni minus ylowast

( 1113857 minus yni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus D2

a12

a211113944

m

i1ylowast y

ni+1

yni

+y

niminus1

yni

minus 21113888 1113889 + o(1) + o ρ2( 1113857

minusa12

a211113944

m

i1yi minus y

lowasti( 1113857

2+ 1113944

m

i1a12 x

ni minus xlowast

( 1113857 yni minus ylowast

( 1113857 minusa12d2

a211113944

m

i1y

ni minus ylowast

( 1113857 un2i minus u

lowast2( 1113857 minus D2

a12

a211113944

mminus1

i1

middot

yn

i+1yn

i

1113971

minus

yn

iminus1yn

i

1113971

⎛⎝ ⎞⎠

2

minus D2a12

a21

yn

m

yn1

1113971

minus

yn1

ynm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ2( 1113857

(28)

ΔVn2 V

n+12 minus V

n2

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowast ln

yn+1i

yni

1113888 1113889

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowasty

n+1i minus y

ni

yni

1113888 1113889 + o(1)

a12

a211113944

m

i1y

n+1i minus y

ni1113872 1113873 1 minus

ylowast

yni

1113888 1113889 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni minus y

ni1113872 1113873 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni 1 minus y

ni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus y

ni + D2nabla

2y

ni1113872 1113873 + o(1) + o ρ2( 1113857

a12

a211113944

m

i1y

ni minus ylowast

( 1113857 minus yni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus D2

a12

a211113944

m

i1ylowast y

ni+1

yni

+y

niminus1

yni

minus 21113888 1113889 + o(1) + o ρ2( 1113857

minusa12

a211113944

m

i1yi minus y

lowasti( 1113857

2+ a12 1113944

m

i1x

ni minus xlowast

( 1113857 yni minus ylowast

( 1113857 minusa12d2

a211113944

m

i1y

ni minus ylowast

( 1113857 un2i minus u

lowast2( 1113857 minus D2

a12

a211113944

mminus1

i1

yn

i+1yn

i

1113971

minus

yn

iminus1yn

i

1113971

⎛⎝ ⎞⎠

2

minus D2a12

a21

yn

m

yn1

1113971

minus

yn1

ynm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ2( 1113857

(29)

Complexity 5

where ρ2

(xni minus xlowast)2 + (yn

i minus ylowast)2 + (un2i minus ulowast2 )2

1113969

Let

Vn3

d1

2 1 minus η1( 1113857e1u

n1i minus u

lowast1( 1113857

2 (30)

-en we can obtain

ΔVn3 V

n+13 minus V

n3

d1

2 1 minus η1( 1113857e11113944

m

i1u

n+11i minus u

n1i1113872 1113873 u

n+11i + u

n1i minus 2u

lowast11113872 1113873

d1

2 1 minus η1( 1113857e11113944

m

i1minusη1u

n1i + e1x

ni( 1113857 2 minus η1( 1113857u

n1i(

+ e1xni minus 2u

lowast1 1113857

d1

2 1 minus η1( 1113857e11113944

m

i1minusη1 u

n1i minus u

lowast1( 1113857 + e1 x

ni minus xlowast

( 1113857( 1113857

middot 2 minus η1( 1113857 un1i minus u

lowast1( 1113857 + e1 x

ni minus xlowast

( 1113857( 1113857

minusd1η1 2 minus η1( 1113857

2 1 minus η1( 1113857e11113944

m

i1u

n1i minus u

lowast1( 1113857

2

+d1e1

2 1 minus η1( 11138571113944

m

i1x

ni minus xlowast

( 11138572

+ d1 xni minus xlowast

( 1113857 un1i minus u

lowast1( 1113857

(31)

Let

Vn4

d2a12

2 1 minus η2( 1113857e2a21u

n2i minus u

lowast2( 1113857

2 (32)

-en we can obtain

ΔVn4 V

n+14 minus V

n4

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1u

n+12i minus u

n2i1113872 1113873 u

n+12i + u

n2i minus 2u

lowast21113872 1113873

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1minusη2u

n2i + e2y

ni( 1113857 minusη2( 1113857u

n2i(

+ e2yni minus 2u

lowast2 1113857

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1minusη2 u

n2i minus u

lowast2( 1113857(

+ e2 yni minus ylowast

( 11138571113857 2 minus η2( 1113857 un2i minus u

lowast2( 1113857 + e2 y

ni minus ylowast

( 1113857( 1113857

minusd2a12η2 2 minus η2( 1113857

2 1 minus η2( 1113857e2a211113944

m

i1u

n2i minus u

lowast2( 1113857

2

+d2a12e2

2 1 minus η2( 1113857a211113944

m

i1y

ni minus ylowast

( 11138572

+a12d2

a21

middot yni minus ylowast

( 1113857 un2i minus u

lowast2( 1113857

(33)

Let

Vn

Vn1 + V

n2 + V

n3 + V

n4 (34)

-en

ΔVn V

n+1minus V

n

le minus1 +d1e1

2 1 minus η1( 11138571113888 1113889 1113944

m

i1x

ni minus xlowast

( 11138572

+ minusa12

a21+

d2a12e2

2 1 minus η2( 1113857a211113888 1113889 1113944

m

i1y

ni minus ylowast

( 11138572

+d1η1 η1 minus 2( 1113857

2 1 minus η1( 1113857e11113944

m

i1u

n1i minus u

lowast1( 1113857

2

+d2a12η2 η2 minus 2( 1113857

2 1 minus η2( 1113857e2a211113944

m

i1u

n2i minus u

lowast2( 1113857

2

minus D1 1113944

mminus1

i1

xn

i+1xn

i

1113971

minus

xn

iminus1xn

i

1113971

⎛⎝ ⎞⎠

2

minus D1

xn

m

xn1

1113971

minus

xn1

xnm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ1( 1113857

minus D2a12

a211113944

mminus1

i1

yn

i+1yn

i

1113971

minus

yn

iminus1yn

i

1113971

⎛⎝ ⎞⎠

2

minus D2a12

a21

yn

m

yn1

1113971

minus

yn1

ynm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ2( 1113857

(35)

If (d1e12(1 minus η1))le 1 (d2e22(1 minus η2))le 1 (d1η1(η1 minus

2)2(1 minus η1)e1)le 0 and (d2a12η2(η2 minus 2)2(1 minus η2)e2a21)le 0or(d1e12(1 minus η1))le 1 (d2e22(1 minus η2))le 1 η1 lt 1 andη2 lt 1hold ΔVn le 0 -e proof is completed

4 Example and Numerical Simulations

In the following example we will show the feasibility of ourmain results and discuss the effects of feedback controlsTake i 2 in the system we obtain a model with feedbackcontrols as follows

xn+1i x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857 + D1nabla

2x

ni

yn+1i y

ni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni

un+11i 1 minus η1( 1113857u

n1i + e1x

ni

un+12i 1 minus η2( 1113857u

n2i + e2y

ni

i 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)

with the periodic boundary conditions

6 Complexity

xn0 x

n2

xn1 x

n3

yn0 y

n2

yn1 y

n3

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)

To illustrate our purposes the parameter values arechosen as follows (the choice of parameter values is hypo-thetical with appropriate units and not based on data) r1

12 r2 1a12 08a21 03d1 1d2 1D1 03D2 04

e1 07 e2 06η1 05 andη2 06 then there is only aunique positive equilibrium Elowast(xlowast1 xlowast2 ylowast1 ylowast2 ulowast11u

lowast12 ulowast21

ulowast22) (03181 03181 05487 05487 04453 0445305487

05487) It is easy to see that the conditions in -eorem 2 areverified Dynamic behaviors of systems (36) and (37) with theinitial conditions are shown in Figure 1 and three sets ofdifferent initial conditions are listed in Table 1-e simulations

can illustrate the fact that the positive equilibrium is globallyasymptotically stable

To explore clearly the dynamical behavior of systems(36) and (37) we investigate the effect of diffusion pa-rameter d2 by keeping other parameters of the systemfixed Figure 2 exhibits in detail an interesting situation when

050

045

040

035

030

025

0 10 20 30 40n

50 60

xn1

un11

Solu

tions

xn a

nd u

n 1

(a)

050

045

040

035

030

025

0 10 20 30 40n

50 60

xn2

un12

Solu

tions

xn a

nd u

n 1

(b)

0 10 20 30 40n

50 60

Solu

tions

yn an

d un 2

06

05

04

03

02yn1

un21

(c)

0 10 20 30 40n

50 60

Solu

tions

yn an

d un 2

06

05

04

03

02yn2

un22

(d)

Figure 1 Dynamic behaviors of systems (36) and (37) with three sets of different initial conditions when r1 12

r2 1 a12 08 a21 03 e1 07 e2 06 η1 05 η2 06 d1 1 d2 1 D1 03 andD2 04

Table 1 Different initial values for xn1 xn

2 yn1 yn

2 un11 un

12 un21 un

22

x01 x0

2 y01 y0

2 u011 u0

12 u021 u0

22

1 022 035 027 022 040 046 045 0522 033 026 023 028 047 042 048 0433 025 023 021 030 043 041 053 056 050

055

060

045

040

035

0 10 20 30 40n

50 60

xn1

un11

Solu

tions

xn a

nd u

n 1

Figure 2 Dynamic behaviors of xn1 and un

11 for systems (36) and(37) with initial conditions (036 034 036 034 028 031

028 031 059 052 045 054) when d2 08

Complexity 7

d2 08(r1 12 r2 1 a12 08 a21 03 d1 1 D1

03 D2 04 e1 07 e2 06 η1 05 η2 06) in whichthe solutions converge faster than Figure 1 With the increaseof d2 the solutions converge more slowly as depicted inFigure 3 For the sake of convenience only the dynamicalbehavior of xn

1 and un11 with a group of initial values is shown

in Figures 2 and 3 -e simulation results of adjustingfeedback control coefficients d1 e1 and e2 are omitted

5 Conclusions and Discussion

-is paper investigates the global asymptomatic stability ofthe unique positive equilibrium of a discrete diffusion modelwith the periodic boundary conditions and feedback control-e condition to ensure the nonnegativity and boundednessof the solutions of the discrete model is discussed and theglobally asymptotical stability of the positive equilibrium isproved -rough comparing numerical simulations wenotice that when we improve the feedback control coeffi-cients the solutions will converge more slowly It followsthat we can adjust the rate of convergence by choosingsuitable values of feedback control variables Such work mayalso be applied to other discrete diffusion models

It should be noted that there is only a basic conditionobtained to guarantee the existence of positive solutionJudgement sentence xn

i yni un

1i un2i ge 0 should be added into

the simulation programs under the conditions oferj minus 2Dj ge 0 and ηj le 1 j 1 2 Further improvements areneeded

In this study all of the coefficients of the model systemare constant in many situations they can be assumed to benonconstant bounded nonnegative sequences such as pe-riodic positive sequences which can reflect the seasonalfluctuations [19] On the other hand time delays can have agreat influence on species populations It is worthy toconsider the nonautonomous space-time discrete Lot-kandashVolterra system with feedback control

Different control schemes such as switching controlconstraint control and sliding control can be applied to thesystem models Many interesting results can be obtained

Based on the backstepping recursive technique a neuralnetwork-based finite-time control strategy is proposed for aclass of non-strict-feedback nonlinear systems [31] and anevent-triggered robust fuzzy adaptive prescribed perfor-mance finite-time control strategy is proposed for a class ofstrict-feedback nonlinear systems with external disturbances[32] How to apply these control schemes on the discretediffusion models may be worth considering

It is well known that noise disturbance is unavoidable inreal systems and it has an important effect on the stability ofsystems Also noise can be used to stabilize a given unstablesystem or to make a system even more stable when thesystem is already stable which reveals that the stochasticfeedback control can stabilize and destabilize the deter-ministic systems [33 34]-erefore it will be interesting andchallenging to investigate stabilization or destabilization ofnonlinear discrete space-time systems by stochastic feedbackcontrol in our future work

Data Availability

No data were used to support this study

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was supported by the Applied Study Program(grant nos 171006901B 60204 and WH18012)

References

[1] J Xu and Z Teng ldquoPermanence for a nonautonomous dis-crete single-species system with delays and feedback controlrdquoApplied Mathematics Letters vol 23 no 9 pp 949ndash954 2010

[2] O S Board An Ecosystem Services Approach to Assessing theImpacts of the Deepwater Horizon Oil Spill in the Gulf ofMexico National Academies Press Washington DC USA2013

[3] V Tiwari J P Tripathi R K Upadhyay Y-PWu J-SWangand G-Q Sun ldquoPredator-prey interaction system with mu-tually interfering predator role of feedback controlrdquo AppliedMathematical Modelling vol 87 pp 222ndash244 2020

[4] X Li X Yang and T Huang ldquoPersistence of delayed co-operative models impulsive control methodrdquo AppliedMathematics and Computation vol 342 pp 130ndash146 2019

[5] T Luo ldquoStabilization of multi-group models with multipledispersal and stochastic perturbation via feedback controlbased on discrete-time state observationsrdquo Applied Mathe-matics and Computation vol 354 pp 396ndash410 2019

[6] W Qin X Tan M Tosato and X Liu ldquo-reshold controlstrategy for a non-smooth Filippov ecosystem with groupdefenserdquo Applied Mathematics and Computation vol 362Article ID 124532 2019

[7] J Xu Z Teng and H Jiang ldquoPermanence and globalattractivity for discrete nonautonomous two-species lotka-volterra competitive system with delays and feedback con-trolsrdquo Periodica Mathematica Hungarica vol 63 no 1pp 19ndash45 2011

030

028

026

024

022

020

018

0 10 20 30 40 50 60 70 80n

xn1

un11

Solu

tions

xn a

nd u

n 1

Figure 3 Dynamic behaviors of xn1 and un

11 for systems (36) and(37) with initial conditions (016 021 016 021 021 026 021 026 025 022 045 050) when d2 12

8 Complexity

[8] C Walter ldquoStability of controlled biological systemsrdquo Journalof -eoretical Biology vol 23 no 1 pp 23ndash38 1969

[9] P Wang Z Zhao and W Li ldquoGlobal stability analysis fordiscrete-time coupled systems with both time delay andmultiple dispersal and its applicationrdquo Neurocomputingvol 244 pp 42ndash52 2017

[10] Y Kuang ldquoGlobal stability in delay differential systemswithout dominating instantaneous negative feedbacksrdquoJournal of Differential Equations vol 119 no 2 pp 503ndash5321995

[11] Y Yan and E-O N Ekaka-a ldquoStabilizing a mathematicalmodel of population systemrdquo Journal of the Franklin Institutevol 348 no 10 pp 2744ndash2758 2011

[12] Y Muroya ldquoGlobal stability of a delayed nonlinear lotka-volterra system with feedback controls and patch structurerdquoApplied Mathematics and Computation vol 239 pp 60ndash732014

[13] J L Liu and W C Zhao ldquoDynamic analysis of stochasticlotkandashvolterra predator-prey model with discrete delays andfeedback controlrdquo Complexity vol 2019 p 15 Article ID4873290 2019

[14] Q Zhu and H Wang ldquoOutput feedback stabilization ofstochastic feedforward systems with unknown control coef-ficients and unknown output functionrdquo Automatica vol 87pp 166ndash175 2018

[15] Q Zhu ldquoStabilization of stochastic nonlinear delay systemswith exogenous disturbances and the event-triggered feed-back controlrdquo IEEE Transactions on Automatic Controlvol 64 no 9 pp 3764ndash3771 2019

[16] K Kiss and E Gyurkovics ldquoLMI approach to global stabilityanalysis of stochastic delayed lotka-volterra modelsrdquo AppliedMathematics Letters vol 104 pp 106227 1ndash6 2020

[17] X Chen and C Fengde ldquoStable periodic solution of a discreteperiodic lotka-volterra competition system with a feedbackcontrolrdquo Applied Mathematics and Computation vol 181no 2 pp 1446ndash1454 2006

[18] C Niu and X Chen ldquoAlmost periodic sequence solutions of adiscrete lotka-volterra competitive system with feedbackcontrolrdquo Nonlinear Analysis Real World Applications vol 10no 5 pp 3152ndash3161 2009

[19] X Liao S Zhou and Y Chen ldquoPermanence and globalstability in a discrete n-species competition system withfeedback controlsrdquo Nonlinear Analysis Real World Appli-cations vol 9 no 4 pp 1661ndash1671 2008

[20] A M Turing ldquo-e chemical basis of morphogenesisrdquo Philo-Sophical Transactions of the Royal Society B Biological Sci-ences vol 237 no 641 pp 37ndash72 1952

[21] K Gopalsamy and P X Weng ldquoFeedback regulation of lo-gistic growthrdquo International Journal of Mathematics andMathematical Sciences vol 16 no 1 pp 177ndash192 1993

[22] L Xu S S Lou P Q Xu and G Zhang ldquoFeedback controland parameter invasion for a discrete competitive lot-kandashvolterra systemrdquoDiscrete Dynamics in Nature and Societyvol 2018 p 8 Article ID 7473208 2018

[23] X W Li and H J Gao ldquoRobust finite frequency Hinfinfilteringfor uncertain 2-D roesser systemsrdquo Automatica vol 48pp 1163ndash1170 2012

[24] X W Li H J Gao and C H Wang ldquoGeneralized kal-manndashyakubovichndashpopov lemma for 2-D FM LSS modelrdquoIEEE Transaction on Automatic Control vol 57 no 12pp 3090ndash3193 2012

[25] J Zhou Y Yang and T Zhang ldquoGlobal dynamics of a re-action-diffusion waterborne pathogen model with general

incidence raterdquo Journal of Mathematical Analysis and Ap-plications vol 466 no 1 pp 835ndash859 2018

[26] Y Yang and J Zhou ldquoGlobal stability of a discrete virusdynamics model with diffusion and general infection func-tionrdquo International Journal of Computer Mathematics vol 96no 9 pp 1752ndash1762 2019

[27] P Liu and S N Elaydi ldquoDiscrete competitive and cooperativemodels of lotka-volterra typerdquo Journal of ComputationalAnalysis and Applications vol 3 no 1 pp 53ndash73 2001

[28] L L Meng and Y T Han ldquoBifurcation chaos and patternformation for the discrete predator-prey reaction-diffusionmodelrdquo Discrete Dynamics in Nature and Society vol 2019p 9 Article ID 9592878 2019

[29] R M May and G F Oster ldquoBifurcations and dynamiccomplexity in simple ecological modelsrdquo -e AmericanNaturalist vol 110 no 974 pp 573ndash599 1976

[30] Z Zhou and X Zou ldquoStable periodic solutions in a discreteperiodic logistic equationrdquo Applied Mathematics Lettersvol 16 no 2 pp 165ndash171 2003

[31] K K Sun J B Qiu H R Karimi and H J Gao ldquoA novelfinite-time control for nonstrict feedback saturated nonlinearsystems with tracking error constraintrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash12 2020

[32] K K Sun J B Qiu H R Karimi and Y L Fu ldquoEvent-triggered robust fuzzy adaptive finite-time control of non-linear systems with prescribed performancerdquo IEEE Transac-tions on Fuzzy Systems 2020

[33] X Mao ldquoAlmost sure exponential stabilization by discrete-time stochastic feedback controlrdquo IEEE Transactions onAutomatic Control vol 61 no 6 pp 1619ndash1624 2016

[34] Q Zhu and T Huang ldquoStability analysis for a class of sto-chastic delay nonlinear systems driven by G-Brownian mo-tionrdquo Systems amp Control Letters vol 140 Article ID 1046999 pages 2020

Complexity 9

from which it is true that xni ge 0 holds for all n with

x0i gt 0 u0

i gt 0 i 1 2 m if er1 minus 2D1 ge 0 and appropriateparameters a12 d1 are selected

Similarly from the second equation of system (7) we getthat yn

i ge 0 holds for all n with x0i gt 0 u0

i gt 0 i 1 2 mif er2 minus 2D2 ge 0 and appropriate parameters a21 d2 areselected

If ηj le 1 j 1 2 unji ge 0 can also hold by means of the

third and fourth equations of system (7)Next we will show the boundedness of the solutions

1113944

m

i1x

n+1i 1113944

m

i1x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857 + D1nabla

2x

ni1113872 1113873

1113944

m

i1x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857

le 1113944m

i1x

ni exp r1 minus x

ni( 1113857

(18)

From Lemma 1 we can obtain

limn⟶infin

sup1113944m

i1x

ni le 1113944

m

i1r1 mr1 (19)

Similarly we can also obtain

1113944

m

i1y

n+1i 1113944

m

i1y

ni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni1113872 1113873

1113944m

i1y

ni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857

le 1113944m

i1y

ni exp r2 + a21Mx minus y

ni( 1113857

(20)

where Mx supnisinZxni -en

limn⟶infin

sup1113944m

i1y

ni le 1113944

m

i1r2 + a21Mx( 1113857 m r2 + a21Mx( 1113857

(21)

From Lemma 2 by means of (20) and (21) and η1 η2 le 1we can obtain

limn⟶infin

sup un1i le

e1Mx

η1 (22)

limn⟶infin

sup un2i le

e2My

η2 (23)

where My supnisinZyni -e proof is finished

3 Global Stability

In this section we devote ourselves to studying the globalasymptotic stability of the unique positive equilibrium ElowastBy using global Lyapunov function we derive the sufficient

conditions under which the positive equilibrium is globallyasymptotically stable

Denote

H(1) erj minus 2Dj ge 0 j 1 2

H(2)djej

2 1 minus ηj1113872 1113873le 1 ηj le 1 j 1 2

(24)

Assume xni1113864 1113865

nisinZ+

iisin[1m] yni1113864 1113865

nisinZ+

iisin[1m] are positive solutions ofsystems (7) and (8) we can establish the following result

Theorem 2 Assume H(1) and H(2) hold the positiveequilibrium Elowast of systems (7) and (8) is globally asymptot-ically stable

Proof Let

Vn1 1113944

m

i1x

ni minus xlowast

minus xlowastln

xni

xlowast1113888 1113889 (25)

-en we can obtain

ΔVn1 V

n+11 minus V

n1

1113944m

i1x

n+1i minus x

ni minus xlowastln

xn+1i

xni

1113888 1113889

1113944

m

i1x

n+1i minus x

ni minus xlowastx

n+1i minus x

ni

xni

1113888 1113889 + o(1)

1113944m

i1x

n+1i minus x

ni1113872 1113873 1 minus

xlowast

xni

1113888 1113889 + o(1)

1113944m

i11 minus

xlowast

xni

1113888 1113889 xni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857(

+ D1nabla2x

ni minus x

ni 1113873 + o(1)

1113944m

i11 minus

xlowast

xni

1113888 1113889 xni 1 minus x

ni minus xlowast

( 1113857 minus a12 yni minus ylowast

( 1113857((

minus d1 un1i minus u

lowast1( 11138571113857 + D1nabla

2x

ni minus x

ni 1113873

minus D1 1113944

m

i1xlowast x

ni+1

xni

+x

niminus1

xni

minus 21113888 1113889 + o(1) + o ρ1( 1113857

minus 1113944m

i1xi minus x

lowasti( 1113857

2minus a12 1113944

m

i1x

ni minus xlowast

( 1113857 yni minus ylowast

( 1113857

minus d1 1113944

m

i1x

ni minus xlowast

( 1113857 un1i minus u

lowast1( 1113857 minus D1x

lowast1113944

mminus1

i1

middot

xn

i+1xn

i

1113971

minus

xn

iminus1xn

i

1113971

⎛⎝ ⎞⎠

2

minus D1xlowast

xn

m

xn1

1113971

minus

xn1

xnm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ1( 1113857

(26)

where ρ1

(xni minus xlowast)2 + (yn

i minus ylowast)2 + (un1i minus ulowast1 )2

1113969

4 Complexity

Let

Vn2

a12

a211113944

m

i1y

ni minus ylowast

minus ylowastln

yni

ylowast1113888 1113889 (27)

-en we can obtain

ΔVn2 V

n+12 minus V

n2

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowast ln

yn+1i

yni

1113888 1113889

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowasty

n+1i minus y

ni

yni

1113888 1113889 + o(1)

a12

a211113944

m

i1y

n+1i minus y

ni1113872 1113873 1 minus

ylowast

yni

1113888 1113889 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni minus y

ni1113872 1113873 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni 1 minus y

ni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus y

ni + D2nabla

2y

ni1113872 1113873 + o(1) + o ρ2( 1113857

a12

a211113944

m

i1y

ni minus ylowast

( 1113857 minus yni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus D2

a12

a211113944

m

i1ylowast y

ni+1

yni

+y

niminus1

yni

minus 21113888 1113889 + o(1) + o ρ2( 1113857

minusa12

a211113944

m

i1yi minus y

lowasti( 1113857

2+ 1113944

m

i1a12 x

ni minus xlowast

( 1113857 yni minus ylowast

( 1113857 minusa12d2

a211113944

m

i1y

ni minus ylowast

( 1113857 un2i minus u

lowast2( 1113857 minus D2

a12

a211113944

mminus1

i1

middot

yn

i+1yn

i

1113971

minus

yn

iminus1yn

i

1113971

⎛⎝ ⎞⎠

2

minus D2a12

a21

yn

m

yn1

1113971

minus

yn1

ynm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ2( 1113857

(28)

ΔVn2 V

n+12 minus V

n2

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowast ln

yn+1i

yni

1113888 1113889

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowasty

n+1i minus y

ni

yni

1113888 1113889 + o(1)

a12

a211113944

m

i1y

n+1i minus y

ni1113872 1113873 1 minus

ylowast

yni

1113888 1113889 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni minus y

ni1113872 1113873 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni 1 minus y

ni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus y

ni + D2nabla

2y

ni1113872 1113873 + o(1) + o ρ2( 1113857

a12

a211113944

m

i1y

ni minus ylowast

( 1113857 minus yni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus D2

a12

a211113944

m

i1ylowast y

ni+1

yni

+y

niminus1

yni

minus 21113888 1113889 + o(1) + o ρ2( 1113857

minusa12

a211113944

m

i1yi minus y

lowasti( 1113857

2+ a12 1113944

m

i1x

ni minus xlowast

( 1113857 yni minus ylowast

( 1113857 minusa12d2

a211113944

m

i1y

ni minus ylowast

( 1113857 un2i minus u

lowast2( 1113857 minus D2

a12

a211113944

mminus1

i1

yn

i+1yn

i

1113971

minus

yn

iminus1yn

i

1113971

⎛⎝ ⎞⎠

2

minus D2a12

a21

yn

m

yn1

1113971

minus

yn1

ynm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ2( 1113857

(29)

Complexity 5

where ρ2

(xni minus xlowast)2 + (yn

i minus ylowast)2 + (un2i minus ulowast2 )2

1113969

Let

Vn3

d1

2 1 minus η1( 1113857e1u

n1i minus u

lowast1( 1113857

2 (30)

-en we can obtain

ΔVn3 V

n+13 minus V

n3

d1

2 1 minus η1( 1113857e11113944

m

i1u

n+11i minus u

n1i1113872 1113873 u

n+11i + u

n1i minus 2u

lowast11113872 1113873

d1

2 1 minus η1( 1113857e11113944

m

i1minusη1u

n1i + e1x

ni( 1113857 2 minus η1( 1113857u

n1i(

+ e1xni minus 2u

lowast1 1113857

d1

2 1 minus η1( 1113857e11113944

m

i1minusη1 u

n1i minus u

lowast1( 1113857 + e1 x

ni minus xlowast

( 1113857( 1113857

middot 2 minus η1( 1113857 un1i minus u

lowast1( 1113857 + e1 x

ni minus xlowast

( 1113857( 1113857

minusd1η1 2 minus η1( 1113857

2 1 minus η1( 1113857e11113944

m

i1u

n1i minus u

lowast1( 1113857

2

+d1e1

2 1 minus η1( 11138571113944

m

i1x

ni minus xlowast

( 11138572

+ d1 xni minus xlowast

( 1113857 un1i minus u

lowast1( 1113857

(31)

Let

Vn4

d2a12

2 1 minus η2( 1113857e2a21u

n2i minus u

lowast2( 1113857

2 (32)

-en we can obtain

ΔVn4 V

n+14 minus V

n4

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1u

n+12i minus u

n2i1113872 1113873 u

n+12i + u

n2i minus 2u

lowast21113872 1113873

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1minusη2u

n2i + e2y

ni( 1113857 minusη2( 1113857u

n2i(

+ e2yni minus 2u

lowast2 1113857

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1minusη2 u

n2i minus u

lowast2( 1113857(

+ e2 yni minus ylowast

( 11138571113857 2 minus η2( 1113857 un2i minus u

lowast2( 1113857 + e2 y

ni minus ylowast

( 1113857( 1113857

minusd2a12η2 2 minus η2( 1113857

2 1 minus η2( 1113857e2a211113944

m

i1u

n2i minus u

lowast2( 1113857

2

+d2a12e2

2 1 minus η2( 1113857a211113944

m

i1y

ni minus ylowast

( 11138572

+a12d2

a21

middot yni minus ylowast

( 1113857 un2i minus u

lowast2( 1113857

(33)

Let

Vn

Vn1 + V

n2 + V

n3 + V

n4 (34)

-en

ΔVn V

n+1minus V

n

le minus1 +d1e1

2 1 minus η1( 11138571113888 1113889 1113944

m

i1x

ni minus xlowast

( 11138572

+ minusa12

a21+

d2a12e2

2 1 minus η2( 1113857a211113888 1113889 1113944

m

i1y

ni minus ylowast

( 11138572

+d1η1 η1 minus 2( 1113857

2 1 minus η1( 1113857e11113944

m

i1u

n1i minus u

lowast1( 1113857

2

+d2a12η2 η2 minus 2( 1113857

2 1 minus η2( 1113857e2a211113944

m

i1u

n2i minus u

lowast2( 1113857

2

minus D1 1113944

mminus1

i1

xn

i+1xn

i

1113971

minus

xn

iminus1xn

i

1113971

⎛⎝ ⎞⎠

2

minus D1

xn

m

xn1

1113971

minus

xn1

xnm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ1( 1113857

minus D2a12

a211113944

mminus1

i1

yn

i+1yn

i

1113971

minus

yn

iminus1yn

i

1113971

⎛⎝ ⎞⎠

2

minus D2a12

a21

yn

m

yn1

1113971

minus

yn1

ynm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ2( 1113857

(35)

If (d1e12(1 minus η1))le 1 (d2e22(1 minus η2))le 1 (d1η1(η1 minus

2)2(1 minus η1)e1)le 0 and (d2a12η2(η2 minus 2)2(1 minus η2)e2a21)le 0or(d1e12(1 minus η1))le 1 (d2e22(1 minus η2))le 1 η1 lt 1 andη2 lt 1hold ΔVn le 0 -e proof is completed

4 Example and Numerical Simulations

In the following example we will show the feasibility of ourmain results and discuss the effects of feedback controlsTake i 2 in the system we obtain a model with feedbackcontrols as follows

xn+1i x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857 + D1nabla

2x

ni

yn+1i y

ni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni

un+11i 1 minus η1( 1113857u

n1i + e1x

ni

un+12i 1 minus η2( 1113857u

n2i + e2y

ni

i 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)

with the periodic boundary conditions

6 Complexity

xn0 x

n2

xn1 x

n3

yn0 y

n2

yn1 y

n3

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)

To illustrate our purposes the parameter values arechosen as follows (the choice of parameter values is hypo-thetical with appropriate units and not based on data) r1

12 r2 1a12 08a21 03d1 1d2 1D1 03D2 04

e1 07 e2 06η1 05 andη2 06 then there is only aunique positive equilibrium Elowast(xlowast1 xlowast2 ylowast1 ylowast2 ulowast11u

lowast12 ulowast21

ulowast22) (03181 03181 05487 05487 04453 0445305487

05487) It is easy to see that the conditions in -eorem 2 areverified Dynamic behaviors of systems (36) and (37) with theinitial conditions are shown in Figure 1 and three sets ofdifferent initial conditions are listed in Table 1-e simulations

can illustrate the fact that the positive equilibrium is globallyasymptotically stable

To explore clearly the dynamical behavior of systems(36) and (37) we investigate the effect of diffusion pa-rameter d2 by keeping other parameters of the systemfixed Figure 2 exhibits in detail an interesting situation when

050

045

040

035

030

025

0 10 20 30 40n

50 60

xn1

un11

Solu

tions

xn a

nd u

n 1

(a)

050

045

040

035

030

025

0 10 20 30 40n

50 60

xn2

un12

Solu

tions

xn a

nd u

n 1

(b)

0 10 20 30 40n

50 60

Solu

tions

yn an

d un 2

06

05

04

03

02yn1

un21

(c)

0 10 20 30 40n

50 60

Solu

tions

yn an

d un 2

06

05

04

03

02yn2

un22

(d)

Figure 1 Dynamic behaviors of systems (36) and (37) with three sets of different initial conditions when r1 12

r2 1 a12 08 a21 03 e1 07 e2 06 η1 05 η2 06 d1 1 d2 1 D1 03 andD2 04

Table 1 Different initial values for xn1 xn

2 yn1 yn

2 un11 un

12 un21 un

22

x01 x0

2 y01 y0

2 u011 u0

12 u021 u0

22

1 022 035 027 022 040 046 045 0522 033 026 023 028 047 042 048 0433 025 023 021 030 043 041 053 056 050

055

060

045

040

035

0 10 20 30 40n

50 60

xn1

un11

Solu

tions

xn a

nd u

n 1

Figure 2 Dynamic behaviors of xn1 and un

11 for systems (36) and(37) with initial conditions (036 034 036 034 028 031

028 031 059 052 045 054) when d2 08

Complexity 7

d2 08(r1 12 r2 1 a12 08 a21 03 d1 1 D1

03 D2 04 e1 07 e2 06 η1 05 η2 06) in whichthe solutions converge faster than Figure 1 With the increaseof d2 the solutions converge more slowly as depicted inFigure 3 For the sake of convenience only the dynamicalbehavior of xn

1 and un11 with a group of initial values is shown

in Figures 2 and 3 -e simulation results of adjustingfeedback control coefficients d1 e1 and e2 are omitted

5 Conclusions and Discussion

-is paper investigates the global asymptomatic stability ofthe unique positive equilibrium of a discrete diffusion modelwith the periodic boundary conditions and feedback control-e condition to ensure the nonnegativity and boundednessof the solutions of the discrete model is discussed and theglobally asymptotical stability of the positive equilibrium isproved -rough comparing numerical simulations wenotice that when we improve the feedback control coeffi-cients the solutions will converge more slowly It followsthat we can adjust the rate of convergence by choosingsuitable values of feedback control variables Such work mayalso be applied to other discrete diffusion models

It should be noted that there is only a basic conditionobtained to guarantee the existence of positive solutionJudgement sentence xn

i yni un

1i un2i ge 0 should be added into

the simulation programs under the conditions oferj minus 2Dj ge 0 and ηj le 1 j 1 2 Further improvements areneeded

In this study all of the coefficients of the model systemare constant in many situations they can be assumed to benonconstant bounded nonnegative sequences such as pe-riodic positive sequences which can reflect the seasonalfluctuations [19] On the other hand time delays can have agreat influence on species populations It is worthy toconsider the nonautonomous space-time discrete Lot-kandashVolterra system with feedback control

Different control schemes such as switching controlconstraint control and sliding control can be applied to thesystem models Many interesting results can be obtained

Based on the backstepping recursive technique a neuralnetwork-based finite-time control strategy is proposed for aclass of non-strict-feedback nonlinear systems [31] and anevent-triggered robust fuzzy adaptive prescribed perfor-mance finite-time control strategy is proposed for a class ofstrict-feedback nonlinear systems with external disturbances[32] How to apply these control schemes on the discretediffusion models may be worth considering

It is well known that noise disturbance is unavoidable inreal systems and it has an important effect on the stability ofsystems Also noise can be used to stabilize a given unstablesystem or to make a system even more stable when thesystem is already stable which reveals that the stochasticfeedback control can stabilize and destabilize the deter-ministic systems [33 34]-erefore it will be interesting andchallenging to investigate stabilization or destabilization ofnonlinear discrete space-time systems by stochastic feedbackcontrol in our future work

Data Availability

No data were used to support this study

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was supported by the Applied Study Program(grant nos 171006901B 60204 and WH18012)

References

[1] J Xu and Z Teng ldquoPermanence for a nonautonomous dis-crete single-species system with delays and feedback controlrdquoApplied Mathematics Letters vol 23 no 9 pp 949ndash954 2010

[2] O S Board An Ecosystem Services Approach to Assessing theImpacts of the Deepwater Horizon Oil Spill in the Gulf ofMexico National Academies Press Washington DC USA2013

[3] V Tiwari J P Tripathi R K Upadhyay Y-PWu J-SWangand G-Q Sun ldquoPredator-prey interaction system with mu-tually interfering predator role of feedback controlrdquo AppliedMathematical Modelling vol 87 pp 222ndash244 2020

[4] X Li X Yang and T Huang ldquoPersistence of delayed co-operative models impulsive control methodrdquo AppliedMathematics and Computation vol 342 pp 130ndash146 2019

[5] T Luo ldquoStabilization of multi-group models with multipledispersal and stochastic perturbation via feedback controlbased on discrete-time state observationsrdquo Applied Mathe-matics and Computation vol 354 pp 396ndash410 2019

[6] W Qin X Tan M Tosato and X Liu ldquo-reshold controlstrategy for a non-smooth Filippov ecosystem with groupdefenserdquo Applied Mathematics and Computation vol 362Article ID 124532 2019

[7] J Xu Z Teng and H Jiang ldquoPermanence and globalattractivity for discrete nonautonomous two-species lotka-volterra competitive system with delays and feedback con-trolsrdquo Periodica Mathematica Hungarica vol 63 no 1pp 19ndash45 2011

030

028

026

024

022

020

018

0 10 20 30 40 50 60 70 80n

xn1

un11

Solu

tions

xn a

nd u

n 1

Figure 3 Dynamic behaviors of xn1 and un

11 for systems (36) and(37) with initial conditions (016 021 016 021 021 026 021 026 025 022 045 050) when d2 12

8 Complexity

[8] C Walter ldquoStability of controlled biological systemsrdquo Journalof -eoretical Biology vol 23 no 1 pp 23ndash38 1969

[9] P Wang Z Zhao and W Li ldquoGlobal stability analysis fordiscrete-time coupled systems with both time delay andmultiple dispersal and its applicationrdquo Neurocomputingvol 244 pp 42ndash52 2017

[10] Y Kuang ldquoGlobal stability in delay differential systemswithout dominating instantaneous negative feedbacksrdquoJournal of Differential Equations vol 119 no 2 pp 503ndash5321995

[11] Y Yan and E-O N Ekaka-a ldquoStabilizing a mathematicalmodel of population systemrdquo Journal of the Franklin Institutevol 348 no 10 pp 2744ndash2758 2011

[12] Y Muroya ldquoGlobal stability of a delayed nonlinear lotka-volterra system with feedback controls and patch structurerdquoApplied Mathematics and Computation vol 239 pp 60ndash732014

[13] J L Liu and W C Zhao ldquoDynamic analysis of stochasticlotkandashvolterra predator-prey model with discrete delays andfeedback controlrdquo Complexity vol 2019 p 15 Article ID4873290 2019

[14] Q Zhu and H Wang ldquoOutput feedback stabilization ofstochastic feedforward systems with unknown control coef-ficients and unknown output functionrdquo Automatica vol 87pp 166ndash175 2018

[15] Q Zhu ldquoStabilization of stochastic nonlinear delay systemswith exogenous disturbances and the event-triggered feed-back controlrdquo IEEE Transactions on Automatic Controlvol 64 no 9 pp 3764ndash3771 2019

[16] K Kiss and E Gyurkovics ldquoLMI approach to global stabilityanalysis of stochastic delayed lotka-volterra modelsrdquo AppliedMathematics Letters vol 104 pp 106227 1ndash6 2020

[17] X Chen and C Fengde ldquoStable periodic solution of a discreteperiodic lotka-volterra competition system with a feedbackcontrolrdquo Applied Mathematics and Computation vol 181no 2 pp 1446ndash1454 2006

[18] C Niu and X Chen ldquoAlmost periodic sequence solutions of adiscrete lotka-volterra competitive system with feedbackcontrolrdquo Nonlinear Analysis Real World Applications vol 10no 5 pp 3152ndash3161 2009

[19] X Liao S Zhou and Y Chen ldquoPermanence and globalstability in a discrete n-species competition system withfeedback controlsrdquo Nonlinear Analysis Real World Appli-cations vol 9 no 4 pp 1661ndash1671 2008

[20] A M Turing ldquo-e chemical basis of morphogenesisrdquo Philo-Sophical Transactions of the Royal Society B Biological Sci-ences vol 237 no 641 pp 37ndash72 1952

[21] K Gopalsamy and P X Weng ldquoFeedback regulation of lo-gistic growthrdquo International Journal of Mathematics andMathematical Sciences vol 16 no 1 pp 177ndash192 1993

[22] L Xu S S Lou P Q Xu and G Zhang ldquoFeedback controland parameter invasion for a discrete competitive lot-kandashvolterra systemrdquoDiscrete Dynamics in Nature and Societyvol 2018 p 8 Article ID 7473208 2018

[23] X W Li and H J Gao ldquoRobust finite frequency Hinfinfilteringfor uncertain 2-D roesser systemsrdquo Automatica vol 48pp 1163ndash1170 2012

[24] X W Li H J Gao and C H Wang ldquoGeneralized kal-manndashyakubovichndashpopov lemma for 2-D FM LSS modelrdquoIEEE Transaction on Automatic Control vol 57 no 12pp 3090ndash3193 2012

[25] J Zhou Y Yang and T Zhang ldquoGlobal dynamics of a re-action-diffusion waterborne pathogen model with general

incidence raterdquo Journal of Mathematical Analysis and Ap-plications vol 466 no 1 pp 835ndash859 2018

[26] Y Yang and J Zhou ldquoGlobal stability of a discrete virusdynamics model with diffusion and general infection func-tionrdquo International Journal of Computer Mathematics vol 96no 9 pp 1752ndash1762 2019

[27] P Liu and S N Elaydi ldquoDiscrete competitive and cooperativemodels of lotka-volterra typerdquo Journal of ComputationalAnalysis and Applications vol 3 no 1 pp 53ndash73 2001

[28] L L Meng and Y T Han ldquoBifurcation chaos and patternformation for the discrete predator-prey reaction-diffusionmodelrdquo Discrete Dynamics in Nature and Society vol 2019p 9 Article ID 9592878 2019

[29] R M May and G F Oster ldquoBifurcations and dynamiccomplexity in simple ecological modelsrdquo -e AmericanNaturalist vol 110 no 974 pp 573ndash599 1976

[30] Z Zhou and X Zou ldquoStable periodic solutions in a discreteperiodic logistic equationrdquo Applied Mathematics Lettersvol 16 no 2 pp 165ndash171 2003

[31] K K Sun J B Qiu H R Karimi and H J Gao ldquoA novelfinite-time control for nonstrict feedback saturated nonlinearsystems with tracking error constraintrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash12 2020

[32] K K Sun J B Qiu H R Karimi and Y L Fu ldquoEvent-triggered robust fuzzy adaptive finite-time control of non-linear systems with prescribed performancerdquo IEEE Transac-tions on Fuzzy Systems 2020

[33] X Mao ldquoAlmost sure exponential stabilization by discrete-time stochastic feedback controlrdquo IEEE Transactions onAutomatic Control vol 61 no 6 pp 1619ndash1624 2016

[34] Q Zhu and T Huang ldquoStability analysis for a class of sto-chastic delay nonlinear systems driven by G-Brownian mo-tionrdquo Systems amp Control Letters vol 140 Article ID 1046999 pages 2020

Complexity 9

Let

Vn2

a12

a211113944

m

i1y

ni minus ylowast

minus ylowastln

yni

ylowast1113888 1113889 (27)

-en we can obtain

ΔVn2 V

n+12 minus V

n2

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowast ln

yn+1i

yni

1113888 1113889

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowasty

n+1i minus y

ni

yni

1113888 1113889 + o(1)

a12

a211113944

m

i1y

n+1i minus y

ni1113872 1113873 1 minus

ylowast

yni

1113888 1113889 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni minus y

ni1113872 1113873 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni 1 minus y

ni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus y

ni + D2nabla

2y

ni1113872 1113873 + o(1) + o ρ2( 1113857

a12

a211113944

m

i1y

ni minus ylowast

( 1113857 minus yni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus D2

a12

a211113944

m

i1ylowast y

ni+1

yni

+y

niminus1

yni

minus 21113888 1113889 + o(1) + o ρ2( 1113857

minusa12

a211113944

m

i1yi minus y

lowasti( 1113857

2+ 1113944

m

i1a12 x

ni minus xlowast

( 1113857 yni minus ylowast

( 1113857 minusa12d2

a211113944

m

i1y

ni minus ylowast

( 1113857 un2i minus u

lowast2( 1113857 minus D2

a12

a211113944

mminus1

i1

middot

yn

i+1yn

i

1113971

minus

yn

iminus1yn

i

1113971

⎛⎝ ⎞⎠

2

minus D2a12

a21

yn

m

yn1

1113971

minus

yn1

ynm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ2( 1113857

(28)

ΔVn2 V

n+12 minus V

n2

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowast ln

yn+1i

yni

1113888 1113889

a12

a211113944

m

i1y

n+1i minus y

ni minus ylowasty

n+1i minus y

ni

yni

1113888 1113889 + o(1)

a12

a211113944

m

i1y

n+1i minus y

ni1113872 1113873 1 minus

ylowast

yni

1113888 1113889 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni minus y

ni1113872 1113873 + o(1)

a12

a211113944

m

i11 minus

ylowast

yni

1113888 1113889 yni 1 minus y

ni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus y

ni + D2nabla

2y

ni1113872 1113873 + o(1) + o ρ2( 1113857

a12

a211113944

m

i1y

ni minus ylowast

( 1113857 minus yni minus ylowast

( 1113857 + a21 xni minus xlowast

( 1113857 minus d2 un2i minus u

lowast2( 1113857( 1113857 minus D2

a12

a211113944

m

i1ylowast y

ni+1

yni

+y

niminus1

yni

minus 21113888 1113889 + o(1) + o ρ2( 1113857

minusa12

a211113944

m

i1yi minus y

lowasti( 1113857

2+ a12 1113944

m

i1x

ni minus xlowast

( 1113857 yni minus ylowast

( 1113857 minusa12d2

a211113944

m

i1y

ni minus ylowast

( 1113857 un2i minus u

lowast2( 1113857 minus D2

a12

a211113944

mminus1

i1

yn

i+1yn

i

1113971

minus

yn

iminus1yn

i

1113971

⎛⎝ ⎞⎠

2

minus D2a12

a21

yn

m

yn1

1113971

minus

yn1

ynm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ2( 1113857

(29)

Complexity 5

where ρ2

(xni minus xlowast)2 + (yn

i minus ylowast)2 + (un2i minus ulowast2 )2

1113969

Let

Vn3

d1

2 1 minus η1( 1113857e1u

n1i minus u

lowast1( 1113857

2 (30)

-en we can obtain

ΔVn3 V

n+13 minus V

n3

d1

2 1 minus η1( 1113857e11113944

m

i1u

n+11i minus u

n1i1113872 1113873 u

n+11i + u

n1i minus 2u

lowast11113872 1113873

d1

2 1 minus η1( 1113857e11113944

m

i1minusη1u

n1i + e1x

ni( 1113857 2 minus η1( 1113857u

n1i(

+ e1xni minus 2u

lowast1 1113857

d1

2 1 minus η1( 1113857e11113944

m

i1minusη1 u

n1i minus u

lowast1( 1113857 + e1 x

ni minus xlowast

( 1113857( 1113857

middot 2 minus η1( 1113857 un1i minus u

lowast1( 1113857 + e1 x

ni minus xlowast

( 1113857( 1113857

minusd1η1 2 minus η1( 1113857

2 1 minus η1( 1113857e11113944

m

i1u

n1i minus u

lowast1( 1113857

2

+d1e1

2 1 minus η1( 11138571113944

m

i1x

ni minus xlowast

( 11138572

+ d1 xni minus xlowast

( 1113857 un1i minus u

lowast1( 1113857

(31)

Let

Vn4

d2a12

2 1 minus η2( 1113857e2a21u

n2i minus u

lowast2( 1113857

2 (32)

-en we can obtain

ΔVn4 V

n+14 minus V

n4

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1u

n+12i minus u

n2i1113872 1113873 u

n+12i + u

n2i minus 2u

lowast21113872 1113873

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1minusη2u

n2i + e2y

ni( 1113857 minusη2( 1113857u

n2i(

+ e2yni minus 2u

lowast2 1113857

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1minusη2 u

n2i minus u

lowast2( 1113857(

+ e2 yni minus ylowast

( 11138571113857 2 minus η2( 1113857 un2i minus u

lowast2( 1113857 + e2 y

ni minus ylowast

( 1113857( 1113857

minusd2a12η2 2 minus η2( 1113857

2 1 minus η2( 1113857e2a211113944

m

i1u

n2i minus u

lowast2( 1113857

2

+d2a12e2

2 1 minus η2( 1113857a211113944

m

i1y

ni minus ylowast

( 11138572

+a12d2

a21

middot yni minus ylowast

( 1113857 un2i minus u

lowast2( 1113857

(33)

Let

Vn

Vn1 + V

n2 + V

n3 + V

n4 (34)

-en

ΔVn V

n+1minus V

n

le minus1 +d1e1

2 1 minus η1( 11138571113888 1113889 1113944

m

i1x

ni minus xlowast

( 11138572

+ minusa12

a21+

d2a12e2

2 1 minus η2( 1113857a211113888 1113889 1113944

m

i1y

ni minus ylowast

( 11138572

+d1η1 η1 minus 2( 1113857

2 1 minus η1( 1113857e11113944

m

i1u

n1i minus u

lowast1( 1113857

2

+d2a12η2 η2 minus 2( 1113857

2 1 minus η2( 1113857e2a211113944

m

i1u

n2i minus u

lowast2( 1113857

2

minus D1 1113944

mminus1

i1

xn

i+1xn

i

1113971

minus

xn

iminus1xn

i

1113971

⎛⎝ ⎞⎠

2

minus D1

xn

m

xn1

1113971

minus

xn1

xnm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ1( 1113857

minus D2a12

a211113944

mminus1

i1

yn

i+1yn

i

1113971

minus

yn

iminus1yn

i

1113971

⎛⎝ ⎞⎠

2

minus D2a12

a21

yn

m

yn1

1113971

minus

yn1

ynm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ2( 1113857

(35)

If (d1e12(1 minus η1))le 1 (d2e22(1 minus η2))le 1 (d1η1(η1 minus

2)2(1 minus η1)e1)le 0 and (d2a12η2(η2 minus 2)2(1 minus η2)e2a21)le 0or(d1e12(1 minus η1))le 1 (d2e22(1 minus η2))le 1 η1 lt 1 andη2 lt 1hold ΔVn le 0 -e proof is completed

4 Example and Numerical Simulations

In the following example we will show the feasibility of ourmain results and discuss the effects of feedback controlsTake i 2 in the system we obtain a model with feedbackcontrols as follows

xn+1i x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857 + D1nabla

2x

ni

yn+1i y

ni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni

un+11i 1 minus η1( 1113857u

n1i + e1x

ni

un+12i 1 minus η2( 1113857u

n2i + e2y

ni

i 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)

with the periodic boundary conditions

6 Complexity

xn0 x

n2

xn1 x

n3

yn0 y

n2

yn1 y

n3

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)

To illustrate our purposes the parameter values arechosen as follows (the choice of parameter values is hypo-thetical with appropriate units and not based on data) r1

12 r2 1a12 08a21 03d1 1d2 1D1 03D2 04

e1 07 e2 06η1 05 andη2 06 then there is only aunique positive equilibrium Elowast(xlowast1 xlowast2 ylowast1 ylowast2 ulowast11u

lowast12 ulowast21

ulowast22) (03181 03181 05487 05487 04453 0445305487

05487) It is easy to see that the conditions in -eorem 2 areverified Dynamic behaviors of systems (36) and (37) with theinitial conditions are shown in Figure 1 and three sets ofdifferent initial conditions are listed in Table 1-e simulations

can illustrate the fact that the positive equilibrium is globallyasymptotically stable

To explore clearly the dynamical behavior of systems(36) and (37) we investigate the effect of diffusion pa-rameter d2 by keeping other parameters of the systemfixed Figure 2 exhibits in detail an interesting situation when

050

045

040

035

030

025

0 10 20 30 40n

50 60

xn1

un11

Solu

tions

xn a

nd u

n 1

(a)

050

045

040

035

030

025

0 10 20 30 40n

50 60

xn2

un12

Solu

tions

xn a

nd u

n 1

(b)

0 10 20 30 40n

50 60

Solu

tions

yn an

d un 2

06

05

04

03

02yn1

un21

(c)

0 10 20 30 40n

50 60

Solu

tions

yn an

d un 2

06

05

04

03

02yn2

un22

(d)

Figure 1 Dynamic behaviors of systems (36) and (37) with three sets of different initial conditions when r1 12

r2 1 a12 08 a21 03 e1 07 e2 06 η1 05 η2 06 d1 1 d2 1 D1 03 andD2 04

Table 1 Different initial values for xn1 xn

2 yn1 yn

2 un11 un

12 un21 un

22

x01 x0

2 y01 y0

2 u011 u0

12 u021 u0

22

1 022 035 027 022 040 046 045 0522 033 026 023 028 047 042 048 0433 025 023 021 030 043 041 053 056 050

055

060

045

040

035

0 10 20 30 40n

50 60

xn1

un11

Solu

tions

xn a

nd u

n 1

Figure 2 Dynamic behaviors of xn1 and un

11 for systems (36) and(37) with initial conditions (036 034 036 034 028 031

028 031 059 052 045 054) when d2 08

Complexity 7

d2 08(r1 12 r2 1 a12 08 a21 03 d1 1 D1

03 D2 04 e1 07 e2 06 η1 05 η2 06) in whichthe solutions converge faster than Figure 1 With the increaseof d2 the solutions converge more slowly as depicted inFigure 3 For the sake of convenience only the dynamicalbehavior of xn

1 and un11 with a group of initial values is shown

in Figures 2 and 3 -e simulation results of adjustingfeedback control coefficients d1 e1 and e2 are omitted

5 Conclusions and Discussion

-is paper investigates the global asymptomatic stability ofthe unique positive equilibrium of a discrete diffusion modelwith the periodic boundary conditions and feedback control-e condition to ensure the nonnegativity and boundednessof the solutions of the discrete model is discussed and theglobally asymptotical stability of the positive equilibrium isproved -rough comparing numerical simulations wenotice that when we improve the feedback control coeffi-cients the solutions will converge more slowly It followsthat we can adjust the rate of convergence by choosingsuitable values of feedback control variables Such work mayalso be applied to other discrete diffusion models

It should be noted that there is only a basic conditionobtained to guarantee the existence of positive solutionJudgement sentence xn

i yni un

1i un2i ge 0 should be added into

the simulation programs under the conditions oferj minus 2Dj ge 0 and ηj le 1 j 1 2 Further improvements areneeded

In this study all of the coefficients of the model systemare constant in many situations they can be assumed to benonconstant bounded nonnegative sequences such as pe-riodic positive sequences which can reflect the seasonalfluctuations [19] On the other hand time delays can have agreat influence on species populations It is worthy toconsider the nonautonomous space-time discrete Lot-kandashVolterra system with feedback control

Different control schemes such as switching controlconstraint control and sliding control can be applied to thesystem models Many interesting results can be obtained

Based on the backstepping recursive technique a neuralnetwork-based finite-time control strategy is proposed for aclass of non-strict-feedback nonlinear systems [31] and anevent-triggered robust fuzzy adaptive prescribed perfor-mance finite-time control strategy is proposed for a class ofstrict-feedback nonlinear systems with external disturbances[32] How to apply these control schemes on the discretediffusion models may be worth considering

It is well known that noise disturbance is unavoidable inreal systems and it has an important effect on the stability ofsystems Also noise can be used to stabilize a given unstablesystem or to make a system even more stable when thesystem is already stable which reveals that the stochasticfeedback control can stabilize and destabilize the deter-ministic systems [33 34]-erefore it will be interesting andchallenging to investigate stabilization or destabilization ofnonlinear discrete space-time systems by stochastic feedbackcontrol in our future work

Data Availability

No data were used to support this study

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was supported by the Applied Study Program(grant nos 171006901B 60204 and WH18012)

References

[1] J Xu and Z Teng ldquoPermanence for a nonautonomous dis-crete single-species system with delays and feedback controlrdquoApplied Mathematics Letters vol 23 no 9 pp 949ndash954 2010

[2] O S Board An Ecosystem Services Approach to Assessing theImpacts of the Deepwater Horizon Oil Spill in the Gulf ofMexico National Academies Press Washington DC USA2013

[3] V Tiwari J P Tripathi R K Upadhyay Y-PWu J-SWangand G-Q Sun ldquoPredator-prey interaction system with mu-tually interfering predator role of feedback controlrdquo AppliedMathematical Modelling vol 87 pp 222ndash244 2020

[4] X Li X Yang and T Huang ldquoPersistence of delayed co-operative models impulsive control methodrdquo AppliedMathematics and Computation vol 342 pp 130ndash146 2019

[5] T Luo ldquoStabilization of multi-group models with multipledispersal and stochastic perturbation via feedback controlbased on discrete-time state observationsrdquo Applied Mathe-matics and Computation vol 354 pp 396ndash410 2019

[6] W Qin X Tan M Tosato and X Liu ldquo-reshold controlstrategy for a non-smooth Filippov ecosystem with groupdefenserdquo Applied Mathematics and Computation vol 362Article ID 124532 2019

[7] J Xu Z Teng and H Jiang ldquoPermanence and globalattractivity for discrete nonautonomous two-species lotka-volterra competitive system with delays and feedback con-trolsrdquo Periodica Mathematica Hungarica vol 63 no 1pp 19ndash45 2011

030

028

026

024

022

020

018

0 10 20 30 40 50 60 70 80n

xn1

un11

Solu

tions

xn a

nd u

n 1

Figure 3 Dynamic behaviors of xn1 and un

11 for systems (36) and(37) with initial conditions (016 021 016 021 021 026 021 026 025 022 045 050) when d2 12

8 Complexity

[8] C Walter ldquoStability of controlled biological systemsrdquo Journalof -eoretical Biology vol 23 no 1 pp 23ndash38 1969

[9] P Wang Z Zhao and W Li ldquoGlobal stability analysis fordiscrete-time coupled systems with both time delay andmultiple dispersal and its applicationrdquo Neurocomputingvol 244 pp 42ndash52 2017

[10] Y Kuang ldquoGlobal stability in delay differential systemswithout dominating instantaneous negative feedbacksrdquoJournal of Differential Equations vol 119 no 2 pp 503ndash5321995

[11] Y Yan and E-O N Ekaka-a ldquoStabilizing a mathematicalmodel of population systemrdquo Journal of the Franklin Institutevol 348 no 10 pp 2744ndash2758 2011

[12] Y Muroya ldquoGlobal stability of a delayed nonlinear lotka-volterra system with feedback controls and patch structurerdquoApplied Mathematics and Computation vol 239 pp 60ndash732014

[13] J L Liu and W C Zhao ldquoDynamic analysis of stochasticlotkandashvolterra predator-prey model with discrete delays andfeedback controlrdquo Complexity vol 2019 p 15 Article ID4873290 2019

[14] Q Zhu and H Wang ldquoOutput feedback stabilization ofstochastic feedforward systems with unknown control coef-ficients and unknown output functionrdquo Automatica vol 87pp 166ndash175 2018

[15] Q Zhu ldquoStabilization of stochastic nonlinear delay systemswith exogenous disturbances and the event-triggered feed-back controlrdquo IEEE Transactions on Automatic Controlvol 64 no 9 pp 3764ndash3771 2019

[16] K Kiss and E Gyurkovics ldquoLMI approach to global stabilityanalysis of stochastic delayed lotka-volterra modelsrdquo AppliedMathematics Letters vol 104 pp 106227 1ndash6 2020

[17] X Chen and C Fengde ldquoStable periodic solution of a discreteperiodic lotka-volterra competition system with a feedbackcontrolrdquo Applied Mathematics and Computation vol 181no 2 pp 1446ndash1454 2006

[18] C Niu and X Chen ldquoAlmost periodic sequence solutions of adiscrete lotka-volterra competitive system with feedbackcontrolrdquo Nonlinear Analysis Real World Applications vol 10no 5 pp 3152ndash3161 2009

[19] X Liao S Zhou and Y Chen ldquoPermanence and globalstability in a discrete n-species competition system withfeedback controlsrdquo Nonlinear Analysis Real World Appli-cations vol 9 no 4 pp 1661ndash1671 2008

[20] A M Turing ldquo-e chemical basis of morphogenesisrdquo Philo-Sophical Transactions of the Royal Society B Biological Sci-ences vol 237 no 641 pp 37ndash72 1952

[21] K Gopalsamy and P X Weng ldquoFeedback regulation of lo-gistic growthrdquo International Journal of Mathematics andMathematical Sciences vol 16 no 1 pp 177ndash192 1993

[22] L Xu S S Lou P Q Xu and G Zhang ldquoFeedback controland parameter invasion for a discrete competitive lot-kandashvolterra systemrdquoDiscrete Dynamics in Nature and Societyvol 2018 p 8 Article ID 7473208 2018

[23] X W Li and H J Gao ldquoRobust finite frequency Hinfinfilteringfor uncertain 2-D roesser systemsrdquo Automatica vol 48pp 1163ndash1170 2012

[24] X W Li H J Gao and C H Wang ldquoGeneralized kal-manndashyakubovichndashpopov lemma for 2-D FM LSS modelrdquoIEEE Transaction on Automatic Control vol 57 no 12pp 3090ndash3193 2012

[25] J Zhou Y Yang and T Zhang ldquoGlobal dynamics of a re-action-diffusion waterborne pathogen model with general

incidence raterdquo Journal of Mathematical Analysis and Ap-plications vol 466 no 1 pp 835ndash859 2018

[26] Y Yang and J Zhou ldquoGlobal stability of a discrete virusdynamics model with diffusion and general infection func-tionrdquo International Journal of Computer Mathematics vol 96no 9 pp 1752ndash1762 2019

[27] P Liu and S N Elaydi ldquoDiscrete competitive and cooperativemodels of lotka-volterra typerdquo Journal of ComputationalAnalysis and Applications vol 3 no 1 pp 53ndash73 2001

[28] L L Meng and Y T Han ldquoBifurcation chaos and patternformation for the discrete predator-prey reaction-diffusionmodelrdquo Discrete Dynamics in Nature and Society vol 2019p 9 Article ID 9592878 2019

[29] R M May and G F Oster ldquoBifurcations and dynamiccomplexity in simple ecological modelsrdquo -e AmericanNaturalist vol 110 no 974 pp 573ndash599 1976

[30] Z Zhou and X Zou ldquoStable periodic solutions in a discreteperiodic logistic equationrdquo Applied Mathematics Lettersvol 16 no 2 pp 165ndash171 2003

[31] K K Sun J B Qiu H R Karimi and H J Gao ldquoA novelfinite-time control for nonstrict feedback saturated nonlinearsystems with tracking error constraintrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash12 2020

[32] K K Sun J B Qiu H R Karimi and Y L Fu ldquoEvent-triggered robust fuzzy adaptive finite-time control of non-linear systems with prescribed performancerdquo IEEE Transac-tions on Fuzzy Systems 2020

[33] X Mao ldquoAlmost sure exponential stabilization by discrete-time stochastic feedback controlrdquo IEEE Transactions onAutomatic Control vol 61 no 6 pp 1619ndash1624 2016

[34] Q Zhu and T Huang ldquoStability analysis for a class of sto-chastic delay nonlinear systems driven by G-Brownian mo-tionrdquo Systems amp Control Letters vol 140 Article ID 1046999 pages 2020

Complexity 9

where ρ2

(xni minus xlowast)2 + (yn

i minus ylowast)2 + (un2i minus ulowast2 )2

1113969

Let

Vn3

d1

2 1 minus η1( 1113857e1u

n1i minus u

lowast1( 1113857

2 (30)

-en we can obtain

ΔVn3 V

n+13 minus V

n3

d1

2 1 minus η1( 1113857e11113944

m

i1u

n+11i minus u

n1i1113872 1113873 u

n+11i + u

n1i minus 2u

lowast11113872 1113873

d1

2 1 minus η1( 1113857e11113944

m

i1minusη1u

n1i + e1x

ni( 1113857 2 minus η1( 1113857u

n1i(

+ e1xni minus 2u

lowast1 1113857

d1

2 1 minus η1( 1113857e11113944

m

i1minusη1 u

n1i minus u

lowast1( 1113857 + e1 x

ni minus xlowast

( 1113857( 1113857

middot 2 minus η1( 1113857 un1i minus u

lowast1( 1113857 + e1 x

ni minus xlowast

( 1113857( 1113857

minusd1η1 2 minus η1( 1113857

2 1 minus η1( 1113857e11113944

m

i1u

n1i minus u

lowast1( 1113857

2

+d1e1

2 1 minus η1( 11138571113944

m

i1x

ni minus xlowast

( 11138572

+ d1 xni minus xlowast

( 1113857 un1i minus u

lowast1( 1113857

(31)

Let

Vn4

d2a12

2 1 minus η2( 1113857e2a21u

n2i minus u

lowast2( 1113857

2 (32)

-en we can obtain

ΔVn4 V

n+14 minus V

n4

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1u

n+12i minus u

n2i1113872 1113873 u

n+12i + u

n2i minus 2u

lowast21113872 1113873

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1minusη2u

n2i + e2y

ni( 1113857 minusη2( 1113857u

n2i(

+ e2yni minus 2u

lowast2 1113857

d2a12

2 1 minus η2( 1113857e2a211113944

m

i1minusη2 u

n2i minus u

lowast2( 1113857(

+ e2 yni minus ylowast

( 11138571113857 2 minus η2( 1113857 un2i minus u

lowast2( 1113857 + e2 y

ni minus ylowast

( 1113857( 1113857

minusd2a12η2 2 minus η2( 1113857

2 1 minus η2( 1113857e2a211113944

m

i1u

n2i minus u

lowast2( 1113857

2

+d2a12e2

2 1 minus η2( 1113857a211113944

m

i1y

ni minus ylowast

( 11138572

+a12d2

a21

middot yni minus ylowast

( 1113857 un2i minus u

lowast2( 1113857

(33)

Let

Vn

Vn1 + V

n2 + V

n3 + V

n4 (34)

-en

ΔVn V

n+1minus V

n

le minus1 +d1e1

2 1 minus η1( 11138571113888 1113889 1113944

m

i1x

ni minus xlowast

( 11138572

+ minusa12

a21+

d2a12e2

2 1 minus η2( 1113857a211113888 1113889 1113944

m

i1y

ni minus ylowast

( 11138572

+d1η1 η1 minus 2( 1113857

2 1 minus η1( 1113857e11113944

m

i1u

n1i minus u

lowast1( 1113857

2

+d2a12η2 η2 minus 2( 1113857

2 1 minus η2( 1113857e2a211113944

m

i1u

n2i minus u

lowast2( 1113857

2

minus D1 1113944

mminus1

i1

xn

i+1xn

i

1113971

minus

xn

iminus1xn

i

1113971

⎛⎝ ⎞⎠

2

minus D1

xn

m

xn1

1113971

minus

xn1

xnm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ1( 1113857

minus D2a12

a211113944

mminus1

i1

yn

i+1yn

i

1113971

minus

yn

iminus1yn

i

1113971

⎛⎝ ⎞⎠

2

minus D2a12

a21

yn

m

yn1

1113971

minus

yn1

ynm

1113971

⎛⎝ ⎞⎠

2

+ o(1) + o ρ2( 1113857

(35)

If (d1e12(1 minus η1))le 1 (d2e22(1 minus η2))le 1 (d1η1(η1 minus

2)2(1 minus η1)e1)le 0 and (d2a12η2(η2 minus 2)2(1 minus η2)e2a21)le 0or(d1e12(1 minus η1))le 1 (d2e22(1 minus η2))le 1 η1 lt 1 andη2 lt 1hold ΔVn le 0 -e proof is completed

4 Example and Numerical Simulations

In the following example we will show the feasibility of ourmain results and discuss the effects of feedback controlsTake i 2 in the system we obtain a model with feedbackcontrols as follows

xn+1i x

ni exp r1 minus x

ni minus a12y

ni minus d1u

n1i( 1113857 + D1nabla

2x

ni

yn+1i y

ni exp r2 + a21x

ni minus y

ni minus d2u

n2i( 1113857 + D2nabla

2y

ni

un+11i 1 minus η1( 1113857u

n1i + e1x

ni

un+12i 1 minus η2( 1113857u

n2i + e2y

ni

i 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)

with the periodic boundary conditions

6 Complexity

xn0 x

n2

xn1 x

n3

yn0 y

n2

yn1 y

n3

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)

To illustrate our purposes the parameter values arechosen as follows (the choice of parameter values is hypo-thetical with appropriate units and not based on data) r1

12 r2 1a12 08a21 03d1 1d2 1D1 03D2 04

e1 07 e2 06η1 05 andη2 06 then there is only aunique positive equilibrium Elowast(xlowast1 xlowast2 ylowast1 ylowast2 ulowast11u

lowast12 ulowast21

ulowast22) (03181 03181 05487 05487 04453 0445305487

05487) It is easy to see that the conditions in -eorem 2 areverified Dynamic behaviors of systems (36) and (37) with theinitial conditions are shown in Figure 1 and three sets ofdifferent initial conditions are listed in Table 1-e simulations

can illustrate the fact that the positive equilibrium is globallyasymptotically stable

To explore clearly the dynamical behavior of systems(36) and (37) we investigate the effect of diffusion pa-rameter d2 by keeping other parameters of the systemfixed Figure 2 exhibits in detail an interesting situation when

050

045

040

035

030

025

0 10 20 30 40n

50 60

xn1

un11

Solu

tions

xn a

nd u

n 1

(a)

050

045

040

035

030

025

0 10 20 30 40n

50 60

xn2

un12

Solu

tions

xn a

nd u

n 1

(b)

0 10 20 30 40n

50 60

Solu

tions

yn an

d un 2

06

05

04

03

02yn1

un21

(c)

0 10 20 30 40n

50 60

Solu

tions

yn an

d un 2

06

05

04

03

02yn2

un22

(d)

Figure 1 Dynamic behaviors of systems (36) and (37) with three sets of different initial conditions when r1 12

r2 1 a12 08 a21 03 e1 07 e2 06 η1 05 η2 06 d1 1 d2 1 D1 03 andD2 04

Table 1 Different initial values for xn1 xn

2 yn1 yn

2 un11 un

12 un21 un

22

x01 x0

2 y01 y0

2 u011 u0

12 u021 u0

22

1 022 035 027 022 040 046 045 0522 033 026 023 028 047 042 048 0433 025 023 021 030 043 041 053 056 050

055

060

045

040

035

0 10 20 30 40n

50 60

xn1

un11

Solu

tions

xn a

nd u

n 1

Figure 2 Dynamic behaviors of xn1 and un

11 for systems (36) and(37) with initial conditions (036 034 036 034 028 031

028 031 059 052 045 054) when d2 08

Complexity 7

d2 08(r1 12 r2 1 a12 08 a21 03 d1 1 D1

03 D2 04 e1 07 e2 06 η1 05 η2 06) in whichthe solutions converge faster than Figure 1 With the increaseof d2 the solutions converge more slowly as depicted inFigure 3 For the sake of convenience only the dynamicalbehavior of xn

1 and un11 with a group of initial values is shown

in Figures 2 and 3 -e simulation results of adjustingfeedback control coefficients d1 e1 and e2 are omitted

5 Conclusions and Discussion

-is paper investigates the global asymptomatic stability ofthe unique positive equilibrium of a discrete diffusion modelwith the periodic boundary conditions and feedback control-e condition to ensure the nonnegativity and boundednessof the solutions of the discrete model is discussed and theglobally asymptotical stability of the positive equilibrium isproved -rough comparing numerical simulations wenotice that when we improve the feedback control coeffi-cients the solutions will converge more slowly It followsthat we can adjust the rate of convergence by choosingsuitable values of feedback control variables Such work mayalso be applied to other discrete diffusion models

It should be noted that there is only a basic conditionobtained to guarantee the existence of positive solutionJudgement sentence xn

i yni un

1i un2i ge 0 should be added into

the simulation programs under the conditions oferj minus 2Dj ge 0 and ηj le 1 j 1 2 Further improvements areneeded

In this study all of the coefficients of the model systemare constant in many situations they can be assumed to benonconstant bounded nonnegative sequences such as pe-riodic positive sequences which can reflect the seasonalfluctuations [19] On the other hand time delays can have agreat influence on species populations It is worthy toconsider the nonautonomous space-time discrete Lot-kandashVolterra system with feedback control

Different control schemes such as switching controlconstraint control and sliding control can be applied to thesystem models Many interesting results can be obtained

Based on the backstepping recursive technique a neuralnetwork-based finite-time control strategy is proposed for aclass of non-strict-feedback nonlinear systems [31] and anevent-triggered robust fuzzy adaptive prescribed perfor-mance finite-time control strategy is proposed for a class ofstrict-feedback nonlinear systems with external disturbances[32] How to apply these control schemes on the discretediffusion models may be worth considering

It is well known that noise disturbance is unavoidable inreal systems and it has an important effect on the stability ofsystems Also noise can be used to stabilize a given unstablesystem or to make a system even more stable when thesystem is already stable which reveals that the stochasticfeedback control can stabilize and destabilize the deter-ministic systems [33 34]-erefore it will be interesting andchallenging to investigate stabilization or destabilization ofnonlinear discrete space-time systems by stochastic feedbackcontrol in our future work

Data Availability

No data were used to support this study

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was supported by the Applied Study Program(grant nos 171006901B 60204 and WH18012)

References

[1] J Xu and Z Teng ldquoPermanence for a nonautonomous dis-crete single-species system with delays and feedback controlrdquoApplied Mathematics Letters vol 23 no 9 pp 949ndash954 2010

[2] O S Board An Ecosystem Services Approach to Assessing theImpacts of the Deepwater Horizon Oil Spill in the Gulf ofMexico National Academies Press Washington DC USA2013

[3] V Tiwari J P Tripathi R K Upadhyay Y-PWu J-SWangand G-Q Sun ldquoPredator-prey interaction system with mu-tually interfering predator role of feedback controlrdquo AppliedMathematical Modelling vol 87 pp 222ndash244 2020

[4] X Li X Yang and T Huang ldquoPersistence of delayed co-operative models impulsive control methodrdquo AppliedMathematics and Computation vol 342 pp 130ndash146 2019

[5] T Luo ldquoStabilization of multi-group models with multipledispersal and stochastic perturbation via feedback controlbased on discrete-time state observationsrdquo Applied Mathe-matics and Computation vol 354 pp 396ndash410 2019

[6] W Qin X Tan M Tosato and X Liu ldquo-reshold controlstrategy for a non-smooth Filippov ecosystem with groupdefenserdquo Applied Mathematics and Computation vol 362Article ID 124532 2019

[7] J Xu Z Teng and H Jiang ldquoPermanence and globalattractivity for discrete nonautonomous two-species lotka-volterra competitive system with delays and feedback con-trolsrdquo Periodica Mathematica Hungarica vol 63 no 1pp 19ndash45 2011

030

028

026

024

022

020

018

0 10 20 30 40 50 60 70 80n

xn1

un11

Solu

tions

xn a

nd u

n 1

Figure 3 Dynamic behaviors of xn1 and un

11 for systems (36) and(37) with initial conditions (016 021 016 021 021 026 021 026 025 022 045 050) when d2 12

8 Complexity

[8] C Walter ldquoStability of controlled biological systemsrdquo Journalof -eoretical Biology vol 23 no 1 pp 23ndash38 1969

[9] P Wang Z Zhao and W Li ldquoGlobal stability analysis fordiscrete-time coupled systems with both time delay andmultiple dispersal and its applicationrdquo Neurocomputingvol 244 pp 42ndash52 2017

[10] Y Kuang ldquoGlobal stability in delay differential systemswithout dominating instantaneous negative feedbacksrdquoJournal of Differential Equations vol 119 no 2 pp 503ndash5321995

[11] Y Yan and E-O N Ekaka-a ldquoStabilizing a mathematicalmodel of population systemrdquo Journal of the Franklin Institutevol 348 no 10 pp 2744ndash2758 2011

[12] Y Muroya ldquoGlobal stability of a delayed nonlinear lotka-volterra system with feedback controls and patch structurerdquoApplied Mathematics and Computation vol 239 pp 60ndash732014

[13] J L Liu and W C Zhao ldquoDynamic analysis of stochasticlotkandashvolterra predator-prey model with discrete delays andfeedback controlrdquo Complexity vol 2019 p 15 Article ID4873290 2019

[14] Q Zhu and H Wang ldquoOutput feedback stabilization ofstochastic feedforward systems with unknown control coef-ficients and unknown output functionrdquo Automatica vol 87pp 166ndash175 2018

[15] Q Zhu ldquoStabilization of stochastic nonlinear delay systemswith exogenous disturbances and the event-triggered feed-back controlrdquo IEEE Transactions on Automatic Controlvol 64 no 9 pp 3764ndash3771 2019

[16] K Kiss and E Gyurkovics ldquoLMI approach to global stabilityanalysis of stochastic delayed lotka-volterra modelsrdquo AppliedMathematics Letters vol 104 pp 106227 1ndash6 2020

[17] X Chen and C Fengde ldquoStable periodic solution of a discreteperiodic lotka-volterra competition system with a feedbackcontrolrdquo Applied Mathematics and Computation vol 181no 2 pp 1446ndash1454 2006

[18] C Niu and X Chen ldquoAlmost periodic sequence solutions of adiscrete lotka-volterra competitive system with feedbackcontrolrdquo Nonlinear Analysis Real World Applications vol 10no 5 pp 3152ndash3161 2009

[19] X Liao S Zhou and Y Chen ldquoPermanence and globalstability in a discrete n-species competition system withfeedback controlsrdquo Nonlinear Analysis Real World Appli-cations vol 9 no 4 pp 1661ndash1671 2008

[20] A M Turing ldquo-e chemical basis of morphogenesisrdquo Philo-Sophical Transactions of the Royal Society B Biological Sci-ences vol 237 no 641 pp 37ndash72 1952

[21] K Gopalsamy and P X Weng ldquoFeedback regulation of lo-gistic growthrdquo International Journal of Mathematics andMathematical Sciences vol 16 no 1 pp 177ndash192 1993

[22] L Xu S S Lou P Q Xu and G Zhang ldquoFeedback controland parameter invasion for a discrete competitive lot-kandashvolterra systemrdquoDiscrete Dynamics in Nature and Societyvol 2018 p 8 Article ID 7473208 2018

[23] X W Li and H J Gao ldquoRobust finite frequency Hinfinfilteringfor uncertain 2-D roesser systemsrdquo Automatica vol 48pp 1163ndash1170 2012

[24] X W Li H J Gao and C H Wang ldquoGeneralized kal-manndashyakubovichndashpopov lemma for 2-D FM LSS modelrdquoIEEE Transaction on Automatic Control vol 57 no 12pp 3090ndash3193 2012

[25] J Zhou Y Yang and T Zhang ldquoGlobal dynamics of a re-action-diffusion waterborne pathogen model with general

incidence raterdquo Journal of Mathematical Analysis and Ap-plications vol 466 no 1 pp 835ndash859 2018

[26] Y Yang and J Zhou ldquoGlobal stability of a discrete virusdynamics model with diffusion and general infection func-tionrdquo International Journal of Computer Mathematics vol 96no 9 pp 1752ndash1762 2019

[27] P Liu and S N Elaydi ldquoDiscrete competitive and cooperativemodels of lotka-volterra typerdquo Journal of ComputationalAnalysis and Applications vol 3 no 1 pp 53ndash73 2001

[28] L L Meng and Y T Han ldquoBifurcation chaos and patternformation for the discrete predator-prey reaction-diffusionmodelrdquo Discrete Dynamics in Nature and Society vol 2019p 9 Article ID 9592878 2019

[29] R M May and G F Oster ldquoBifurcations and dynamiccomplexity in simple ecological modelsrdquo -e AmericanNaturalist vol 110 no 974 pp 573ndash599 1976

[30] Z Zhou and X Zou ldquoStable periodic solutions in a discreteperiodic logistic equationrdquo Applied Mathematics Lettersvol 16 no 2 pp 165ndash171 2003

[31] K K Sun J B Qiu H R Karimi and H J Gao ldquoA novelfinite-time control for nonstrict feedback saturated nonlinearsystems with tracking error constraintrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash12 2020

[32] K K Sun J B Qiu H R Karimi and Y L Fu ldquoEvent-triggered robust fuzzy adaptive finite-time control of non-linear systems with prescribed performancerdquo IEEE Transac-tions on Fuzzy Systems 2020

[33] X Mao ldquoAlmost sure exponential stabilization by discrete-time stochastic feedback controlrdquo IEEE Transactions onAutomatic Control vol 61 no 6 pp 1619ndash1624 2016

[34] Q Zhu and T Huang ldquoStability analysis for a class of sto-chastic delay nonlinear systems driven by G-Brownian mo-tionrdquo Systems amp Control Letters vol 140 Article ID 1046999 pages 2020

Complexity 9

xn0 x

n2

xn1 x

n3

yn0 y

n2

yn1 y

n3

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)

To illustrate our purposes the parameter values arechosen as follows (the choice of parameter values is hypo-thetical with appropriate units and not based on data) r1

12 r2 1a12 08a21 03d1 1d2 1D1 03D2 04

e1 07 e2 06η1 05 andη2 06 then there is only aunique positive equilibrium Elowast(xlowast1 xlowast2 ylowast1 ylowast2 ulowast11u

lowast12 ulowast21

ulowast22) (03181 03181 05487 05487 04453 0445305487

05487) It is easy to see that the conditions in -eorem 2 areverified Dynamic behaviors of systems (36) and (37) with theinitial conditions are shown in Figure 1 and three sets ofdifferent initial conditions are listed in Table 1-e simulations

can illustrate the fact that the positive equilibrium is globallyasymptotically stable

To explore clearly the dynamical behavior of systems(36) and (37) we investigate the effect of diffusion pa-rameter d2 by keeping other parameters of the systemfixed Figure 2 exhibits in detail an interesting situation when

050

045

040

035

030

025

0 10 20 30 40n

50 60

xn1

un11

Solu

tions

xn a

nd u

n 1

(a)

050

045

040

035

030

025

0 10 20 30 40n

50 60

xn2

un12

Solu

tions

xn a

nd u

n 1

(b)

0 10 20 30 40n

50 60

Solu

tions

yn an

d un 2

06

05

04

03

02yn1

un21

(c)

0 10 20 30 40n

50 60

Solu

tions

yn an

d un 2

06

05

04

03

02yn2

un22

(d)

Figure 1 Dynamic behaviors of systems (36) and (37) with three sets of different initial conditions when r1 12

r2 1 a12 08 a21 03 e1 07 e2 06 η1 05 η2 06 d1 1 d2 1 D1 03 andD2 04

Table 1 Different initial values for xn1 xn

2 yn1 yn

2 un11 un

12 un21 un

22

x01 x0

2 y01 y0

2 u011 u0

12 u021 u0

22

1 022 035 027 022 040 046 045 0522 033 026 023 028 047 042 048 0433 025 023 021 030 043 041 053 056 050

055

060

045

040

035

0 10 20 30 40n

50 60

xn1

un11

Solu

tions

xn a

nd u

n 1

Figure 2 Dynamic behaviors of xn1 and un

11 for systems (36) and(37) with initial conditions (036 034 036 034 028 031

028 031 059 052 045 054) when d2 08

Complexity 7

d2 08(r1 12 r2 1 a12 08 a21 03 d1 1 D1

03 D2 04 e1 07 e2 06 η1 05 η2 06) in whichthe solutions converge faster than Figure 1 With the increaseof d2 the solutions converge more slowly as depicted inFigure 3 For the sake of convenience only the dynamicalbehavior of xn

1 and un11 with a group of initial values is shown

in Figures 2 and 3 -e simulation results of adjustingfeedback control coefficients d1 e1 and e2 are omitted

5 Conclusions and Discussion

-is paper investigates the global asymptomatic stability ofthe unique positive equilibrium of a discrete diffusion modelwith the periodic boundary conditions and feedback control-e condition to ensure the nonnegativity and boundednessof the solutions of the discrete model is discussed and theglobally asymptotical stability of the positive equilibrium isproved -rough comparing numerical simulations wenotice that when we improve the feedback control coeffi-cients the solutions will converge more slowly It followsthat we can adjust the rate of convergence by choosingsuitable values of feedback control variables Such work mayalso be applied to other discrete diffusion models

It should be noted that there is only a basic conditionobtained to guarantee the existence of positive solutionJudgement sentence xn

i yni un

1i un2i ge 0 should be added into

the simulation programs under the conditions oferj minus 2Dj ge 0 and ηj le 1 j 1 2 Further improvements areneeded

In this study all of the coefficients of the model systemare constant in many situations they can be assumed to benonconstant bounded nonnegative sequences such as pe-riodic positive sequences which can reflect the seasonalfluctuations [19] On the other hand time delays can have agreat influence on species populations It is worthy toconsider the nonautonomous space-time discrete Lot-kandashVolterra system with feedback control

Different control schemes such as switching controlconstraint control and sliding control can be applied to thesystem models Many interesting results can be obtained

Based on the backstepping recursive technique a neuralnetwork-based finite-time control strategy is proposed for aclass of non-strict-feedback nonlinear systems [31] and anevent-triggered robust fuzzy adaptive prescribed perfor-mance finite-time control strategy is proposed for a class ofstrict-feedback nonlinear systems with external disturbances[32] How to apply these control schemes on the discretediffusion models may be worth considering

It is well known that noise disturbance is unavoidable inreal systems and it has an important effect on the stability ofsystems Also noise can be used to stabilize a given unstablesystem or to make a system even more stable when thesystem is already stable which reveals that the stochasticfeedback control can stabilize and destabilize the deter-ministic systems [33 34]-erefore it will be interesting andchallenging to investigate stabilization or destabilization ofnonlinear discrete space-time systems by stochastic feedbackcontrol in our future work

Data Availability

No data were used to support this study

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was supported by the Applied Study Program(grant nos 171006901B 60204 and WH18012)

References

[1] J Xu and Z Teng ldquoPermanence for a nonautonomous dis-crete single-species system with delays and feedback controlrdquoApplied Mathematics Letters vol 23 no 9 pp 949ndash954 2010

[2] O S Board An Ecosystem Services Approach to Assessing theImpacts of the Deepwater Horizon Oil Spill in the Gulf ofMexico National Academies Press Washington DC USA2013

[3] V Tiwari J P Tripathi R K Upadhyay Y-PWu J-SWangand G-Q Sun ldquoPredator-prey interaction system with mu-tually interfering predator role of feedback controlrdquo AppliedMathematical Modelling vol 87 pp 222ndash244 2020

[4] X Li X Yang and T Huang ldquoPersistence of delayed co-operative models impulsive control methodrdquo AppliedMathematics and Computation vol 342 pp 130ndash146 2019

[5] T Luo ldquoStabilization of multi-group models with multipledispersal and stochastic perturbation via feedback controlbased on discrete-time state observationsrdquo Applied Mathe-matics and Computation vol 354 pp 396ndash410 2019

[6] W Qin X Tan M Tosato and X Liu ldquo-reshold controlstrategy for a non-smooth Filippov ecosystem with groupdefenserdquo Applied Mathematics and Computation vol 362Article ID 124532 2019

[7] J Xu Z Teng and H Jiang ldquoPermanence and globalattractivity for discrete nonautonomous two-species lotka-volterra competitive system with delays and feedback con-trolsrdquo Periodica Mathematica Hungarica vol 63 no 1pp 19ndash45 2011

030

028

026

024

022

020

018

0 10 20 30 40 50 60 70 80n

xn1

un11

Solu

tions

xn a

nd u

n 1

Figure 3 Dynamic behaviors of xn1 and un

11 for systems (36) and(37) with initial conditions (016 021 016 021 021 026 021 026 025 022 045 050) when d2 12

8 Complexity

[8] C Walter ldquoStability of controlled biological systemsrdquo Journalof -eoretical Biology vol 23 no 1 pp 23ndash38 1969

[9] P Wang Z Zhao and W Li ldquoGlobal stability analysis fordiscrete-time coupled systems with both time delay andmultiple dispersal and its applicationrdquo Neurocomputingvol 244 pp 42ndash52 2017

[10] Y Kuang ldquoGlobal stability in delay differential systemswithout dominating instantaneous negative feedbacksrdquoJournal of Differential Equations vol 119 no 2 pp 503ndash5321995

[11] Y Yan and E-O N Ekaka-a ldquoStabilizing a mathematicalmodel of population systemrdquo Journal of the Franklin Institutevol 348 no 10 pp 2744ndash2758 2011

[12] Y Muroya ldquoGlobal stability of a delayed nonlinear lotka-volterra system with feedback controls and patch structurerdquoApplied Mathematics and Computation vol 239 pp 60ndash732014

[13] J L Liu and W C Zhao ldquoDynamic analysis of stochasticlotkandashvolterra predator-prey model with discrete delays andfeedback controlrdquo Complexity vol 2019 p 15 Article ID4873290 2019

[14] Q Zhu and H Wang ldquoOutput feedback stabilization ofstochastic feedforward systems with unknown control coef-ficients and unknown output functionrdquo Automatica vol 87pp 166ndash175 2018

[15] Q Zhu ldquoStabilization of stochastic nonlinear delay systemswith exogenous disturbances and the event-triggered feed-back controlrdquo IEEE Transactions on Automatic Controlvol 64 no 9 pp 3764ndash3771 2019

[16] K Kiss and E Gyurkovics ldquoLMI approach to global stabilityanalysis of stochastic delayed lotka-volterra modelsrdquo AppliedMathematics Letters vol 104 pp 106227 1ndash6 2020

[17] X Chen and C Fengde ldquoStable periodic solution of a discreteperiodic lotka-volterra competition system with a feedbackcontrolrdquo Applied Mathematics and Computation vol 181no 2 pp 1446ndash1454 2006

[18] C Niu and X Chen ldquoAlmost periodic sequence solutions of adiscrete lotka-volterra competitive system with feedbackcontrolrdquo Nonlinear Analysis Real World Applications vol 10no 5 pp 3152ndash3161 2009

[19] X Liao S Zhou and Y Chen ldquoPermanence and globalstability in a discrete n-species competition system withfeedback controlsrdquo Nonlinear Analysis Real World Appli-cations vol 9 no 4 pp 1661ndash1671 2008

[20] A M Turing ldquo-e chemical basis of morphogenesisrdquo Philo-Sophical Transactions of the Royal Society B Biological Sci-ences vol 237 no 641 pp 37ndash72 1952

[21] K Gopalsamy and P X Weng ldquoFeedback regulation of lo-gistic growthrdquo International Journal of Mathematics andMathematical Sciences vol 16 no 1 pp 177ndash192 1993

[22] L Xu S S Lou P Q Xu and G Zhang ldquoFeedback controland parameter invasion for a discrete competitive lot-kandashvolterra systemrdquoDiscrete Dynamics in Nature and Societyvol 2018 p 8 Article ID 7473208 2018

[23] X W Li and H J Gao ldquoRobust finite frequency Hinfinfilteringfor uncertain 2-D roesser systemsrdquo Automatica vol 48pp 1163ndash1170 2012

[24] X W Li H J Gao and C H Wang ldquoGeneralized kal-manndashyakubovichndashpopov lemma for 2-D FM LSS modelrdquoIEEE Transaction on Automatic Control vol 57 no 12pp 3090ndash3193 2012

[25] J Zhou Y Yang and T Zhang ldquoGlobal dynamics of a re-action-diffusion waterborne pathogen model with general

incidence raterdquo Journal of Mathematical Analysis and Ap-plications vol 466 no 1 pp 835ndash859 2018

[26] Y Yang and J Zhou ldquoGlobal stability of a discrete virusdynamics model with diffusion and general infection func-tionrdquo International Journal of Computer Mathematics vol 96no 9 pp 1752ndash1762 2019

[27] P Liu and S N Elaydi ldquoDiscrete competitive and cooperativemodels of lotka-volterra typerdquo Journal of ComputationalAnalysis and Applications vol 3 no 1 pp 53ndash73 2001

[28] L L Meng and Y T Han ldquoBifurcation chaos and patternformation for the discrete predator-prey reaction-diffusionmodelrdquo Discrete Dynamics in Nature and Society vol 2019p 9 Article ID 9592878 2019

[29] R M May and G F Oster ldquoBifurcations and dynamiccomplexity in simple ecological modelsrdquo -e AmericanNaturalist vol 110 no 974 pp 573ndash599 1976

[30] Z Zhou and X Zou ldquoStable periodic solutions in a discreteperiodic logistic equationrdquo Applied Mathematics Lettersvol 16 no 2 pp 165ndash171 2003

[31] K K Sun J B Qiu H R Karimi and H J Gao ldquoA novelfinite-time control for nonstrict feedback saturated nonlinearsystems with tracking error constraintrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash12 2020

[32] K K Sun J B Qiu H R Karimi and Y L Fu ldquoEvent-triggered robust fuzzy adaptive finite-time control of non-linear systems with prescribed performancerdquo IEEE Transac-tions on Fuzzy Systems 2020

[33] X Mao ldquoAlmost sure exponential stabilization by discrete-time stochastic feedback controlrdquo IEEE Transactions onAutomatic Control vol 61 no 6 pp 1619ndash1624 2016

[34] Q Zhu and T Huang ldquoStability analysis for a class of sto-chastic delay nonlinear systems driven by G-Brownian mo-tionrdquo Systems amp Control Letters vol 140 Article ID 1046999 pages 2020

Complexity 9

d2 08(r1 12 r2 1 a12 08 a21 03 d1 1 D1

03 D2 04 e1 07 e2 06 η1 05 η2 06) in whichthe solutions converge faster than Figure 1 With the increaseof d2 the solutions converge more slowly as depicted inFigure 3 For the sake of convenience only the dynamicalbehavior of xn

1 and un11 with a group of initial values is shown

in Figures 2 and 3 -e simulation results of adjustingfeedback control coefficients d1 e1 and e2 are omitted

5 Conclusions and Discussion

-is paper investigates the global asymptomatic stability ofthe unique positive equilibrium of a discrete diffusion modelwith the periodic boundary conditions and feedback control-e condition to ensure the nonnegativity and boundednessof the solutions of the discrete model is discussed and theglobally asymptotical stability of the positive equilibrium isproved -rough comparing numerical simulations wenotice that when we improve the feedback control coeffi-cients the solutions will converge more slowly It followsthat we can adjust the rate of convergence by choosingsuitable values of feedback control variables Such work mayalso be applied to other discrete diffusion models

It should be noted that there is only a basic conditionobtained to guarantee the existence of positive solutionJudgement sentence xn

i yni un

1i un2i ge 0 should be added into

the simulation programs under the conditions oferj minus 2Dj ge 0 and ηj le 1 j 1 2 Further improvements areneeded

In this study all of the coefficients of the model systemare constant in many situations they can be assumed to benonconstant bounded nonnegative sequences such as pe-riodic positive sequences which can reflect the seasonalfluctuations [19] On the other hand time delays can have agreat influence on species populations It is worthy toconsider the nonautonomous space-time discrete Lot-kandashVolterra system with feedback control

Different control schemes such as switching controlconstraint control and sliding control can be applied to thesystem models Many interesting results can be obtained

Based on the backstepping recursive technique a neuralnetwork-based finite-time control strategy is proposed for aclass of non-strict-feedback nonlinear systems [31] and anevent-triggered robust fuzzy adaptive prescribed perfor-mance finite-time control strategy is proposed for a class ofstrict-feedback nonlinear systems with external disturbances[32] How to apply these control schemes on the discretediffusion models may be worth considering

It is well known that noise disturbance is unavoidable inreal systems and it has an important effect on the stability ofsystems Also noise can be used to stabilize a given unstablesystem or to make a system even more stable when thesystem is already stable which reveals that the stochasticfeedback control can stabilize and destabilize the deter-ministic systems [33 34]-erefore it will be interesting andchallenging to investigate stabilization or destabilization ofnonlinear discrete space-time systems by stochastic feedbackcontrol in our future work

Data Availability

No data were used to support this study

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was supported by the Applied Study Program(grant nos 171006901B 60204 and WH18012)

References

[1] J Xu and Z Teng ldquoPermanence for a nonautonomous dis-crete single-species system with delays and feedback controlrdquoApplied Mathematics Letters vol 23 no 9 pp 949ndash954 2010

[2] O S Board An Ecosystem Services Approach to Assessing theImpacts of the Deepwater Horizon Oil Spill in the Gulf ofMexico National Academies Press Washington DC USA2013

[3] V Tiwari J P Tripathi R K Upadhyay Y-PWu J-SWangand G-Q Sun ldquoPredator-prey interaction system with mu-tually interfering predator role of feedback controlrdquo AppliedMathematical Modelling vol 87 pp 222ndash244 2020

[4] X Li X Yang and T Huang ldquoPersistence of delayed co-operative models impulsive control methodrdquo AppliedMathematics and Computation vol 342 pp 130ndash146 2019

[5] T Luo ldquoStabilization of multi-group models with multipledispersal and stochastic perturbation via feedback controlbased on discrete-time state observationsrdquo Applied Mathe-matics and Computation vol 354 pp 396ndash410 2019

[6] W Qin X Tan M Tosato and X Liu ldquo-reshold controlstrategy for a non-smooth Filippov ecosystem with groupdefenserdquo Applied Mathematics and Computation vol 362Article ID 124532 2019

[7] J Xu Z Teng and H Jiang ldquoPermanence and globalattractivity for discrete nonautonomous two-species lotka-volterra competitive system with delays and feedback con-trolsrdquo Periodica Mathematica Hungarica vol 63 no 1pp 19ndash45 2011

030

028

026

024

022

020

018

0 10 20 30 40 50 60 70 80n

xn1

un11

Solu

tions

xn a

nd u

n 1

Figure 3 Dynamic behaviors of xn1 and un

11 for systems (36) and(37) with initial conditions (016 021 016 021 021 026 021 026 025 022 045 050) when d2 12

8 Complexity

[8] C Walter ldquoStability of controlled biological systemsrdquo Journalof -eoretical Biology vol 23 no 1 pp 23ndash38 1969

[9] P Wang Z Zhao and W Li ldquoGlobal stability analysis fordiscrete-time coupled systems with both time delay andmultiple dispersal and its applicationrdquo Neurocomputingvol 244 pp 42ndash52 2017

[10] Y Kuang ldquoGlobal stability in delay differential systemswithout dominating instantaneous negative feedbacksrdquoJournal of Differential Equations vol 119 no 2 pp 503ndash5321995

[11] Y Yan and E-O N Ekaka-a ldquoStabilizing a mathematicalmodel of population systemrdquo Journal of the Franklin Institutevol 348 no 10 pp 2744ndash2758 2011

[12] Y Muroya ldquoGlobal stability of a delayed nonlinear lotka-volterra system with feedback controls and patch structurerdquoApplied Mathematics and Computation vol 239 pp 60ndash732014

[13] J L Liu and W C Zhao ldquoDynamic analysis of stochasticlotkandashvolterra predator-prey model with discrete delays andfeedback controlrdquo Complexity vol 2019 p 15 Article ID4873290 2019

[14] Q Zhu and H Wang ldquoOutput feedback stabilization ofstochastic feedforward systems with unknown control coef-ficients and unknown output functionrdquo Automatica vol 87pp 166ndash175 2018

[15] Q Zhu ldquoStabilization of stochastic nonlinear delay systemswith exogenous disturbances and the event-triggered feed-back controlrdquo IEEE Transactions on Automatic Controlvol 64 no 9 pp 3764ndash3771 2019

[16] K Kiss and E Gyurkovics ldquoLMI approach to global stabilityanalysis of stochastic delayed lotka-volterra modelsrdquo AppliedMathematics Letters vol 104 pp 106227 1ndash6 2020

[17] X Chen and C Fengde ldquoStable periodic solution of a discreteperiodic lotka-volterra competition system with a feedbackcontrolrdquo Applied Mathematics and Computation vol 181no 2 pp 1446ndash1454 2006

[18] C Niu and X Chen ldquoAlmost periodic sequence solutions of adiscrete lotka-volterra competitive system with feedbackcontrolrdquo Nonlinear Analysis Real World Applications vol 10no 5 pp 3152ndash3161 2009

[19] X Liao S Zhou and Y Chen ldquoPermanence and globalstability in a discrete n-species competition system withfeedback controlsrdquo Nonlinear Analysis Real World Appli-cations vol 9 no 4 pp 1661ndash1671 2008

[20] A M Turing ldquo-e chemical basis of morphogenesisrdquo Philo-Sophical Transactions of the Royal Society B Biological Sci-ences vol 237 no 641 pp 37ndash72 1952

[21] K Gopalsamy and P X Weng ldquoFeedback regulation of lo-gistic growthrdquo International Journal of Mathematics andMathematical Sciences vol 16 no 1 pp 177ndash192 1993

[22] L Xu S S Lou P Q Xu and G Zhang ldquoFeedback controland parameter invasion for a discrete competitive lot-kandashvolterra systemrdquoDiscrete Dynamics in Nature and Societyvol 2018 p 8 Article ID 7473208 2018

[23] X W Li and H J Gao ldquoRobust finite frequency Hinfinfilteringfor uncertain 2-D roesser systemsrdquo Automatica vol 48pp 1163ndash1170 2012

[24] X W Li H J Gao and C H Wang ldquoGeneralized kal-manndashyakubovichndashpopov lemma for 2-D FM LSS modelrdquoIEEE Transaction on Automatic Control vol 57 no 12pp 3090ndash3193 2012

[25] J Zhou Y Yang and T Zhang ldquoGlobal dynamics of a re-action-diffusion waterborne pathogen model with general

incidence raterdquo Journal of Mathematical Analysis and Ap-plications vol 466 no 1 pp 835ndash859 2018

[26] Y Yang and J Zhou ldquoGlobal stability of a discrete virusdynamics model with diffusion and general infection func-tionrdquo International Journal of Computer Mathematics vol 96no 9 pp 1752ndash1762 2019

[27] P Liu and S N Elaydi ldquoDiscrete competitive and cooperativemodels of lotka-volterra typerdquo Journal of ComputationalAnalysis and Applications vol 3 no 1 pp 53ndash73 2001

[28] L L Meng and Y T Han ldquoBifurcation chaos and patternformation for the discrete predator-prey reaction-diffusionmodelrdquo Discrete Dynamics in Nature and Society vol 2019p 9 Article ID 9592878 2019

[29] R M May and G F Oster ldquoBifurcations and dynamiccomplexity in simple ecological modelsrdquo -e AmericanNaturalist vol 110 no 974 pp 573ndash599 1976

[30] Z Zhou and X Zou ldquoStable periodic solutions in a discreteperiodic logistic equationrdquo Applied Mathematics Lettersvol 16 no 2 pp 165ndash171 2003

[31] K K Sun J B Qiu H R Karimi and H J Gao ldquoA novelfinite-time control for nonstrict feedback saturated nonlinearsystems with tracking error constraintrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash12 2020

[32] K K Sun J B Qiu H R Karimi and Y L Fu ldquoEvent-triggered robust fuzzy adaptive finite-time control of non-linear systems with prescribed performancerdquo IEEE Transac-tions on Fuzzy Systems 2020

[33] X Mao ldquoAlmost sure exponential stabilization by discrete-time stochastic feedback controlrdquo IEEE Transactions onAutomatic Control vol 61 no 6 pp 1619ndash1624 2016

[34] Q Zhu and T Huang ldquoStability analysis for a class of sto-chastic delay nonlinear systems driven by G-Brownian mo-tionrdquo Systems amp Control Letters vol 140 Article ID 1046999 pages 2020

Complexity 9

[8] C Walter ldquoStability of controlled biological systemsrdquo Journalof -eoretical Biology vol 23 no 1 pp 23ndash38 1969

[9] P Wang Z Zhao and W Li ldquoGlobal stability analysis fordiscrete-time coupled systems with both time delay andmultiple dispersal and its applicationrdquo Neurocomputingvol 244 pp 42ndash52 2017

[10] Y Kuang ldquoGlobal stability in delay differential systemswithout dominating instantaneous negative feedbacksrdquoJournal of Differential Equations vol 119 no 2 pp 503ndash5321995

[11] Y Yan and E-O N Ekaka-a ldquoStabilizing a mathematicalmodel of population systemrdquo Journal of the Franklin Institutevol 348 no 10 pp 2744ndash2758 2011

[12] Y Muroya ldquoGlobal stability of a delayed nonlinear lotka-volterra system with feedback controls and patch structurerdquoApplied Mathematics and Computation vol 239 pp 60ndash732014

[13] J L Liu and W C Zhao ldquoDynamic analysis of stochasticlotkandashvolterra predator-prey model with discrete delays andfeedback controlrdquo Complexity vol 2019 p 15 Article ID4873290 2019

[14] Q Zhu and H Wang ldquoOutput feedback stabilization ofstochastic feedforward systems with unknown control coef-ficients and unknown output functionrdquo Automatica vol 87pp 166ndash175 2018

[15] Q Zhu ldquoStabilization of stochastic nonlinear delay systemswith exogenous disturbances and the event-triggered feed-back controlrdquo IEEE Transactions on Automatic Controlvol 64 no 9 pp 3764ndash3771 2019

[16] K Kiss and E Gyurkovics ldquoLMI approach to global stabilityanalysis of stochastic delayed lotka-volterra modelsrdquo AppliedMathematics Letters vol 104 pp 106227 1ndash6 2020

[17] X Chen and C Fengde ldquoStable periodic solution of a discreteperiodic lotka-volterra competition system with a feedbackcontrolrdquo Applied Mathematics and Computation vol 181no 2 pp 1446ndash1454 2006

[18] C Niu and X Chen ldquoAlmost periodic sequence solutions of adiscrete lotka-volterra competitive system with feedbackcontrolrdquo Nonlinear Analysis Real World Applications vol 10no 5 pp 3152ndash3161 2009

[19] X Liao S Zhou and Y Chen ldquoPermanence and globalstability in a discrete n-species competition system withfeedback controlsrdquo Nonlinear Analysis Real World Appli-cations vol 9 no 4 pp 1661ndash1671 2008

[20] A M Turing ldquo-e chemical basis of morphogenesisrdquo Philo-Sophical Transactions of the Royal Society B Biological Sci-ences vol 237 no 641 pp 37ndash72 1952

[21] K Gopalsamy and P X Weng ldquoFeedback regulation of lo-gistic growthrdquo International Journal of Mathematics andMathematical Sciences vol 16 no 1 pp 177ndash192 1993

[22] L Xu S S Lou P Q Xu and G Zhang ldquoFeedback controland parameter invasion for a discrete competitive lot-kandashvolterra systemrdquoDiscrete Dynamics in Nature and Societyvol 2018 p 8 Article ID 7473208 2018

[23] X W Li and H J Gao ldquoRobust finite frequency Hinfinfilteringfor uncertain 2-D roesser systemsrdquo Automatica vol 48pp 1163ndash1170 2012

[24] X W Li H J Gao and C H Wang ldquoGeneralized kal-manndashyakubovichndashpopov lemma for 2-D FM LSS modelrdquoIEEE Transaction on Automatic Control vol 57 no 12pp 3090ndash3193 2012

[25] J Zhou Y Yang and T Zhang ldquoGlobal dynamics of a re-action-diffusion waterborne pathogen model with general

incidence raterdquo Journal of Mathematical Analysis and Ap-plications vol 466 no 1 pp 835ndash859 2018

[26] Y Yang and J Zhou ldquoGlobal stability of a discrete virusdynamics model with diffusion and general infection func-tionrdquo International Journal of Computer Mathematics vol 96no 9 pp 1752ndash1762 2019

[27] P Liu and S N Elaydi ldquoDiscrete competitive and cooperativemodels of lotka-volterra typerdquo Journal of ComputationalAnalysis and Applications vol 3 no 1 pp 53ndash73 2001

[28] L L Meng and Y T Han ldquoBifurcation chaos and patternformation for the discrete predator-prey reaction-diffusionmodelrdquo Discrete Dynamics in Nature and Society vol 2019p 9 Article ID 9592878 2019

[29] R M May and G F Oster ldquoBifurcations and dynamiccomplexity in simple ecological modelsrdquo -e AmericanNaturalist vol 110 no 974 pp 573ndash599 1976

[30] Z Zhou and X Zou ldquoStable periodic solutions in a discreteperiodic logistic equationrdquo Applied Mathematics Lettersvol 16 no 2 pp 165ndash171 2003

[31] K K Sun J B Qiu H R Karimi and H J Gao ldquoA novelfinite-time control for nonstrict feedback saturated nonlinearsystems with tracking error constraintrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash12 2020

[32] K K Sun J B Qiu H R Karimi and Y L Fu ldquoEvent-triggered robust fuzzy adaptive finite-time control of non-linear systems with prescribed performancerdquo IEEE Transac-tions on Fuzzy Systems 2020

[33] X Mao ldquoAlmost sure exponential stabilization by discrete-time stochastic feedback controlrdquo IEEE Transactions onAutomatic Control vol 61 no 6 pp 1619ndash1624 2016

[34] Q Zhu and T Huang ldquoStability analysis for a class of sto-chastic delay nonlinear systems driven by G-Brownian mo-tionrdquo Systems amp Control Letters vol 140 Article ID 1046999 pages 2020

Complexity 9