# Global Dynamics, Bifurcation Analysis, and Chaos in a Discrete Kolmogorov Model with Piecewise-Constant Argument

Authors: Abdul Qadeer Khan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The local behavior with topological classifications, bifurcation analysis, chaos control, boundedness, and global attractivity of the discrete-time Kolmogorov model with piecewise-constant argument are investigated. It is explored that Kolmogorov model has trivial and two semitrival fixed points for all involved parameters, but it has an interior fixed point under definite parametric condition. Then, by linear stability theory, local dynamics with different topological classifications are investigated around trivial, semitrival, and interior fixed points. Further for the discrete Kolmogorov model, existence of periodic points is also investigated. It is also investigated the occurrence of bifurcations at interior fixed point and proved that at interior fixed point, there exists no bifurcation, except flip bifurcation by bifurcation theory. Next, feedback control method is utilized to stabilize chaos existing in discrete Kolmogorov model. Boundedness and global attractivity of the discrete Kolmogorov model are also investigated. Finally, obtained results are numerically verified.


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## Body

## 1. Introduction

### 1.1. Motivation and Literature Review

There are many symbiotic interactions existing in nature between two or more species in an ecosystem. Mutualism is an example of such interaction where interacted species get benefit from each other. For example, termites eat cellulose of wood but cannot digest it, and flagellates reside in the termite’s gut decomposing the cellulose of food and thus providing nutrients to termites. On the other hand, termite’s gut provides food and shelter to flagellates. It is pointed out that in theoretical ecology, mutualist behavior of symbiosis or mutualism is very significant [1]. This field is not widely studied as the other fields of mathematical biology even for two species, although its importance is equal to the other competitive interactions such as host-parasitoid and prey-predator interactions. So, this topic of mutualism system seems interesting to study. For instance, May [2] suggested the two-species hybrid continuous-time Kolmogorov model represented by the following system of differential equations:(1)dxdt=r1xt1−xtβ1+α1yt,dydt=r2yt1−ytβ2+α2xt,where t represents the greatest integer in t, and parameters r1,r2,α1,α2,β1, and β2 are positive numbers. Moreover, due to the effect of piecewise-constant argument, we have integrated both sides of the Kolmogorov model, which is depicted in (1). On interval t∈n,n+1, (1) can be written as(2)dxtdt−r1xt=−r1xt2β1+α1yn,dytdt−r2yt=−r2yt2β2+α2xn,where for n=0,1,…, and t⟶n+1 resulted in the following solution:(3)xn+1=er1xn1+er1−1/β1+α1ynxn,yn+1=er2yn1+er2−1/β2+α2xnyn.It is important here to mention that discrete-time models directed by difference equations are more appropriate than the continuous ones in the case where populations have nonoverlapping generations, and also these models provide efficient computational results as compared to continuous models. Due to that reason in recent years, many scholars, researchers, and scientists have studied the dynamics of biological systems such as ratio-dependent predator-prey system, discrete hyperchaotic system, and host-parasitoid model. For instance, Bhattacharya and Saha [3] studied the dynamics characteristics of discrete Kolmogorov system. Cheng and Cao [4] explored dynamic characteristics of a discrete ratio-dependent prey-predator model. Jing and Yang [5] explored the dynamic characteristics of discrete prey-predator model. Kangalgil and Topsakal [6] explored dynamics characteristics of a discrete prey-predator system. Ran et al. [7] explored the Neimark-Sacker bifurcation of a stochastic discrete hyperchaotic system. Beddington et al. [8] explored dynamic characteristics in prey-predator models. Chen [9] studied the global dynamics and permanence of a discrete multispecies system. Lu and Zhang [10] investigated global attractivity and permanence of a discrete system with Holling type-II functional response. Fang and Chen [11] studied permanence of a discrete multispecies Lotka–Volterra model with delays. Fang et al. [12] investigated the dynamic characteristics of a discrete system. Jana and Samanta [13] studied prey-predator system in discrete-time scale using interval parameters.


### 1.2. Objective, Contributions, and Novelties

Motivated from aforementioned studies, the objective of the present work is to explore the global dynamics, bifurcations, and chaos in a discrete Kolmogorov model with piecewise-constant argument (3). More precisely, our main finding in this paper includesTopological classifications at fixed points of the discrete Kolmogorov model (3) by linear stability theoryExploration of periodic points of the discrete Kolmogorov model (3)Flip bifurcation analysis at interior fixed point by bifurcation theory.Investigation of chaos by feedback control methodTo explore boundedness and global attractivity of discrete Kolmogorov modelValidation of obtained results numerically


### 1.3. Paper Structure

The rest of the paper is structured as follows: Section2 relates with the investigation of topological classifications of discrete Kolmogorov model (3) at fixed points. In Section 3, periodic points of prime period-1 and period-2,3,…,n of the discrete Kolmogorov model (3) are explored, whereas comprehensive analysis of the bifurcation at fixed point S is explored in Section 4. In Section 5, chaos control is explored by feedback control method, whereas Section 6 is about the presentation of numerical simulations to validate obtained results. In Section 7, boundedness and global dynamics are explored. The conclusion and future work are given in Section 8.


## 1.1. Motivation and Literature Review

There are many symbiotic interactions existing in nature between two or more species in an ecosystem. Mutualism is an example of such interaction where interacted species get benefit from each other. For example, termites eat cellulose of wood but cannot digest it, and flagellates reside in the termite’s gut decomposing the cellulose of food and thus providing nutrients to termites. On the other hand, termite’s gut provides food and shelter to flagellates. It is pointed out that in theoretical ecology, mutualist behavior of symbiosis or mutualism is very significant [1]. This field is not widely studied as the other fields of mathematical biology even for two species, although its importance is equal to the other competitive interactions such as host-parasitoid and prey-predator interactions. So, this topic of mutualism system seems interesting to study. For instance, May [2] suggested the two-species hybrid continuous-time Kolmogorov model represented by the following system of differential equations:(1)dxdt=r1xt1−xtβ1+α1yt,dydt=r2yt1−ytβ2+α2xt,where t represents the greatest integer in t, and parameters r1,r2,α1,α2,β1, and β2 are positive numbers. Moreover, due to the effect of piecewise-constant argument, we have integrated both sides of the Kolmogorov model, which is depicted in (1). On interval t∈n,n+1, (1) can be written as(2)dxtdt−r1xt=−r1xt2β1+α1yn,dytdt−r2yt=−r2yt2β2+α2xn,where for n=0,1,…, and t⟶n+1 resulted in the following solution:(3)xn+1=er1xn1+er1−1/β1+α1ynxn,yn+1=er2yn1+er2−1/β2+α2xnyn.It is important here to mention that discrete-time models directed by difference equations are more appropriate than the continuous ones in the case where populations have nonoverlapping generations, and also these models provide efficient computational results as compared to continuous models. Due to that reason in recent years, many scholars, researchers, and scientists have studied the dynamics of biological systems such as ratio-dependent predator-prey system, discrete hyperchaotic system, and host-parasitoid model. For instance, Bhattacharya and Saha [3] studied the dynamics characteristics of discrete Kolmogorov system. Cheng and Cao [4] explored dynamic characteristics of a discrete ratio-dependent prey-predator model. Jing and Yang [5] explored the dynamic characteristics of discrete prey-predator model. Kangalgil and Topsakal [6] explored dynamics characteristics of a discrete prey-predator system. Ran et al. [7] explored the Neimark-Sacker bifurcation of a stochastic discrete hyperchaotic system. Beddington et al. [8] explored dynamic characteristics in prey-predator models. Chen [9] studied the global dynamics and permanence of a discrete multispecies system. Lu and Zhang [10] investigated global attractivity and permanence of a discrete system with Holling type-II functional response. Fang and Chen [11] studied permanence of a discrete multispecies Lotka–Volterra model with delays. Fang et al. [12] investigated the dynamic characteristics of a discrete system. Jana and Samanta [13] studied prey-predator system in discrete-time scale using interval parameters.


## 1.2. Objective, Contributions, and Novelties

Motivated from aforementioned studies, the objective of the present work is to explore the global dynamics, bifurcations, and chaos in a discrete Kolmogorov model with piecewise-constant argument (3). More precisely, our main finding in this paper includesTopological classifications at fixed points of the discrete Kolmogorov model (3) by linear stability theoryExploration of periodic points of the discrete Kolmogorov model (3)Flip bifurcation analysis at interior fixed point by bifurcation theory.Investigation of chaos by feedback control methodTo explore boundedness and global attractivity of discrete Kolmogorov modelValidation of obtained results numerically


## 1.3. Paper Structure

The rest of the paper is structured as follows: Section2 relates with the investigation of topological classifications of discrete Kolmogorov model (3) at fixed points. In Section 3, periodic points of prime period-1 and period-2,3,…,n of the discrete Kolmogorov model (3) are explored, whereas comprehensive analysis of the bifurcation at fixed point S is explored in Section 4. In Section 5, chaos control is explored by feedback control method, whereas Section 6 is about the presentation of numerical simulations to validate obtained results. In Section 7, boundedness and global dynamics are explored. The conclusion and future work are given in Section 8.


## 2. Topological Classifications of Discrete Kolmogorov Model (3) at Fixed Points

Here, local dynamical properties with topological classifications at fixed points are explored inℝ+2=x,y:x,y≥0. For this, first it is easy to verify that ∀r1,r2,α1,α2,β1,β2 discrete Kolmogorov model (3) has trivial fixed point P=0,0, boundary fixed points Q=0,β2, and R=β1,0, but if α1<1/α2, then it has an interior fixed point S=β1+α1β2/1−α1α2,β2+α2β1/1−α1α2. Additionally, the variational matrix V at fixed point x,y under the following map,(4)f1,f2↦xn+1,yn+1,where(5)f1=er1xβ1+α1yβ1+α1y+er1−1x,f2=er2yβ2+α2xβ2+α2x+er2−1y,is(6)V=er1β1+α1y2β1+α1y+er1−1x2α1er1er1−1x2β1+α1y+er1−1x2α2er2er2−1y2β2+α2x+er2−1y2er2β2+α2x2β2+α2x+er2−1y2.Now, topological classifications at fixed pointsP,Q,R, and S are explored for the completion of this section.


### 2.1. Topological Classifications atP

The variational matrix at fixed pointP is(7)VP=er100er2,whose eigenvalues are(8)λ1=er1,λ2=er2.Thus, the topological classifications at P are summarized as Table 1.Table 1
Topological classifications at fixed pointP.
PropertiesRespective parametric conditionsSourceSinceλ1=er1>1 and λ2=er2>1Remark 1.
Fixed pointP of the discrete Kolmogorov model (3) is never sink, saddle, and nonhyperbolic.


### 2.2. Topological Classifications atQ

The variational matrix at fixed pointQ is(9)VQ=er10α2er2−1er21er2,whose eigenvalues are(10)λ1=er1,λ2=1er2.So, the topological classifications atQ of the discrete Kolmogorov model (3) can be summarized as Table 2.Table 2
Topological classifications at fixed pointQ.
PropertiesRespective parametric conditionsSaddleλ1=er1>1,λ2=1/er2<1Remark 2.
Fixed pointQ of the discrete Kolmogorov model (3) is never sink, source, and nonhyperbolic.


### 2.3. Topological Classifications atR

The variational matrix at fixed pointR is(11)VR=1er1α1er1−1er10er2,whose eigenvalues are(12)λ1=1er1,λ2=er2.Based on eigenvaluesλ1,2, we will summarize the topological classifications at Q of the discrete-time Kolmogorov model (3) as Table 3.Table 3
Topological classifications at fixed pointR.
PropertiesRespective parametric conditionsSaddleλ1=1/er1<1,λ2=er2>1Remark 3.
Fixed pointR of the discrete Kolmogorov model (3) is never sink, source, and nonhyperbolic.


### 2.4. Topological Classifications atS

The variational matrix at fixed pointS is(13)VS=1er1α1er1−1er1α2er2−1er21er2,with corresponding characteristic equation of the form,(14)λ2−ϱ1˜λ+ϱ2˜=0,where(15)ϱ1˜=er1+er2er1+r2,ϱ2˜=1−α1α2er1−1er2−1er1+r2.Finally, roots of (14) are(16)λ1,2=−ϱ1˜±Δ2,where(17)Δ=ϱ1˜2−4ϱ2˜,=er1−er22+4α1α2er1+r2er1−1er2−1er1+r22>0.SinceΔ>0, therefore it is important here to note that fixed point S is never stable focus, unstable focus, and nonhyperbolic. So, we will summarize the topological classifications of discrete Kolmogorov model (3) at S as follows.


## 2.1. Topological Classifications atP

The variational matrix at fixed pointP is(7)VP=er100er2,whose eigenvalues are(8)λ1=er1,λ2=er2.Thus, the topological classifications at P are summarized as Table 1.Table 1
Topological classifications at fixed pointP.
PropertiesRespective parametric conditionsSourceSinceλ1=er1>1 and λ2=er2>1Remark 1.
Fixed pointP of the discrete Kolmogorov model (3) is never sink, saddle, and nonhyperbolic.


## 2.2. Topological Classifications atQ

The variational matrix at fixed pointQ is(9)VQ=er10α2er2−1er21er2,whose eigenvalues are(10)λ1=er1,λ2=1er2.So, the topological classifications atQ of the discrete Kolmogorov model (3) can be summarized as Table 2.Table 2
Topological classifications at fixed pointQ.
PropertiesRespective parametric conditionsSaddleλ1=er1>1,λ2=1/er2<1Remark 2.
Fixed pointQ of the discrete Kolmogorov model (3) is never sink, source, and nonhyperbolic.


## 2.3. Topological Classifications atR

The variational matrix at fixed pointR is(11)VR=1er1α1er1−1er10er2,whose eigenvalues are(12)λ1=1er1,λ2=er2.Based on eigenvaluesλ1,2, we will summarize the topological classifications at Q of the discrete-time Kolmogorov model (3) as Table 3.Table 3
Topological classifications at fixed pointR.
PropertiesRespective parametric conditionsSaddleλ1=1/er1<1,λ2=er2>1Remark 3.
Fixed pointR of the discrete Kolmogorov model (3) is never sink, source, and nonhyperbolic.


## 2.4. Topological Classifications atS

The variational matrix at fixed pointS is(13)VS=1er1α1er1−1er1α2er2−1er21er2,with corresponding characteristic equation of the form,(14)λ2−ϱ1˜λ+ϱ2˜=0,where(15)ϱ1˜=er1+er2er1+r2,ϱ2˜=1−α1α2er1−1er2−1er1+r2.Finally, roots of (14) are(16)λ1,2=−ϱ1˜±Δ2,where(17)Δ=ϱ1˜2−4ϱ2˜,=er1−er22+4α1α2er1+r2er1−1er2−1er1+r22>0.SinceΔ>0, therefore it is important here to note that fixed point S is never stable focus, unstable focus, and nonhyperbolic. So, we will summarize the topological classifications of discrete Kolmogorov model (3) at S as follows.


## 3. Exploration of Periodic Points

In the following proposition, periodic points of prime period-1 of the discrete Kolmogorov model (3) are explored.Proposition 1.
Fixed pointsP,Q,R, and S of the Kolmogorov model (3) are periodic points of prime period-1.Proof.
From (3), defining(18)F≔f1,f2,where f1 and f2 are defined in (5), after some computations, one gets(19)F|P=0,0=P,(20)F|Q=0,β2=Q,(21)F|R=β1,0=R,(22)F|S=β1+α1β2/1−α1α2,β2+α2β1/1−α1α2=S.
From (19)–(22), one can summarize that P,Q,R, and S of the Kolmogorov model (3) are periodic points of prime period-1.In the following proposition, periodic points of period-2,3,…,n of the discrete-time Kolmogorov model (3) are explored.Proposition 2.
Fixed pointsP,Q,R, and S of the Kolmogorov model (3) are periodic points of period-2,3,…,n.Proof.
After some straightforward computations, from (18), one gets(23)F2=er1f1β1+α1f2β1+α1f2+er1−1f1,er2f2β2+α2f1β2+α2f1+er2−1f2⇒F2|P=0,0=P,F3=er1f12β1+α1f22β1+α1f22+er1−1f12,er2f22β2+α2f12β2+α2f12+er2−1f22⇒F3|P=0,0=P,⋮Fn=er1f1n−1β1+α1f2n−1β1+α1f2n−1+er1−1f1n−1,er2f2n−1β2+α2f1n−1β2+α2f1n−1+er2−1f2n−1⇒Fn|P=0,0=P.
From (23), one concludes the required statement. Similarly, one can show that fixed points Q,R, and S of the discrete Kolmogorov model (3) are periodic points of period-2,3,…,n.


## 4. Analysis of Bifurcation

It is easy to note that there exists no bifurcation at fixed pointsP,Q, and R, but in the following subsection, we explore that at S discrete, Kolmogorov model (3) undergoes only flip bifurcation by bifurcation theory [14–20].


### 4.1. Flip Bifurcation aboutS

From Table4, it is noted that equilibrium point S of the discrete Kolmogorov model (3) is nonhyperbolic if α1=1+er1+er2+er1+r2/α2er1−1er2−1. Therefore, eigenvalues of VS at nonhyperbolic condition α1=1+er1+er2+er1+r2/α2er1−1er2−1 are computed and one gets λ1|α1=1+er1+er2+er1+r2/α2er1−1er2−1=−1, but λ2|α1=1+er1+er2+er1+r2/α2er1−1er2−1=er1+er2+er1+r2/er1+r2≠1or−1 which conclude that at fixed point S, Kolmogorov model (3) undergoes flip bifurcation if r1,r2,α1,α2,β1,β2 are located in the following set:(24)ℱ|S=r1,r2,α1,α2,β1,β2,α1=1+er1+er2+er1+r2α2er1−1er2−1.Table 4
Topological classifications at fixed pointS.
PropertiesRespective parametric conditionsStable nodeα1>1+er1+er2+er1+r2/α2er1−1er2−1Unstable node0<α1<1+er1+er2+er1+r2/α2er1−1er2−1Nonhyperbolicα1=1+er1+er2+er1+r2/α2er1−1er2−1Hereafter, in the following, we will present comprehensive flip bifurcation analysis atS of the Kolmogorov model (3).Theorem 1.
Ifr1,r2,α1,α2,β1,β2∈ℱ|S, then the discrete Kolmogorov model (3) undergoes the flip bifurcation.Proof.
It is recalled from Table4 that if α1=1+er1+er2+er1+r2/α2er1−1er2−1, then S is nonhyperbolic where Δ=er1−er22+4α1α2er1+r2er1−1er2−1/er1+r22>0. Moreover, at α1=1+er1+er2+er1+r2/α2er1−1er2−1, one gets λ1|α1=1+er1+er2+er1+r2/α2er1−1er2−1=−1, where as λ2|α1=1+er1+er2+er1+r2/α2er1−1er2−1=er1+er2+er1+r2/er1+r2≠1or−1, which gives the existence of flip bifurcation at S by choosing α1 as a bifurcation parameter. So, if α1 varies in a neighborhood of α1∗, then model (3) takes the form,(25)xn+1=er1xnβ1+α1∗+ϵynβ1+α1∗+ϵyn+er1−1xn,yn+1=er2ynβ2+α2xnβ2+α2xn+er2−1yn.
Now, it is noted that by using transformation,(26)un=xn−x∗,vn=yn−y∗,one can transform S to P where (25) takes the following form:(27)un+1=D11^un+D12^vn+D13^un2+D14^unvn+D15^vn2+γ01unϵ+γ02vnϵ+γ03un2ϵ+γ04unvnϵ+γ05vn2ϵ,vn+1=D21^un+D22^vn+D23^un2+D24^unvn+D25^vn2,where(28)D11^=er1β1+α1∗y∗2β1+α1∗y∗+er1−1x∗2,D12^=er1α1∗er1−1x∗2β1+α1∗y∗+er1−1x∗2,D13^=−er1er1−1β1+α1∗y∗2β1+α1∗y∗+er1−1x∗3,D14^=er1α1∗er1−1β1+α1∗y∗x∗β1+α1∗y∗+er1−1x∗3,D15^=−α1∗2er1er1−1x∗2β1+α1∗y∗+er1−1x∗3,γ01=2er1β1+α1∗y∗y∗β1+α1∗y∗+er1−1x∗2,γ02=er1er1−1x∗2β1+α1∗y∗+er1−1x∗2,γ03=−2er1er1−1β1+α1∗y∗y∗β1+α1∗y∗+er1−1x∗3,γ04=er1er1−1β1+2α1∗y∗x∗β1+α1∗y∗+er1−1x∗3,γ05=−2er1er1−1x∗2α1∗β1+α1∗y∗+er1−1x∗3,D21^=α2er2er2−1y∗2β2+α2x∗+er2−1y∗2,D22^=er2β2+α2x∗2β2+α2x∗+er2−1y∗2,D23^=−α22er2er2−1y∗2β2+α2x∗+er2−1y∗3,D24^=α2er2er2−1β2+α2x∗y∗β2+α2x∗+er2−1y∗3,D25^=−er2er2−1β2+α2x∗2β2+α2x∗+er2−1y∗3.
Now, (27) takes the following form:(29)Xn+1Yn+1=−100λ2XnYn+P^ϵQ^ϵ,where(30)P^ϵ=D13^λ2−D11^−D12D23^D12^1+λ2un2+D14^λ2−D11^−D12^D24^D12^1+λ2unvn+D15^λ2−D11^−D12D25^D12^1+λ2vn2+γ01λ2−D11^D12^1+λ2unϵ+γ02λ2−D11^D12^1+λ2vnϵ+γ03λ2−D11^D12^1+λ2un2ϵ+γ04λ2−D11^D12^1+λ2unvnϵ+γ05λ2−D11^D12^1+λ2vn2ϵ,Q^ϵ=D13^1+D11^+D12D23^D12^1+λ2un2+D14^1+D11^+D12^D24^D12^1+λ2unvn+D15^1+D11^+D12D25^D12^1+λ2vn2+γ011+D11^D12^1+λ2unϵ+γ021+D11^D12^1+λ2vnϵ+γ031+D11^D12^1+λ2un2ϵ+γ041+D11^D12^1+λ2unvnϵ+γ051+D11^D12^1+λ2vn2ϵ,un2=D12^2Xn2+2XnYn+Yn2,unvn=−D12^1+D11^Xn2+D12^λ2−D11^−D12^1+D11^XnYn+D12^λ2−D11^Yn2,vn2=1+D11^2Xn2−21+D11^λ2−D11^XnYn+λ2−D11^2Yn2,unϵ=D12^Xnϵ+D12^Ynϵ,vnϵ=−1−D11^Xnϵ+λ2−D11^Ynϵ,un2ϵ=D12^2Xn2ϵ+2XnYnϵ+Yn2ϵ,unvnϵ=−D12^1+D11^Xn2ϵ+D12^λ2−D11^−D12^1+D11^XnYnϵ+D12^λ2−D11^Yn2ϵ,vn2ϵ=1+D11^2Xn2ϵ−21+D11^λ2−D11^XnYnϵ+λ2−D11^2Yn2ϵ,by(31)unvn=D12^D12^−1−D11^λ2−D11^XnYn.
Now, for (29), center manifold McP about P is examined in a small neighborhood of ϵ, and therefore McP can be expressed as the following expression:(32)McP=Xn,Yn:Yn=h0^ϵ+h1^Xn2+h2^Xnϵ+h3^ϵ3+OXn+ϵ3.
Therefore, the computation yields(33)h0^=0,h1^=1+D11^D12^D13^−D14^1+D11^−D12^D24^1−λ22+D12^2D23^1−λ22+1+D11^2D15^1+D11^+D12^D25^D12^1−λ22,h2^=D12^γ011+D11^−γ021+D11^2D12^1−λ22,h3^=0.
Thus, the map (29) restricting to McP is(34)fxn=−xn+v1^xn2+v2^xnϵ+v3^xn2ϵ+v4^xnϵ2+v5^xn3+OXn+ϵ4,where(35)v1^=11+λ2D12^D13^λ2−D11^−1+D11^D14^λ2−D11^−D12^D24^−D12^2D23^−D25^1+D11^2+D15^λ2−D11^1+D11^2D12^,v2^=11+λ2γ01λ2−D11^−γ02λ2−D11^1+D11^D12^,v3^=11+λ22D12^h2^D13^λ2−D11^−2h2^D12^2D23^+h2^λ2−D11^−h2^1+D11^D14^λ2−D11^−D12^D24^+h1^γ01λ2−D11^−2h2^1+D11^λ2−D11^D15^λ2−D11^−D12^D25^D12^+h1^γ02λ2−D11^2D12^+D12^γ03λ2−D11^−γ041+D11^λ2−D11^+γ051+D11^2D12^λ2−D11^,v4^=h2^1+λ2γ01λ2−D11^+γ02λ2−D11^2D12^,v5^=11+λ22h1^D12^D13^λ2−D11^−D12^D23^+h1^λ2−D11^−h1^1+D11^D14^λ2−D11^−D12^D24^−2h1^λ2−D11^1+D11^D12^D15^λ2−D11^−D12^D25^.
In order to show (34) undergoes flip bifurcation, it is required that Q1,Q2 must be nonzero, i.e.,(36)Q1=∂2f∂xn∂ϵ+12∂f∂ϵ∂2f∂xn2|P,Q2=16∂3f∂xn3+12∂2f∂xn22|P.
After calculating, one gets(37)Q1=α2er1−1e2r2−1er1+er2+2er1+r22β2+α2β1β1α2er1−1er2−1+β21+er1+er2+er1+r2−1+er21+er1+er2+er1+r2≠0,(38)Q2=2h1^1+er1+er2+er1+r2er1+er2+er1+r2er2er1−1er2−1β1α2er1−1er2−1+β21+er1+er2+er1+r2×21−er1er1+1+er2er2−1+er1er1+21+er11+er2er2er1−1+er1β2+α2β121+er1+er2+er1+r2er1er2−1−1+er2er2−1+er1er1−21+er1e2r2−1er21+er1+er2+er1+r2+4er1+er2+er1+r22er21+er2er1−11+er1+er2+er1+r2er1β1α2er1−1er2−1+β21+er1+er2+er1+r2+1β2+α2β11+er11+er1+er2+er1+r2+1+er22−1+er1+er2+er1+r22er2−12,where(39)h1^=2er1+r21+er11+er1+er2+er1+r2er1+r22−er1+er2+er1+r22−1β2+α2β1α2+1+e2r2−e2r1+r2e2r1β1α2er1−1er2−1+1+er1+er2+er1+r2β2−2er1+r21+er1+er2+er1+r2α2er1−1er2−12β2+α2β1e2r1+r2−er1+er2+er1+r22+21+er12er1+r2e2r1+r2−er1+er2+er1+r221−er2e2r2e2r1β1α2er1−1er2−1+1+er1+er2+er1+r2β2−1+er1α2er1−1er2−1β2+α2β1.
In view of (38) and (39), if Q2≠0, then as r1,r2,α1,α2,β1,β2∈ℱ|S, discrete Kolmogorov model (3) undergoes a flip bifurcation. Additionally, period-2 points from S are stable (respectively unstable), if Q2>0respectivelyQ2<0.


## 4.1. Flip Bifurcation aboutS

From Table4, it is noted that equilibrium point S of the discrete Kolmogorov model (3) is nonhyperbolic if α1=1+er1+er2+er1+r2/α2er1−1er2−1. Therefore, eigenvalues of VS at nonhyperbolic condition α1=1+er1+er2+er1+r2/α2er1−1er2−1 are computed and one gets λ1|α1=1+er1+er2+er1+r2/α2er1−1er2−1=−1, but λ2|α1=1+er1+er2+er1+r2/α2er1−1er2−1=er1+er2+er1+r2/er1+r2≠1or−1 which conclude that at fixed point S, Kolmogorov model (3) undergoes flip bifurcation if r1,r2,α1,α2,β1,β2 are located in the following set:(24)ℱ|S=r1,r2,α1,α2,β1,β2,α1=1+er1+er2+er1+r2α2er1−1er2−1.Table 4
Topological classifications at fixed pointS.
PropertiesRespective parametric conditionsStable nodeα1>1+er1+er2+er1+r2/α2er1−1er2−1Unstable node0<α1<1+er1+er2+er1+r2/α2er1−1er2−1Nonhyperbolicα1=1+er1+er2+er1+r2/α2er1−1er2−1Hereafter, in the following, we will present comprehensive flip bifurcation analysis atS of the Kolmogorov model (3).Theorem 1.
Ifr1,r2,α1,α2,β1,β2∈ℱ|S, then the discrete Kolmogorov model (3) undergoes the flip bifurcation.Proof.
It is recalled from Table4 that if α1=1+er1+er2+er1+r2/α2er1−1er2−1, then S is nonhyperbolic where Δ=er1−er22+4α1α2er1+r2er1−1er2−1/er1+r22>0. Moreover, at α1=1+er1+er2+er1+r2/α2er1−1er2−1, one gets λ1|α1=1+er1+er2+er1+r2/α2er1−1er2−1=−1, where as λ2|α1=1+er1+er2+er1+r2/α2er1−1er2−1=er1+er2+er1+r2/er1+r2≠1or−1, which gives the existence of flip bifurcation at S by choosing α1 as a bifurcation parameter. So, if α1 varies in a neighborhood of α1∗, then model (3) takes the form,(25)xn+1=er1xnβ1+α1∗+ϵynβ1+α1∗+ϵyn+er1−1xn,yn+1=er2ynβ2+α2xnβ2+α2xn+er2−1yn.
Now, it is noted that by using transformation,(26)un=xn−x∗,vn=yn−y∗,one can transform S to P where (25) takes the following form:(27)un+1=D11^un+D12^vn+D13^un2+D14^unvn+D15^vn2+γ01unϵ+γ02vnϵ+γ03un2ϵ+γ04unvnϵ+γ05vn2ϵ,vn+1=D21^un+D22^vn+D23^un2+D24^unvn+D25^vn2,where(28)D11^=er1β1+α1∗y∗2β1+α1∗y∗+er1−1x∗2,D12^=er1α1∗er1−1x∗2β1+α1∗y∗+er1−1x∗2,D13^=−er1er1−1β1+α1∗y∗2β1+α1∗y∗+er1−1x∗3,D14^=er1α1∗er1−1β1+α1∗y∗x∗β1+α1∗y∗+er1−1x∗3,D15^=−α1∗2er1er1−1x∗2β1+α1∗y∗+er1−1x∗3,γ01=2er1β1+α1∗y∗y∗β1+α1∗y∗+er1−1x∗2,γ02=er1er1−1x∗2β1+α1∗y∗+er1−1x∗2,γ03=−2er1er1−1β1+α1∗y∗y∗β1+α1∗y∗+er1−1x∗3,γ04=er1er1−1β1+2α1∗y∗x∗β1+α1∗y∗+er1−1x∗3,γ05=−2er1er1−1x∗2α1∗β1+α1∗y∗+er1−1x∗3,D21^=α2er2er2−1y∗2β2+α2x∗+er2−1y∗2,D22^=er2β2+α2x∗2β2+α2x∗+er2−1y∗2,D23^=−α22er2er2−1y∗2β2+α2x∗+er2−1y∗3,D24^=α2er2er2−1β2+α2x∗y∗β2+α2x∗+er2−1y∗3,D25^=−er2er2−1β2+α2x∗2β2+α2x∗+er2−1y∗3.
Now, (27) takes the following form:(29)Xn+1Yn+1=−100λ2XnYn+P^ϵQ^ϵ,where(30)P^ϵ=D13^λ2−D11^−D12D23^D12^1+λ2un2+D14^λ2−D11^−D12^D24^D12^1+λ2unvn+D15^λ2−D11^−D12D25^D12^1+λ2vn2+γ01λ2−D11^D12^1+λ2unϵ+γ02λ2−D11^D12^1+λ2vnϵ+γ03λ2−D11^D12^1+λ2un2ϵ+γ04λ2−D11^D12^1+λ2unvnϵ+γ05λ2−D11^D12^1+λ2vn2ϵ,Q^ϵ=D13^1+D11^+D12D23^D12^1+λ2un2+D14^1+D11^+D12^D24^D12^1+λ2unvn+D15^1+D11^+D12D25^D12^1+λ2vn2+γ011+D11^D12^1+λ2unϵ+γ021+D11^D12^1+λ2vnϵ+γ031+D11^D12^1+λ2un2ϵ+γ041+D11^D12^1+λ2unvnϵ+γ051+D11^D12^1+λ2vn2ϵ,un2=D12^2Xn2+2XnYn+Yn2,unvn=−D12^1+D11^Xn2+D12^λ2−D11^−D12^1+D11^XnYn+D12^λ2−D11^Yn2,vn2=1+D11^2Xn2−21+D11^λ2−D11^XnYn+λ2−D11^2Yn2,unϵ=D12^Xnϵ+D12^Ynϵ,vnϵ=−1−D11^Xnϵ+λ2−D11^Ynϵ,un2ϵ=D12^2Xn2ϵ+2XnYnϵ+Yn2ϵ,unvnϵ=−D12^1+D11^Xn2ϵ+D12^λ2−D11^−D12^1+D11^XnYnϵ+D12^λ2−D11^Yn2ϵ,vn2ϵ=1+D11^2Xn2ϵ−21+D11^λ2−D11^XnYnϵ+λ2−D11^2Yn2ϵ,by(31)unvn=D12^D12^−1−D11^λ2−D11^XnYn.
Now, for (29), center manifold McP about P is examined in a small neighborhood of ϵ, and therefore McP can be expressed as the following expression:(32)McP=Xn,Yn:Yn=h0^ϵ+h1^Xn2+h2^Xnϵ+h3^ϵ3+OXn+ϵ3.
Therefore, the computation yields(33)h0^=0,h1^=1+D11^D12^D13^−D14^1+D11^−D12^D24^1−λ22+D12^2D23^1−λ22+1+D11^2D15^1+D11^+D12^D25^D12^1−λ22,h2^=D12^γ011+D11^−γ021+D11^2D12^1−λ22,h3^=0.
Thus, the map (29) restricting to McP is(34)fxn=−xn+v1^xn2+v2^xnϵ+v3^xn2ϵ+v4^xnϵ2+v5^xn3+OXn+ϵ4,where(35)v1^=11+λ2D12^D13^λ2−D11^−1+D11^D14^λ2−D11^−D12^D24^−D12^2D23^−D25^1+D11^2+D15^λ2−D11^1+D11^2D12^,v2^=11+λ2γ01λ2−D11^−γ02λ2−D11^1+D11^D12^,v3^=11+λ22D12^h2^D13^λ2−D11^−2h2^D12^2D23^+h2^λ2−D11^−h2^1+D11^D14^λ2−D11^−D12^D24^+h1^γ01λ2−D11^−2h2^1+D11^λ2−D11^D15^λ2−D11^−D12^D25^D12^+h1^γ02λ2−D11^2D12^+D12^γ03λ2−D11^−γ041+D11^λ2−D11^+γ051+D11^2D12^λ2−D11^,v4^=h2^1+λ2γ01λ2−D11^+γ02λ2−D11^2D12^,v5^=11+λ22h1^D12^D13^λ2−D11^−D12^D23^+h1^λ2−D11^−h1^1+D11^D14^λ2−D11^−D12^D24^−2h1^λ2−D11^1+D11^D12^D15^λ2−D11^−D12^D25^.
In order to show (34) undergoes flip bifurcation, it is required that Q1,Q2 must be nonzero, i.e.,(36)Q1=∂2f∂xn∂ϵ+12∂f∂ϵ∂2f∂xn2|P,Q2=16∂3f∂xn3+12∂2f∂xn22|P.
After calculating, one gets(37)Q1=α2er1−1e2r2−1er1+er2+2er1+r22β2+α2β1β1α2er1−1er2−1+β21+er1+er2+er1+r2−1+er21+er1+er2+er1+r2≠0,(38)Q2=2h1^1+er1+er2+er1+r2er1+er2+er1+r2er2er1−1er2−1β1α2er1−1er2−1+β21+er1+er2+er1+r2×21−er1er1+1+er2er2−1+er1er1+21+er11+er2er2er1−1+er1β2+α2β121+er1+er2+er1+r2er1er2−1−1+er2er2−1+er1er1−21+er1e2r2−1er21+er1+er2+er1+r2+4er1+er2+er1+r22er21+er2er1−11+er1+er2+er1+r2er1β1α2er1−1er2−1+β21+er1+er2+er1+r2+1β2+α2β11+er11+er1+er2+er1+r2+1+er22−1+er1+er2+er1+r22er2−12,where(39)h1^=2er1+r21+er11+er1+er2+er1+r2er1+r22−er1+er2+er1+r22−1β2+α2β1α2+1+e2r2−e2r1+r2e2r1β1α2er1−1er2−1+1+er1+er2+er1+r2β2−2er1+r21+er1+er2+er1+r2α2er1−1er2−12β2+α2β1e2r1+r2−er1+er2+er1+r22+21+er12er1+r2e2r1+r2−er1+er2+er1+r221−er2e2r2e2r1β1α2er1−1er2−1+1+er1+er2+er1+r2β2−1+er1α2er1−1er2−1β2+α2β1.
In view of (38) and (39), if Q2≠0, then as r1,r2,α1,α2,β1,β2∈ℱ|S, discrete Kolmogorov model (3) undergoes a flip bifurcation. Additionally, period-2 points from S are stable (respectively unstable), if Q2>0respectivelyQ2<0.


## 5. Chaos Control

This section is purely dedicated for the exploration of chaos control in the sense of state feedback control method [21, 22]. After adding control force ut, discrete Kolmogorov model (3) takes the form,(40)xn+1=er1xnβ1+α1ynβ1+α1yn+er1−1xn+un,yn+1=er2ynβ2+α2xnβ2+α2xn+er2−1yn,where un=−l1xn−β1+α1β2/1−α1α2−l2yn−β2+α2β1/1−α1α2. It is noted here that l1,l2 denotes feedback gains. Moreover, VS for the controlled system (40) is(41)VSC=1er1−l1α1er1−1er1−l2α2er2−1er21er2.Ifλ1,2 denotes the characteristic roots of VSC at S, then(42)λ1+λ2=−l1+1er1+1er2,(43)λ1λ2=−1er1l1+α2er2−1er2l2+1er1+r2−α1α2er1−1er2−1er1+r2.Now, the answer ofλ1=±1 and λ1λ2=1 determines the strains of marginal stability, and these prerequisites give that λ1,2<1. If λ1λ2=1, then from (43), one gets(44)L1:−1er1l1+α2er2−1er2l2+1er1+r2−α1α2er1−1er2−1er1+r2−1=0.Ifλ1=1, then from (42) and (43), one gets(45)L2:1−1er1l1+α2er2−1er2l2+1er1+r2−1er1−1er2−α1α2er1−1er2−1er1+r2+1=0.Finally, ifλ1=−1, then from (42) and (43), one gets(46)L3:−1+1er1l1+α2er2−1er2l2+1er1+r2+1er1+1er2−α1α2er1−1er2−1er1+r2+1=0.Therefore, from (44)–(46), lines L1,L2, and L3 in l1,l2-plane give the triangular region, which further gives the fact that λ1,2<1.


## 6. Numerical Simulations

We present numerical simulations to validate the corresponding obtained results. For instance, ifr1=5.25,r2=5.3,α2=0.4,β1=2.9,β2=2.9, then from nonhyperbolic condition, one gets α1=1+er1+er2+er1+r2/α2er1−1er2−1=2.5517238126301827. From theoretical discussion, S=4.284090909090909,4.613636363636363 of the discrete Kolmogorov model (3) is a stable node if α1>2.5517238126301827. So, if one chose the bifurcation value α1=2.652>2.5517238126301827, then it is clear from Figure 1(a) that fixed point S=4.284090909090909,4.613636363636363 of Kolmogorov model (3) is a stable node. Moreover, for some more parametric values, if α1=2.6587,2.7,2.765>2.5517238126301827, then Figures 1(b)–1(d) also indicate that the fixed point S=4.284090909090909,4.613636363636363 of Kolmogorov model (3) is also a stable node. Further, if 0<α1<2.5517238126301827, then fixed point S=4.284090909090909,4.613636363636363 becomes unstable and meanwhile flip bifurcation occurs, i.e., if α1=0.967<2.5517238126301827, then by mathematical computation from (37), one gets Q1=−0.19445133540974902≠0. Moreover, from (38), one gets Q2=0.5853373936670135>0, which indicates the fact that stable period-2 points bifurcate from S, and hence flip bifurcation diagram along with maximum Lyapunov exponents are plotted and presented in Figure 2. Moreover, 3D flip bifurcation diagrams are presented in Figure 3. Finally, Figure 4 shows the complex dynamics with orbits of period-8,9,10,14.Figure 1
Stable focus of the Kolmogorov model (3). (a) α1=2.652 with (0.24, 0.25), (b) α1=2.6587 with (0.294, 0.285), (c) α1=2.7 with (0.09, 0.09), and (d) α1=2.765 with (0.092, 0.093).
(a)(b)(c)(d)Figure 2
(a, b) Flip bifurcation diagram of the Kolmogorov model (3) with α1∈0.01,2.9. (c) Maximum Lyapunov exponents corresponding with 0.14,0.15.
(a)(b)(c)Figure 3
3D flip bifurcation diagram of the Kolmogorov model (3) with α1∈0.01,2.9.
(a)(b)(c)(d)Figure 4
Complex dynamics of the Kolmogorov model (3). (a)α1=0.967 with (2.9, 3.8), (b) α1=1.235 with (3.7, 3.8), (c) α1=1.7 with (2.9, 1.8), and (d) α1=1.956 with (3.7, 3.8).
(a)(b)(c)(d)Hereafter, we will prove the validity of obtained results in Section5. For instance, if r1=5.25,r2=5.3,α1=1.98,α2=0.4,β1=2.9,β2=2.9, then from (44)–(46), one gets(47)L1:−2.179602154437681−0.005332258840143003l1+0.3979708098304923l2=0,(48)L2:−0.19000738870159323+0.994667741159857l1+0.3979708098304923l2=0,(49)L3:−0.16919692017376872−1.005332258840143l1+0.3979708098304923l2=0.Hence, lines that are presented in (47)–(49) determine triangular region that gives λ1,2<1(see Figure 5).Figure 5
Region of stability whereλ1,2<1.


## 7. Boundedness and Global Attractivity of the Kolmogorov Model (3)

Boundedness with construction of invariant rectangle and global dynamics at interior fixed point of the discrete Kolmogorov model (3) are explored in this section.


### 7.1. Boundedness Along with Construction of Invariant Rectangle

Theorem 2.
Every positive solutionxn,ynn=0∞ of the Kolmogorov model (3) is bounded if(50)0<α1<er1−1er2−1α2er1+r2.Proof.
Ifxn,ynn=0∞ is positive solution of the discrete Kolmogorov model (3), then(51)xn+1≤er1er1−1β1+α1yn,yn+1≤er2er2−1β2+α2xn.
From2nd inequality of (51), one gets(52)yn≤er2er2−1β1+α2xn−1.
Utilizing (52) in the 1st inequality of (51), one gets(53)xn+1≤β1er1er1−1+α1β2er1+r2er1−1er2−1+α1α2er1+r2er1−1er2−1xn−1,which further gives the following solution:(54)xn+1≤c1α1α2er1+r2er1−1er2−1n+c2−α1α2er1+r2er1−1er2−1n+β1er1/er1−1+α1β2er1+r2/er1−1er2−11−α1α2er1+r2/er1−1er2−1.
Now, from the1st inequality of (51), one gets(55)xn≤er1er1−1β1+α1yn−1.
Utilizing (55) in the 2nd inequality of (51), one gets(56)yn+1≤β2er2er2−1+α2β1er1+r2er1−1er2−1+α1α2er1+r2er1−1er2−1yn−1,whose solution is(57)yn+1≤c3α1α2er1+r2er1−1er2−1n+c4−α1α2er1+r2er1−1er2−1n+β2er2/er2−1+α2β1er1+r2/er1−1er2−11−α1α2er1+r2/er1−1er2−1.
If (50) holds, then from (54) and (57), one gets(58)0≤xn≤β1er1/er1−1+α1β2er1+r2/er1−1er2−11−α1α2er1+r2/er1−1er2−1,0≤yn≤β2er2/er2−1+α2β1er1+r2/er1−1er2−11−α1α2er1+r2/er1−1er2−1.Theorem 3.
For the discrete Kolmogorov model (3), the invariant rectangle is 0,β1er1/er1−1+α1β2er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1×0,β2er2/er2−1+α2β1er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1.Proof.
Followed by induction.


### 7.2. Attractivity atS

From (5), it is noted that f1 is increasing in x∈0,β1er1/er1−1+α1β2er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1resp⋅y∈0,β2er2/er2−1+α2β1er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1 for some fixed y∈0,β2er2/er2−1+α2β1er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1resp⋅x∈0,β1er1/er1−1+α1β2er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1. Also, f2 is increasing in x∈0,β1er1/er1−1+α1β2er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1resp⋅y∈0,β2er2/er2−1+α2β1er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1 for fixed y∈0,β2er2/er2−1+α2β1er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1resp⋅x∈0,β1er1/er1−1+α1β2er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1. This motivates the study of global attractivity at S of discrete Kolmogorov model (3), as follows.Theorem 4.
S of discrete Kolmogorov model (3) is the global attractor.Proof.
Sincef1 and f2satisfied the above monotonic characteristics and so ifm1,M1,m2,M2is the solution of following system:(59)m1=f1m1,m2,M1=f1M1,M2,m2=f2m1,m2,M2=f2M1,M2.
In view of (3) and (59), one gets(60)m1=er1m1β1+α1m2β1+α1m2+er1−1m1,M1=er1M1β1+α1M2β1+α1M2+er1−1M1,(61)m2=er2m2β2+α2m1β2+α2m1+er2−1m2,M2=er2M2β2+α2M1β2+α2M1+er2−1M2.
From (60), one has(62)β1+α1m2+er1−1m1=er1β1+α1m2,β1+α1M2+er1−1M1=er1β1+α1M2.
From (61), one has(63)β2+α2m1+er1−1m2=er2β2+α2m1,β2+α2M1+er2−1M2=er2β2+α2M1.
From (62), one gets(64)m1−M1=α1m2−M2.
Similarly, from (63), one gets(65)m2−M2=α2m1−M1.
Form (64) and (65), one gets α1α2−1m1−M1=0, which implies that m1=M1. Finally, from (65), one gets m2=M2. So, S of discrete Kolmogorov model (3) is a global attractor by Theorem 1.16 of [23].


## 7.1. Boundedness Along with Construction of Invariant Rectangle

Theorem 2.
Every positive solutionxn,ynn=0∞ of the Kolmogorov model (3) is bounded if(50)0<α1<er1−1er2−1α2er1+r2.Proof.
Ifxn,ynn=0∞ is positive solution of the discrete Kolmogorov model (3), then(51)xn+1≤er1er1−1β1+α1yn,yn+1≤er2er2−1β2+α2xn.
From2nd inequality of (51), one gets(52)yn≤er2er2−1β1+α2xn−1.
Utilizing (52) in the 1st inequality of (51), one gets(53)xn+1≤β1er1er1−1+α1β2er1+r2er1−1er2−1+α1α2er1+r2er1−1er2−1xn−1,which further gives the following solution:(54)xn+1≤c1α1α2er1+r2er1−1er2−1n+c2−α1α2er1+r2er1−1er2−1n+β1er1/er1−1+α1β2er1+r2/er1−1er2−11−α1α2er1+r2/er1−1er2−1.
Now, from the1st inequality of (51), one gets(55)xn≤er1er1−1β1+α1yn−1.
Utilizing (55) in the 2nd inequality of (51), one gets(56)yn+1≤β2er2er2−1+α2β1er1+r2er1−1er2−1+α1α2er1+r2er1−1er2−1yn−1,whose solution is(57)yn+1≤c3α1α2er1+r2er1−1er2−1n+c4−α1α2er1+r2er1−1er2−1n+β2er2/er2−1+α2β1er1+r2/er1−1er2−11−α1α2er1+r2/er1−1er2−1.
If (50) holds, then from (54) and (57), one gets(58)0≤xn≤β1er1/er1−1+α1β2er1+r2/er1−1er2−11−α1α2er1+r2/er1−1er2−1,0≤yn≤β2er2/er2−1+α2β1er1+r2/er1−1er2−11−α1α2er1+r2/er1−1er2−1.Theorem 3.
For the discrete Kolmogorov model (3), the invariant rectangle is 0,β1er1/er1−1+α1β2er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1×0,β2er2/er2−1+α2β1er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1.Proof.
Followed by induction.


## 7.2. Attractivity atS

From (5), it is noted that f1 is increasing in x∈0,β1er1/er1−1+α1β2er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1resp⋅y∈0,β2er2/er2−1+α2β1er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1 for some fixed y∈0,β2er2/er2−1+α2β1er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1resp⋅x∈0,β1er1/er1−1+α1β2er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1. Also, f2 is increasing in x∈0,β1er1/er1−1+α1β2er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1resp⋅y∈0,β2er2/er2−1+α2β1er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1 for fixed y∈0,β2er2/er2−1+α2β1er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1resp⋅x∈0,β1er1/er1−1+α1β2er1+r2/er1−1er2−1/1−α1α2er1+r2/er1−1er2−1. This motivates the study of global attractivity at S of discrete Kolmogorov model (3), as follows.Theorem 4.
S of discrete Kolmogorov model (3) is the global attractor.Proof.
Sincef1 and f2satisfied the above monotonic characteristics and so ifm1,M1,m2,M2is the solution of following system:(59)m1=f1m1,m2,M1=f1M1,M2,m2=f2m1,m2,M2=f2M1,M2.
In view of (3) and (59), one gets(60)m1=er1m1β1+α1m2β1+α1m2+er1−1m1,M1=er1M1β1+α1M2β1+α1M2+er1−1M1,(61)m2=er2m2β2+α2m1β2+α2m1+er2−1m2,M2=er2M2β2+α2M1β2+α2M1+er2−1M2.
From (60), one has(62)β1+α1m2+er1−1m1=er1β1+α1m2,β1+α1M2+er1−1M1=er1β1+α1M2.
From (61), one has(63)β2+α2m1+er1−1m2=er2β2+α2m1,β2+α2M1+er2−1M2=er2β2+α2M1.
From (62), one gets(64)m1−M1=α1m2−M2.
Similarly, from (63), one gets(65)m2−M2=α2m1−M1.
Form (64) and (65), one gets α1α2−1m1−M1=0, which implies that m1=M1. Finally, from (65), one gets m2=M2. So, S of discrete Kolmogorov model (3) is a global attractor by Theorem 1.16 of [23].


## 8. Conclusion and Future Work

The work is about the local dynamical characteristics at fixed points, existence of periodic points, boundedness, global attractivity, chaos control, and bifurcations of a discrete Kolmogorov model with piecewise-constant argument. By existing linear stability theory, it is proved that fixed pointP is never sink, saddle, and nonhyperbolic, but it is a source; Q is never sink, source, and nonhyperbolic, but it is a saddle; R is never sink, source, and nonhyperbolic, but it is a saddle; and finally, S is a stable node if α1>1+er1+er2+er1+r2/α2er1−1er2−1, unstable node if 0<α1<1+er1+er2+er1+r2/α2er1−1er2−1, and nonhyperbolic if α1=1+er1+er2+er1+r2/α2er1−1er2−1, but it is never stable focus, unstable focus, and nonhyperbolic. Further, it is proved that fixed points P,Q,R, and S are periodic having prime period-1. Then, in order to understand under consideration model deeply, we also explored the occurrence of bifurcations at S. It is proved that at S, the model undergoes flip bifurcation if α1,α2,r1,r2,β1,β2∈ℱ|S. Further, state feedback control method is utilized in order to stabilize chaos existing in discrete Kolmogorov model (3). Moreover, numerical simulations are also presented to validate obtained results. Finally, boundedness and global attractivity of the Kolmogorov model (3) at S are also explored. The use of robust control to improve the control process and stabilization for the two-species hybrid Kolmogorov model are our next aim to study.



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