# Global Dynamics and Bifurcations Analysis of a Two-Dimensional Discrete-Time Lotka-Volterra Model

Authors: A. Q. Khan; M. N. Qureshi
Journal: Complexity
 (2018)

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## Abstract

In this paper, global dynamics and bifurcations of a two-dimensional discrete-time Lotka-Volterra model have been studied in the closed first quadrantR2. It is proved that the discrete model has three boundary equilibria and one unique positive equilibrium under certain parametric conditions. We have investigated the local stability of boundary equilibria O(0,0), A(α1-1)/α3,0, B0,(α4-1)/α6 and the unique positive equilibrium C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5), by the method of linearization. It is proved that the discrete model undergoes a period-doubling bifurcation in a small neighborhood of boundary equilibria A(α1-1)/α3,0,B0,(α4-1)/α6 and a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5). Further it is shown that every positive solution of the discrete model is bounded and the set 0,α1/α3×0,α4/α6 is an invariant rectangle. It is proved that if α1<1 and α4<1, then equilibrium O(0,0) of the discrete model is a global attractor. Finally it is proved that the unique positive equilibrium C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) is a global attractor. Some numerical simulations are presented to illustrate theoretical results.


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## Body

## 1. Introduction

In this paper, we study the global dynamics and bifurcations analysis of a two-dimensional discrete-time Lotka-Volterra model in the closed first quadrantR2, which was proposed by Waltman [1]. In this model, if two populations are growing logistically without affecting each other, then their growth can be represented by the following system of two logistic equations:(1)dxdt=r1x1-xk1dydt=r2y1-yk2,where r1,r2,k1,k2 and the initial conditions x0,y0 are positive real numbers. Now, assume that the carrying capacity is a shared resource-each population competes for the resource and thereby interferes with the other. Then the presence of each reduces the intrinsic rate of growth of the other. We refer the reader to [1–5] for detailed discussion on the above assumption. This phenomena can be represented as follows:(2)dxdt=r1x1-xk1-ζ1ydydt=r2y1-yk2-ζ2x,where ζ1,ζ2 are positive constants. It is convenient to change the nondimensional variables by measuring x in units of k1, y in units of k2, and time in units of 1/r1. Then system (2) takes the following form:(3)dxdt=x1-x-λ1ydydt=ry1-y-λ2x,where λi=kiζi for i=1,2 and r=r2/r1. It is clear that for all parametric values, system (3) has three boundary equilibria O(0,0),A(1,0),B(0,1) and a unique positive equilibrium point C(1-λ1)/(1-λ1λ2),(1-λ2)/(1-λ1λ2) if λ1<1,λ2<1,λ1λ2<1. According to continuous dynamical systems theory, it is easy to show that equilibrium O(0,0) is a source but never sink and saddle; equilibrium A(1,0) is a sink if λ2>1 and saddle if λ2<1, but it is never source; B(0,1) is a sink if λ1>1 and saddle if λ1<1, but it is never source, and the unique positive equilibrium point C(1-λ1)/(1-λ1λ2),(1-λ2)/(1-λ1λ2) is locally asymptotically stable.A discrete dynamical system is defined as a system whose state evolves over state space in discrete-time steps according to a fixed rule. These systems are represented by a system of difference equations. This is a well-known fact that difference equations existed before differential equations and have played a fundamental role in the development of the latter. During the last fifty years the theory of difference equations received attention of both mathematicians and users of mathematics and developed greatly, because of its internal mathematical beauty and applicability in almost all branches of modern science such as ecology, population dynamics, queuing problems, statistical problems, stochastic time series, number theory, geometry, neural networks, quanta in radiation, genetics in biology, economics, psychology, sociology, physics, engineering, economics, combinatorial analysis, probability theory, electrical networks, and resource management [6, 7]. Dynamics of a discrete dynamical system is studied by analyzing the behavior of the solution of the system of difference equations representing the system under study. Analyzing the behavior of solutions of a higher-order nonlinear difference equation is very interesting and attracted many researchers in recent times. Behavior of solutions means studying the equilibrium point, boundedness and persistence, existence and uniqueness of positive equilibrium point, local and global stability, periodicity nature of such difference equations or systems of difference equations (see [8–16] and references cited therein).Discrete dynamical systems described by difference equations are more appropriate for population dynamics as compared to continuous ones. Biologists believe that the equilibrium point and its stability analysis is important to understand the population dynamics [17, 18]. Therefore, in this paper, we study the behavior of the following discrete-time Lotka-Volterra model, which is obtained by discretization of continuous-time model (3) followed by forward Euler’s method. Using forward Euler’s method, continuous-time model (3) takes the following form: (4)xn+1-xnh=xn-xnxn+1-λ1xnynyn+1-ynk=ryn-rynyn+1-rλ2xnyn.After some simplification, the above system becomes(5)xn+1=α1xn-α2xnyn1+α3xnyn+1=α4yn-α5xnyn1+α6yn,where α1=1+h,α2=hλ1,α3=h,α4=1+kr,α5=krλ2,α6=kr. It is also noted that the parameters α1,α2,α3,α4,α5,α6 and the initial conditions x0,y0 are positive real numbers.The rest of the paper is organized as follows: In Section2, we study the existence of equilibria of the discrete model. Section 3 deals with the study of local stability of three boundary equilibria and the unique positive equilibrium. Section 4 deals with the study of bifurcations analysis of boundary equilibria and the unique positive equilibrium. Section 5 discusses the boundedness character and the construction of invariant rectangle of the discrete model. Section 6 discusses the global behavior of O(0,0) and the unique positive equilibrium, whereas Section 7 is about numerical simulation to verify the obtained theoretical results. In the last section a brief conclusion is given.


## 2. Existence of Equilibria

In this section, we will study the existence of equilibria of the discrete model (5) in the closed first quadrant R2. The results about the existence of equilibria are summarized as follows.Lemma 1.
Under certain parametric conditions, system (5) has at least three boundary equilibria and one unique positive equilibrium in the closed first quadrant R2. More precisely,(i)
system (5) has a unique boundary equilibrium O(0,0) if α1<1,α4<1,α3α6<α2α5,α6<α2(α4-1)/(α1-1) and α5>α3(α4-1)/(α1-1);(ii)
system (5) has two boundary equilibrium O(0,0),A(α1-1)/α3,0 if α1>1,α4<1,α3α6<α2α5,α6<α2(α4-1)/(α1-1) and α5>α3(α4-1)/(α1-1);(iii)
system (5) has three boundary equilibrium O(0,0),A(α1-1)/α3,0,B0,(α4-1)/α6 if α1>1,α4>1,α3α6<α2α5,α6<α2(α4-1)/(α1-1) and α5>α3(α4-1)/(α1-1);(iv)
system (5) has three boundary equilibrium O(0,0),A(α1-1)/α3,0,B0,(α4-1)/α6 and one interior equilibrium C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) if α1>1,α4>1,α3α6>α2α5,α6>α2(α4-1)/(α1-1) and α5<α3(α4-1)/(α1-1). Additionally if α1>1,α4>1,α3α6>α2α5,α6>α2(α4-1)/(α1-1) and α5<α3(α4-1)/(α1-1), then C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) is the unique positive equilibrium of system (5).Proof.
In order to find equilibria of system (5) in the interior of R2, we need to solve the following algebraic equations:(6)x=α1x-α2xy1+α3xy=α4y-α5xy1+α6y.If x=0,y=0, then (6) are identically satisfied, and hence for all parametric values O(0,0) is the unique equilibrium of this system. If y=0 then the second equation of (6) is identically satisfied, and from the first equation we get x=(α1-1)/α3. Thus if α1>1, then A(α1-1)/α3,0 is one boundary equilibrium point of system (5). If x=0, then first equation of (6) is identically satisfied, and from the second equation we get y=(α4-1)/α6. Thus if α4>1, then B0,(α4-1)/α6 is again a boundary equilibrium of system (5).
On the other hand, we consider the existence of unique positive equilibrium of system (5) in the interior of R2. For this assume that if x≠0 and y≠0, then system (6) takes the following form:(7)1=α1-α2y1+α3x1=α4-α5x1+α6y.Solving system (7) for x and y, one gets (x,y)=((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5). Hence if α1>1,α4>1,α3α6>α2α5,α6>α2(α4-1)/(α1-1) and α5<α3(α4-1)/(α1-1), then (8)Cα1-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1α3α6-α2α5is the unique positive equilibrium of system (5).Remark 2.
The discrete model (5) has three boundary equilibria O(0,0),A(α1-1)/α3,0,B0,(α4-1)/α6 and one interior equilibrium C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) if α1>1,α4>1,α3α6>α2α5,α6>α2(α4-1)/(α1-1) and α5<α3(α4-1)/(α1-1), where α1=1+h,α2=hλ1,α3=h,α4=1+kr,α5=krλ2,α6=kr. Now, using the values αi,i=1,…,6, the equilibria of continuous-time model (3), O(0,0),A(1,0),B(0,1), and C(1-λ1)/(1-λ1λ2),(1-λ2)/(1-λ1λ2) can be recovered with the same conditions on the parameters λ1<1,λ2<1.


## 3. Local Stability

The Jacobian matrixJ(x,y) of linearized system of (5) about equilibrium (x,y) is (9)Jx,y=α1-α2y1+α3x2-α2x1+α3x-α5y1+α6yα4-α5x1+α6y2.


### 3.1. Local Stability of Boundary Equilibria

Hereafter we will study the topological classification of the boundary equilibria. The results regarding the local stability of the boundary equilibria are summarized as follows.Theorem 3.
For equilibrium pointO(0,0), the following statements hold:(i)
The equilibrium pointO(0,0) of system (5) is a sink if α1<1 and α4<1;(ii)
The equilibrium pointO(0,0) of system (5) is a source if α1>1 and α4>1;(iii)
The equilibrium pointO(0,0) of system (5) is a saddle if α1>1 and α4<1;(iv)
The equilibrium pointO(0,0) of system (5) is nonhyperbolic if α1=1 or α4=1.Theorem 4.
For equilibrium pointA(α1-1)/α3,0, the following statements hold:(i)
The equilibriumA(α1-1)/α3,0 of system (5) is a sink if α1>1 and α5<α3(α4+1)/(α1-1);(ii)
The equilibriumA(α1-1)/α3,0 of system (5) is never source;(iii)
The equilibriumA(α1-1)/α3,0 of system (5) is a saddle if α1>1 and α5>α3(α4+1)/(α1-1);(iv)
The equilibriumA(α1-1)/α3,0 of system (5) is nonhyperbolic if α5=α3(α4+1)/(α1-1).Theorem 5.
For equilibrium pointB0,(α4-1)/α6, the following statements hold:(i)
The equilibriumB0,(α4-1)/α6 of system (5) is a sink if α4>1 and α2<(α1+1)α6/α4-1;(ii)
The equilibriumB0,(α4-1)/α6 of system (5) is never source;(iii)
The equilibriumB0,(α4-1)/α6 of system (5) is a saddle if α4>1 and α2>(α1+1)α6/α4-1;(iv)
The equilibriumB0,(α4-1)/α6 of system (5) is nonhyperbolic if α2=(α1+1)α6/α4-1.Now in the following we will study the local stability of the unique positive equilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) by using Remark 1.3.1 of [7].


### 3.2. Local Stability of the Unique Positive Equilibrium

Theorem 6.
For the unique positive equilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5), the following statements hold:(i)
The unique positive equilibrium pointC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is locally asymptotically stable if(10)Θ<α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6;(ii)
The unique positive equilibrium pointC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is unstable if(11)Θ>α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,where(12)Θ=-2α2α3α6α5+α32α62α1+α4+1+α62α3α51-α1+α32α6α21-α4+α22α52α1+1α4+α1-2+α52α2α62+α1α1+1+α22α3α52+α4α4+1+α2α3α6α53+α1+2α4+2α1.Proof.
(i) The Jacobian matrixJC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of linearized system of (5) about C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) is (13)JCα1-1α6-α2α4-1/α3α6-α2α5,α3α4-1+α51-α1/α3α6-α2α5=α3α6-α2α5α1α3α6-α2α5-α2α3α4+α2α3-α2α6α1-1-α22α4-1α1α3α6-α2α5-α2α3α4+α2α3-α3α5α4-1-α52α1-1α3α6α4-α2α5-α6α5α1+α6α5α3α6-α2α5α3α6α4-α2α5-α6α5α1+α6α5.The characteristic polynomial of JC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) about (14)Cα1-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1α3α6-α2α5is (15)Pλ=λ2-Ω1λ+Ω2,where (16)Ω1=A+B-C-D,Ω2=E-F+G+H+I-J-K+L+M-N+O-P+Q+R,A=α3α6α2α3-α2α3α4-α2α5+α1α3α6,B=α3α6-α2α5+α3α4α6+α5α6-α1α5α6,C=α2α5α2α3-α2α3α4-α2α5+α1α3α6,D=α2α5-α2α5+α3α4α6+α5α6-α1α5α6,E=α22α3α5α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,F=2α22α3α4α5α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,G=α22α3α42α5α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,H=α1α22α52α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,I=α22α4α52α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1-α1α5α6,J=α1α22α4α52α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,K=3α2α3α5α6α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,L=α1α2α3α5α6α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,M=α2α3α4α5α6α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,N=α1α2α3α4α5α6α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,O=α2α52α6α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,P=2α1α2α52α6α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,Q=α12α2α52α6α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,R=α32α62α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6.Assume that Θ<α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6, and using Remark 1.3.1 of [7] one gets (17)Ω1+Ω2≤A+B+C+D+E+F+G+H+I+J+K+L+M+N+O+P+Q+R,<Θα2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6<1.Therefore C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is locally asymptotically stable.
(ii) Similarly it is easy to show thatC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is unstable if Θ>α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6.Hereafter we will compute the necessary and sufficient condition(s) for the unique positive equilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) to be locally asymptotically stable, repeller, saddle, and nonhyperbolic, respectively.Theorem 7.
For equilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5), the following statements hold:(i)
EquilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is locally asymptotically stable if and only if(18)α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<1-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<2.(ii)
EquilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is a repeller if and only if (19)α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1>1,α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<1-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1.(iii)
EquilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is a saddle if and only if(20)α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α12+41-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1>0,α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1>1-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1.(iv)
EquilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is nonhyperbolic if and only if (21)α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1=1-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1.Proof.
(i) More precisely the characteristic equation ofJC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) about (22)Cα1-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1α3α6-α2α5is given by (23)λ2-pλ+q=0,where (24)p=α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1q=α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1.Then, it follows from Theorem 1.1.1 of [12] that the unique positive equilibrium(25)Cα1-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1α3α6-α2α5of system (1) is locally asymptotically stable if and only if (26)α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<1-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<2.Similarly, one can prove (ii), (iii), and (iv).


## 3.1. Local Stability of Boundary Equilibria

Hereafter we will study the topological classification of the boundary equilibria. The results regarding the local stability of the boundary equilibria are summarized as follows.Theorem 3.
For equilibrium pointO(0,0), the following statements hold:(i)
The equilibrium pointO(0,0) of system (5) is a sink if α1<1 and α4<1;(ii)
The equilibrium pointO(0,0) of system (5) is a source if α1>1 and α4>1;(iii)
The equilibrium pointO(0,0) of system (5) is a saddle if α1>1 and α4<1;(iv)
The equilibrium pointO(0,0) of system (5) is nonhyperbolic if α1=1 or α4=1.Theorem 4.
For equilibrium pointA(α1-1)/α3,0, the following statements hold:(i)
The equilibriumA(α1-1)/α3,0 of system (5) is a sink if α1>1 and α5<α3(α4+1)/(α1-1);(ii)
The equilibriumA(α1-1)/α3,0 of system (5) is never source;(iii)
The equilibriumA(α1-1)/α3,0 of system (5) is a saddle if α1>1 and α5>α3(α4+1)/(α1-1);(iv)
The equilibriumA(α1-1)/α3,0 of system (5) is nonhyperbolic if α5=α3(α4+1)/(α1-1).Theorem 5.
For equilibrium pointB0,(α4-1)/α6, the following statements hold:(i)
The equilibriumB0,(α4-1)/α6 of system (5) is a sink if α4>1 and α2<(α1+1)α6/α4-1;(ii)
The equilibriumB0,(α4-1)/α6 of system (5) is never source;(iii)
The equilibriumB0,(α4-1)/α6 of system (5) is a saddle if α4>1 and α2>(α1+1)α6/α4-1;(iv)
The equilibriumB0,(α4-1)/α6 of system (5) is nonhyperbolic if α2=(α1+1)α6/α4-1.Now in the following we will study the local stability of the unique positive equilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) by using Remark 1.3.1 of [7].


## 3.2. Local Stability of the Unique Positive Equilibrium

Theorem 6.
For the unique positive equilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5), the following statements hold:(i)
The unique positive equilibrium pointC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is locally asymptotically stable if(10)Θ<α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6;(ii)
The unique positive equilibrium pointC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is unstable if(11)Θ>α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,where(12)Θ=-2α2α3α6α5+α32α62α1+α4+1+α62α3α51-α1+α32α6α21-α4+α22α52α1+1α4+α1-2+α52α2α62+α1α1+1+α22α3α52+α4α4+1+α2α3α6α53+α1+2α4+2α1.Proof.
(i) The Jacobian matrixJC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of linearized system of (5) about C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) is (13)JCα1-1α6-α2α4-1/α3α6-α2α5,α3α4-1+α51-α1/α3α6-α2α5=α3α6-α2α5α1α3α6-α2α5-α2α3α4+α2α3-α2α6α1-1-α22α4-1α1α3α6-α2α5-α2α3α4+α2α3-α3α5α4-1-α52α1-1α3α6α4-α2α5-α6α5α1+α6α5α3α6-α2α5α3α6α4-α2α5-α6α5α1+α6α5.The characteristic polynomial of JC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) about (14)Cα1-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1α3α6-α2α5is (15)Pλ=λ2-Ω1λ+Ω2,where (16)Ω1=A+B-C-D,Ω2=E-F+G+H+I-J-K+L+M-N+O-P+Q+R,A=α3α6α2α3-α2α3α4-α2α5+α1α3α6,B=α3α6-α2α5+α3α4α6+α5α6-α1α5α6,C=α2α5α2α3-α2α3α4-α2α5+α1α3α6,D=α2α5-α2α5+α3α4α6+α5α6-α1α5α6,E=α22α3α5α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,F=2α22α3α4α5α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,G=α22α3α42α5α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,H=α1α22α52α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,I=α22α4α52α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1-α1α5α6,J=α1α22α4α52α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,K=3α2α3α5α6α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,L=α1α2α3α5α6α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,M=α2α3α4α5α6α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,N=α1α2α3α4α5α6α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,O=α2α52α6α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,P=2α1α2α52α6α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,Q=α12α2α52α6α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6,R=α32α62α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6.Assume that Θ<α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6, and using Remark 1.3.1 of [7] one gets (17)Ω1+Ω2≤A+B+C+D+E+F+G+H+I+J+K+L+M+N+O+P+Q+R,<Θα2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6<1.Therefore C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is locally asymptotically stable.
(ii) Similarly it is easy to show thatC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is unstable if Θ>α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6.Hereafter we will compute the necessary and sufficient condition(s) for the unique positive equilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) to be locally asymptotically stable, repeller, saddle, and nonhyperbolic, respectively.Theorem 7.
For equilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5), the following statements hold:(i)
EquilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is locally asymptotically stable if and only if(18)α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<1-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<2.(ii)
EquilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is a repeller if and only if (19)α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1>1,α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<1-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1.(iii)
EquilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is a saddle if and only if(20)α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α12+41-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1>0,α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1>1-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1.(iv)
EquilibriumC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) is nonhyperbolic if and only if (21)α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1=1-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1.Proof.
(i) More precisely the characteristic equation ofJC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) about (22)Cα1-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1α3α6-α2α5is given by (23)λ2-pλ+q=0,where (24)p=α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1q=α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1.Then, it follows from Theorem 1.1.1 of [12] that the unique positive equilibrium(25)Cα1-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1α3α6-α2α5of system (1) is locally asymptotically stable if and only if (26)α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<1-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<2.Similarly, one can prove (ii), (iii), and (iv).


## 4. Bifurcations Analysis

In this section, we will study the bifurcation analysis of discrete model (5) about the equilibria O(0,0),A(α1-1)/α3,0,B0,(α4-1)/α6 and C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5). From theoretical results obtained in Section 3, we conclude the following:(i)
If condition (iv), that is,α1=1, of Theorem 3 holds, then one of the eigenvalues of JO(0,0) about O(0,0) is 1 and so fold bifurcation may occur when parameters vary in a small neighborhood of α1=1. The condition (iv) of Theorem 3 can be rewritten as the following set: (27)FO0,0=α1,α2,α3,α4,α5,α6:α1=1.(ii)
From Theorem4, we can see that one of the eigenvalues of JA(α1-1)/α3,0 about A(α1-1)/α3,0 is -1 and other is neither 1 nor -1 when the parameters of the discrete model (5) are located in the following set: (28)FAα1-1/α3,0=α1,α2,α3,α4,α5,α6:α5=α3α4+1α1-1.Therefore,A(α1-1)/α3,0 can undergo flip or period-doubling bifurcation when all parameters of the discrete model (5) vary in a small neighborhood of FA(α1-1)/α3,0. When the parameters are in FA(α1-1)/α3,0, a center manifold of the discrete model (5) is y=0, and thus (5) restricted to this central manifold is (29)xn+1=α1xn1+α3xn.This shows that the predator becomes extension and prey undergoes period-doubling bifurcation to chaos on choosing bifurcation parameterα5.(iii)
From Theorem5, it is easy to verify that one of the eigenvalues of JB0,(α4-1)/α6 about B0,(α4-1)/α6 is -1 and the other is neither 1 nor -1 when the parameters of the discrete model (5) are located in the following set: (30)FB0,α4-1/α6=α1,α2,α3,α4,α5,α6:α2=α1+1α6α4-1.Therefore,B0,(α4-1)/α6 undergoes period-doubling bifurcation when all parameters of the discrete model (5) vary in a small neighborhood of FB0,(α4-1)/α6. When the parameters are in FB0,(α4-1)/α6, a center manifold of the discrete model (5) is x=0, and thus (5) restricted to this central manifold is (31)yn+1=α4yn1+α6yn.This shows that the prey becomes extension and predator undergoes period-doubling bifurcation to chaos on choosing bifurcation parameterα2.Now before studying the bifurcation analysis of the discrete model (5) about the unique positive equilibrium point C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5), first we will prove that there exist different topological types of this equilibrium point. Recall that eigenvalues of JC(α1-1)α6-α2(α4-1)/α3α6-α2α5,α3(α4-1)+α5(1-α1)/α3α6-α2α5 about (32)Cα1-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1α3α6-α2α5are(33)λ1,2=-p±Δ2,where (34)Δ=p2-4q,=α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α12-4α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1.Hereafter we state the topological classification about C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) as follows.Theorem 8.
ForC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5), the following topological classification holds:(i)
C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) is a locally asymptotically stable focus if Δ<0 and (35)α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<1;(ii)
C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) is an unstable focus if Δ<0 and(36)α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1>1;(iii)
C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) is nonhyperbolic if Δ<0 and(37)α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1=1.We can see that the eigenvalues ofJC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) are a pair of complex conjugates with modulus 1 when parameters of model (5) are located in the following set: (38)NC=α1,α2,α3,α4,α5,α6:Δ<0,α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1=1.Therefore, C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) undergoes Neimark-Sacker bifurcation when all parameters of model (5) vary in a small neighborhood of NC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5).Hereafter we will study the Neimark-Sacker bifurcation of model (5) about (39)Cα1-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1α3α6-α2α5.For given parametersα1,α2,α3,α4,α5,α6, let(40)Fα1,α2,α3,α4,α5,α6=α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1-α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1.It is easy to verify that ∂F(α1,α2,α3,α4,α5,α6)/∂α1<0, then by implicit function theorem we obtain that α1=α1(α2,α3,α4,α5,α6) such that F(α1(α2,α3,α4,α5,α6),α2,α3,α4,α5,α6)=0, and therefore we can choose α1 as a bifurcation parameter.Now consider parameterα1 in a small neighborhood of α1∗; that is, α1=α1∗+ϵ, where ϵ≪1; then system (5) becomes(41)xn+1=α1∗+ϵxn-α2xnyn1+α3xnyn+1=α4yn-α5xnyn1+α6yn.The characteristic equation of JC((α1∗+ϵ-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1∗-ϵ))/(α3α6-α2α5) about (42)Cα1∗+ϵ-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1∗-ϵα3α6-α2α5of system (41) is (43)λ2-pϵλ+qϵ=0,where (44)pϵ=α3α6-α2α5α3α6α1∗+ϵ+α4+α21-α4+α5α61-α1∗-ϵ-2α2α1∗+ϵα3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1∗-ϵ,qϵ=α3α6-α2α52-α2α6α1∗+ϵ-1-α22α4-1α3α5α4-1-α52α1∗+ϵ-1α1∗+ϵα3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1∗-ϵ.The roots of characteristic equation of JC((α1∗+ϵ-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1∗-ϵ))/(α3α6-α2α5) about (45)Cα1∗+ϵ-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1∗-ϵα3α6-α2α5are (46)λ1,2=pϵ±ι4qϵ-p2ϵ2,=α3α6-α2α5α3α6α1∗+ϵ+α4+α21-α4+α5α61-α1∗-ϵ-2α22α1∗+ϵα3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1∗-ϵ±ι2Γε,where (47)Γϵ=4α3α6-α2α52-α2α6α1∗+ϵ-1-α22α4-1α3α5α4-1-α52α1∗+ϵ-1α1∗+ϵα3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1∗-ϵ-α3α6-α2α5α3α6α1∗+ϵ+α4+α21-α4+α5α61-α1∗-ϵ-2α2α1∗+ϵα3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1∗-ϵ2.Moreover λ1,2=q(ϵ) and (dλ1,2/dϵ)ϵ=0≠0. Additionally, we required that when ϵ=0, λ1,2m≠1, m=1,2,3,4, which corresponds to p(0)≠-2,0,1,2. Since p(0)2-4q(0)<0 and q(0)=1, then q(0)2<4 and hence q(0)≠±2. Thus, we only require that q1(0)≠0,1, which holds true by computation.Letun=xn-x∗,vn=yn-y∗; then C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of system (5) transforms into O(0,0). By calculation, we obtain(48)un+1=α1∗+ϵun+x∗-α2un+x∗vn+y∗1+α3un+x∗-x∗vn+1=α4vn+y∗-α5un+x∗vn+y∗1+α6vn+y∗-y∗,where x∗=((α1-1)α6-α2(α4-1))/(α3α6-α2α5) and y∗=(α3(α4-1)+α5(1-α1))/(α3α6-α2α5). Hereafter when ϵ=0, normal form of system (48) is studied. Expanding (48) up to second-order about (un,vn)=(0,0) by Taylor series, we get(49)un+1=m11un+m12vn+m13un2+m14unvn+oun+vn2vn+1=m21un+m22vn+m23unvn+m24vn2+oun+vn2,where (50)m11=α1∗-α2y∗1+α3x∗2,m12=-α2x∗1+α3x∗,m13=-α3α1∗-α2y∗1+α3x∗3,m14=-α21+α3x∗2,m21=-α5y∗1+α6y∗,m22=α4-α5x∗1+α6y∗2,m23=-α51+α6y∗2,m24=-α6α4-α5x∗1+α6y∗3.Now, let(51)η=α3α6-α2α5α3α6α1∗+α4+α21-α4+α5α61-α1∗-2α22α1∗α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1∗,ζ=12Γ0,and invertible matrix T is defined by (52)T=m120η-m11-ζ.Using the following translation (53)unvn=m120η-m11-ζXnYn,(49) gives(54)Xn+1Yn+1=η-ζζηXnYn+FXn,YnGXn,Yn,where(55)FXn,Yn=l11Xn2+l12XnYn+oXn+Yn2,GXn,Yn=l21Xn2+l22XnYn+l23Yn2+oXn+Yn2,l11=m12m13+m14η-m11,l12=-ζm14l21=η-m11ζm12m13+m14η-m11m12-m23-m24η-m11,l22=-m12m14η-m11m12-m23+m24η-m11,l23=-ζm24.In addition, (56)FXnXn0,0=2l11,FXnYn0,0=l12,FYnYn0,0=0,FXnXnXn0,0=FXnXnYn0,0=FXnYnYn0,0=FYnYnYn0,0=0,GXnXn0,0=2l21,GXnYn0,0=l22,GYnYn0,0=2l23,GXnXnXn0,0=GXnXnYn0,0=GXnYnYn0,0=GYnYnYn0,0=0.To guarantee the supercritical Neimark-Sacker bifurcation for (54), we require the following discriminatory quantity, that is, Ψ<0 (see [19–23]):(57)Ψ=-Re⁡1-2λ¯λ¯21-λτ11τ20-12τ112-τ022+Re⁡λ¯τ21,where(58)τ02=18FXnXn-FYnYn+2GXnYn+ιGXnXn-GYnYn+2FXnYn0,0,τ11=14FXnXn+FYnYn+ιGXnXn+GYnYn0,0,τ20=18FXnXn-FYnYn+2GXnYn+ιGXnXn-GYnYn-2FXnYn0,0,τ21=116FXnXnXn+FXnYnYn+GXnXnYn+GYnYnYn+ιGXnXnXn+GXnYnYn-FXnXnYn-FYnYnYn0,0.A calculation reveals(59)τ02=14l11+l22+ιl21-l23+l12,τ11=12l11+ιl21+l23,τ20=14l11+l22+ιl21-l23-l12,τ21=0.Based on this analysis and Neimark-Sacker bifurcation Theorem discussed in [19, 20], we reach the following Theorem.Theorem 9.
IfΨ≠0, then discrete model (5) undergoes a Neimark-Sacker bifurcation about (60)Cα1-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1α3α6-α2α5as the parameters α1,α2,α3,α4,α5,α6 go through NC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5). Additionally, attracting (resp., repelling) invariant closed curve bifurcates from the unique positive equilibrium C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) if Ψ<0 (resp., Ψ>0).


## 5. Boundedness and Construction of Invariant Rectangle

In this section we will study boundedness character and construction of invariant rectangle of positive solution of the discrete model (5).Theorem 10.
For every positive solution{(xn,yn)}n=0∞ of the discrete model (5), the following holds:(i)
Every positive solution of the discrete model (5) is bounded.(ii)
The set0,α1/α3×0,α4/α6 is an invariant rectangle.Proof.
(i) Let{(xn,yn)}n=0∞ be any positive solution of the discrete model (5). From (5), we have (61)xn+1=α1xn-α2xnyn1+α3xn<α1xn1+α3xn<α1α3,yn+1=α4yn-α5xnyn1+α6yn<α4yn1+α6yn<α4α6.Hence, for every solution {(xn,yn)} of the discrete model (5), one has (62)0≤xn<α1α3,0≤yn<α4α6.
(ii) For any positive solution{(xn,yn)}n=0∞ of the discrete model (5) with initial conditions x0∈0,α1/α3 and y0∈0,α4/α6, we have from (5)(63)0≤x1=α1x0-α2x0y01+α3x0<α1x01+α3x0<α1α3,0≤y1=α4y0-α5x0y01+α6y0<α4y01+α6y0<α4α6.Hence, x1∈0,α1/α3 and y1∈0,α4/α6. Similarly, one can show that if xk∈0,α1/α3 and yk∈0,α4/α6, then xk+1∈0,α1/α3 and yk+1∈0,α4/α6.


## 6. Global Stability

Now in the following we will investigate global dynamics of the discrete model (5) about O(0,0) and the unique positive equilibrium C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5).Theorem 11.
Ifα1<1 and α4<1, then equilibrium O(0,0) of the discrete model (5) is globally asymptotically stable.Proof.
According to the conclusion (i) in Lemma1, discrete model (5) has a unique equilibrium O(0,0) in the first quadrant R2 and is a sink by Theorem 3. Moreover every positive solution of the discrete model (5) in the first quadrant R2 satisfies (64)0<xn+1≤α1xn1+α3xn=xn1-α3xn+1-α11+α3xn<xn,0<yn+1≤α4yn1+α6yn=yn1-α6yn+1-α41+α6yn<yn,which leads to limn→∞xn=0 and limn→∞yn=0. Hence the boundary equilibrium O(0,0) of the discrete model (5) is globally asymptotically stable in the first quadrant R2.Hereafter we use Theorem1.16 of [14] to determine the global dynamics of the discrete model (5) about C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5).Theorem 12.
The unique positive equilibrium pointC((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of the discrete model (5) is a global attractor.Proof.
Letf(x,y)=(α1x-α2xy)/(1+α3x) and g(x,y)=(α4y-α5xy)/(1+α6y¯). Then, it is easy to see that f(x,y) is nondecreasing in x and nonincreasing in y for every (x,y)∈0,α1/α3×0,α4/α6. Moreover, g(x,y) is nonincreasing in x and nondecreasing in y for each (x,y)∈0,α1/α3×0,α4/α6. Let (m1,M1,m2,M2) be a positive solution of system (65)m1=fm1,M2,M1=fM1,m2,m2=gM1,m2,M2=gm1,M2.Then, one has(66)m1=α1m1-α2m1M21+α3m1,M1=α1M1-α2M1m21+α3M1,(67)m2=α4m2-α5M1m21+α6m2,M2=α4M2-α5m1M21+α6M2.From (66), one has(68)1+α3m1=α1-α2M2,1+α3M1=α1-α2m2.From (67), one has(69)1+α6m2=α4-α5M1,1+α6M2=α4-α5m1.On subtracting (68), one gets(70)α3m1-M1=α2m2-M2.Similarly, subtracting (69) one gets(71)α6m2-M2=α5m1-M1.From (70) and (71) one gets α6α3-α2α4(m1-M1)=0. Hence, m1=M1, and similarly one has m2=M2. Hence, from Theorem 1.16 of [14] the equilibrium C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) of the discrete model (5) is a global attractor.Corollary 13.
Under the conditions (10) and (18), the unique positive equilibrium point (72)Cα1-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1α3α6-α2α5of the discrete model (5) is globally asymptotically stable.


## 7. Numerical Simulations

In the following we present numerical simulations that represent different types of qualitative behavior of the discrete model (5).Example 1.
Ifα1=2,α2=0.0007,α3=0.23,α4=7,α5=0.3,α6=0.4, then discrete model (5) with initial values x0=0.007,y0=0.0009 can be written as(73)xn+1=2xn-0.0007xnyn1+0.23xnyn+1=7yn-0.3xnyn1+0.4yn.In this case α1=2>1,α4=7>1,α3α6=0.092>α2α5=0.00021, α6=0.4>α2(α4-1)/(α1-1)=0.0042, α5=0.3<α3(α4-1)/(α11-1)=1.38. This shows the correctness of the conditions for the unique positive equilibrium. A straightforward computation shows that condition (10) of Theorem 6, that is, Θ = 0.07435317508000001 < (α2α3-α2α3α4-α2α5+α1α3α6)(-α2α5+α3α4α6+α5α6-α1α5α6) = 0.0957613829600004, under which the unique positive equilibrium point is locally asymptotically stable, holds. Moreover for these arbitrary chosen values of parameters, the necessary and sufficient condition, under which the unique positive equilibrium point is locally asymptotically stable, is also satisfied; that is, |α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2/α1α3α6-α2α5+α2α3(α4-1)α3α6α4-α2α5+α6α5(1-α1)|=0.6702268987306942<1-(α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1)/α1α3α6-α2α5+α2α3(α4-1)α3α6α4-α2α5+α6α5(1-α1)=0.9138643327940412<2. This verifies the condition for which the unique positive equilibrium is locally asymptotically stable. Also 0≤xn<α1/α3=8.695652173913043,0≤yn<α4/α6=17.5, and hence the parametric conditions under which every positive solution is bounded hold true. Moreover, in Figure 1 the plot of xn is shown in Figure 1(a), the plot of yn is shown in Figure 1(b), and attractor of system (73) is shown in Figure 1(c).Figure 1
Plots for system (73).
(a)
Plot ofxn for system (73)
(b)
Plot ofyn for system (73)
(c)
Attractor of system (73)Example 2.
Ifα1=8,α2=0.00007,α3=3.23,α4=7,α5=1.3,α6=3.4, then discrete model (5) with initial values x0=0.007,y0=0.08 can be written as(74)xn+1=8xn-0.00007xnyn1+3.23xnyn+1=7yn-1.3xnyn1+3.4yn.In this case α1=8>1,α4=7>1,α3α6=10.982>α2α5=0.000091, α6=3.4>α2(α4-1)/(α1-1)=0.00006,α5=1.3<α3(α4-1)/(α11-1)=2.76857. This shows the correctness of the conditions of the unique positive equilibrium point. A straightforward computation shows that condition (10) of Theorem 6, that is, Θ = 4035.5030151773317 < (α2α3-α2α3α4-α2α5+α1α3α6)(-α2α5+α3α4α6+α5α6-α1α5α6) = 4035.5030151773317, holds. Moreover the necessary and sufficient condition under which the unique positive equilibrium point of the system is locally asymptotically stable is also satisfied; that is, |α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2/α1α3α6-α2α5+α2α3(α4-1)α3α6α4-α2α5+α6α5(1-α1)| = 0.36407043421653185 < 1-(α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1)/ α1α3α6-α2α5+α2α3(α4-1)α3α6α4-α2α5+α6α5(1-α1) = 0.9701211132646083 < 2. This verifies the condition for which the unique positive equilibrium is locally asymptotically stable. Also 0≤xn<α1/α3=2.476780185758514,0≤yn<α4/α6=2.058823529411765, and hence the parametric conditions under which every positive solution is bounded hold true. Moreover, in Figure 2 the plot of xn is shown in Figure 2(a), the plot of yn is shown in Figure 2(b), and attractor of system (74) is shown in Figure 2(c).Figure 2
Plots for system (74).
(a)
Plot ofxn for system (74)
(b)
Plot ofyn for system (74)
(c)
Attractor of system (74)Example 3.
Ifα1=15,α2=0.057,α3=6.23,α4=7,α5=1.3,α6=3.4, then discrete model (5) with initial values x0=0.008,y0=0.009 can be written as(75)xn+1=15xn-0.057xnyn1+6.23xnyn+1=7yn-1.3xnyn1+3.4yn.In this case α1=15>1,α4=7>1,α3α6=21.182>α2α5=0.0741, α6=3.4>α2(α4-1)/(α1-1=0.0244286,α5=1.3<α3(α4-1)/(α1-1)=2.67. This shows the correctness of the conditions of the unique positive equilibrium point. A computation shows that condition (10) of Theorem 6, that is, Θ = 9280.610848888002 < (α2α3-α2α3α4-α2α5+α1α3α6)(-α2α5+α3α4α6+α5α6-α1α5α6)=27236.10716427601, holds. Moreover for these arbitrary chosen values of parameters the necessary and sufficient condition, under which the unique positive equilibrium point is locally asymptotically stable, is also satisfied; that is, |α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2/α1α3α6-α2α5+α2α3(α4-1)α3α6α4-α2α5+α6α5(1-α1)| = 0.30727881013351915 < 1-(α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1)/α1α3α6-α2α5+α2α3(α4-1)α3α6α4-α2α5+α6α5(1-α1) = 0.9862925899775977 < 2. This verifies the condition for which the unique positive equilibrium is locally asymptotically stable. Also 0≤xn<α1/α3=2.407704654895666,0≤yn<α4/α6=2.058823529411765, and hence the parametric conditions under which every positive solution is bounded hold true. Moreover, in Figure 3 the plot of xn is shown in Figure 3(a), the plot of yn is shown in Figure 3(b), and attractor of system (75) is shown in Figure 3(c).Figure 3
Plots for system (75).
(a)
Plot ofxn for system (75)
(b)
Plot ofyn for system (75)
(c)
Attractor of system (75)


## 8. Conclusion

This work is related to the global dynamics and bifurcations analysis of a two-dimensional discrete-time Lotka-Volterra model in the closed first quadrantR2. We proved that the discrete model (5) has three boundary equilibria O(0,0),A(α1-1)/α3,0,B0,(α4-1)/α6 and the unique positive equilibrium (76)Cα1-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1α3α6-α2α5under certain parametric conditions. The method of linearization is used to prove the local asymptotic stability of these equilibria, and conclusions are presented in Table 1. We proved that boundary equilibrium O(0,0) undergoes fold bifurcation when parameters vary in a small neighborhood of α1=1 and both A(α1-1)/α3,0 and B0,(α4-1)/α6 undergo period-doubling bifurcation when parameters of the discrete model (5) are, respectively, located in the following sets:(77)FAα1-1/α3,0=α1,α2,α3,α4,α5,α6:α5=α3α4+1α1-1,FB0,α4-1/α6=α1,α2,α3,α4,α5,α6:α2=α1+1α6α4-1.We have also shown that C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) undergoes Neimark-Sacker bifurcation when parameters of the discrete model (5) are located in the following set: (78)NC=α1,α2,α3,α4,α5,α6:Δ<0,α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1=1.It is proved that every positive solution of the discrete model (5) is bounded and the set 0,α1/α3×0,α4/α6 is an invariant rectangle. The most interesting aspect in the theory of dynamical systems is to predict the global dynamics about equilibria. In this paper, we proved that if α1<1 and α4<1, then equilibrium O(0,0) of the discrete model (5) is globally asymptotically stable. Furthermore, we have investigated the global stability of the unique positive equilibrium point (79)Cα1-1α6-α2α4-1α3α6-α2α5,α3α4-1+α51-α1α3α6-α2α5of the discrete model (5). Some numerical examples are provided to support our theoretical results. These examples provide experimental verifications of the theoretical discussions.Table 1
Number of equilibria along with qualitative behavior.
EP
Corresponding behavior
O
Sink ifα1<1 and α4<1; source if α1>1 and α4>1; saddle if α1>1 and α4<1; nonhyperbolic if α1=1 or α4=1.
A
Sink ifα1>1 and α5<α3(α4+1)/α1-1; never source; saddle if α1>1 and α5>α3(α4+1)/α1-1; nonhyperbolic if α5=α3(α4+1)/α1-1.
B
Sink ifα4>1 and α2<(α1+1)α6/α4-1; never source; saddle if α4>1 and α2>(α1+1)α6/α4-1; nonhyperbolic if α2=(α1+1)α6/α4-1.
C
Locally asymptotically stable ifΘ<α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6.
Unstable ifΘ>α2α3-α2α3α4-α2α5+α1α3α6-α2α5+α3α4α6+α5α6-α1α5α6.
Locally asymptotically stable if and only if
α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<1-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<2.
Repeller if and only if
α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1>1,
α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<1-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1.
Saddle if and only if
α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α12+41-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1>0,
α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1>1-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1.
Nonhyperbolic if and only if
α3α6-α2α5α3α6α1+α4+α21-α4+α5α61-α1-2α2α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1=1-α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1.
Locally asymptotically stable focus if
α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1<1.
Unstable focus if
α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1>1.
Nonhyperbolic (under which the eigenvalues are a pair of complex conjugates with modulus 1) if
α3α6-α2α52-α2α6α1-1-α22α4-1α3α5α4-1-α52α1-1α1α3α6-α2α5+α2α3α4-1α3α6α4-α2α5+α6α51-α1=1.



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*Source: 7101505-2018-01-21.xml*