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Dynamics of a discretetime predatorprey system
Advances in Difference Equations volume 2016, Article number: 191 (2016)
Abstract
We investigate the dynamics of a discretetime predatorprey system. Firstly, we give necessary and sufficient conditions of the existence and stability of the fixed points. Secondly, we show that the system undergoes a flip bifurcation and a NeimarkSacker bifurcation by using center manifold theorem and bifurcation theory. Furthermore, we present numerical simulations not only to show the consistence with our theoretical analysis, but also to exhibit the complex but interesting dynamical behaviors, such as the period6, 11, 16, 18, 20, 21, 24, 27, and 37 orbits, attracting invariant cycles, quasiperiodic orbits, nice chaotic behaviors, which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discretetime predatorprey system. Finally, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.
Introduction
It is well known that the LotkaVolterra predatorprey model [1, 2] is one of the most important population models. There are also many other predatorprey models of various types that have been extensively investigated, and some of the relevant work may be found in [3–8]. These researches dealing with specific interactions have mainly focused on continuous predatorprey models with two variables. However, discretetime models described are more reasonable than the continuoustime models when populations have nonoverlapping generations. Moreover, using discretetime models is more efficient for computation and numerical simulations [9]. For example, in [10], the authors use the forward Euler discrete scheme to obtain a discretetime predatorprey system and prove that the system undergoes flip bifurcation and NeimarkSacker bifurcation. Recently, the complex dynamics of a discretetime predatorprey system is investigated in [11]. By analysis it is proved that the discretetime model has different properties and structures compared with the continuous one. Such systems discussed as discretetime models can also be found in [12–21] and references therein.
In this paper, we consider the following discretetime predatorprey system:
where x and y represent population densities of a prey and a predator, respectively, and a, b, c, d are positive parameters. Here a represents the natural growth rate of the prey in the absence of predators, b represents the effect of predation on the prey, c represents the natural death rate of the predator in the absence of prey, and d represents the efficiency and propagation rate of the predator in the presence of prey. In [22], the authors investigated the discretetime predatorprey system for \(c=0\), and they proved that there are flip and Hopf bifurcations and there exists a chaotic phenomenon in the sense of Marotto. In this paper, we study system (1) for \(c\neq0\).
Motivation of this paper is to investigate system (1) in detail. Here we derive the conditions of existence for flip bifurcation and NeimarkSacker bifurcation by using bifurcation theory and the center manifold theorem [23, 24]. Numerical simulations are given to support the theoretical results and display new and interesting dynamical behaviors of the system. More specifically, this paper presents the period6, 11, 16, 18, 20, 21, 24, 27, 37 orbits, attracting invariant cycles, quasiperiodic orbits, nice chaotic behaviors, which appear and disappear suddenly, and the new nice types of six and nine coexisting chaotic attractors. The computations of Lyapunov exponents confirm the dynamical behaviors. The results can be useful when the local and global stabilities in discretetime predatorprey systems are concerned.
This paper is organized as follows. In Section 2, we show the existence and stability of fixed points. In Section 3, the sufficient conditions for the existence of codimensionone bifurcations, including flip bifurcation and NeimarkSacker bifurcation, are obtained. In Section 4, numerical simulation results are presented to support the theoretical analysis, and they exhibit new and rich dynamical behaviors. In Section 5, chaos is controlled to an unstable fixed point using the feedback control method. A brief conclusion is given in Section 6.
Existence and stability of fixed points
For system (1), letting \(\bar{u}=x\) and \(\bar{v}=by\), we obtain
For simplicity, we will still use x and y instead of ū and v̄. Thus, system (2) can be rewritten as
We focus ourselves on the dynamical behavior of system (3).
It is easy to see that system (3) has one extinction fixed point \((0,0)\), one exclusion fixed point \((\frac{a1}{a},0)\) for \(a>1\), and one coexistence fixed point \((x^{*},y^{*})=(\frac{1+c}{d},\frac{d(a1)a(1+c)}{d})\) for \(d>\frac {a(1+c)}{a1}\) and \(a>1\). Thus, \((x^{*},y^{*})\) is the unique positive fixed point of system (3).
The following lemma confirms the stability of fixed points of system (3) under some conditions.
Lemma 2.1
For the predatorprey system (3), the following statements are true:

(i)
\((0,0)\) is asymptotically stable if \(0< a,c<1\);

(ii)
\((\frac{a1}{a},0)\) is asymptotically stable if \(1< a\leq3\) and \(\max\{0,\frac{a(c1)}{a1}\}< d<\frac{a(1+c)}{a1}\);

(iii)
\((\frac{1+c}{d},\frac{d(a1)a(1+c)}{d})\) is asymptotically stable if and only if one of the following conditions holds:

(a)
\(1< a\leq3\), \(c>0\) and \(\frac{a(1+c)}{a1}< d<\frac{a(2+c)}{a1}\);

(b)
\(3< a\leq5\), \(c>0\) and \(\frac{a(1+c)(3+c)}{3+ac+ac}< d<\frac {a(2+c)}{a1}\);

(c)
\(5< a<9\), \(0< c<\frac{9a}{a5}\) and \(\frac {a(1+c)(3+c)}{3+ac+ac}< d<\frac{a(2+c)}{a1}\).

(a)
Proof
(i) For the fixed point \((0,0)\), the corresponding characteristic equation is \(\lambda^{2}(ac)\lambdaac=0\), and its roots are \(\lambda_{1} =a\), \(\lambda_{2}=c\). Hence, \((0,0)\) is asymptotically stable when \(0< a,c<1\) and is unstable when \(a>1\) or \(c>1\).
(ii) For the exclusion fixed point \((\frac{a1}{a},0)\) when \(a>1\), linearizing system (3) about \((\frac{a1}{a},0)\), we have the coefficient matrix
Clearly, \(J_{0}\) has characteristic roots \(\lambda_{1}=2a\), \(\lambda_{2}=c+\frac{d(a1)}{a}\). Then \(\lambda_{i}<1\) (\(i=1,2\)) if and only if \(1< a<3\) and \(\max\{0,\frac{a(c1)}{a1}\}< d<\frac{a(1+c)}{a1}\).
We further will prove that when \(a=3\), the exclusion fixed point \((\frac {a1}{a},0)\) is asymptotically stable and when \(d=\frac{a(1+c)}{a1}\), it is unstable by using center manifold theory.
Letting \(u=x\frac{a1}{a}\) and \(v=y\) in (3), we have
Now we consider the first case, that is, \(a=3\) and \(\max\{0,\frac {3(c1)}{2}\}< d<\frac{3(1+c)}{2}\). System (4) becomes
Letting
and using the translation \(\bigl ({\scriptsize\begin{matrix}{} u\cr v \end{matrix}} \bigr )=T \bigl ( {\scriptsize\begin{matrix}{} X\cr Y \end{matrix}} \bigr )\), we can rewrite the map (5) as
where
We assume that a center manifold has the form \(Y=h(X)=\tilde{\alpha} X^{2}+\tilde{\beta} X^{3}+O(X^{4})\). Then it must satisfy
By approximate computation, for the center manifold, we obtain \(\tilde{\alpha}=0\) and \(\tilde{\beta}=0\). Hence, \(h(X)=0\), and on the center manifold \(Y=0\), the new map f̂ is given by
Some computations show that the Schwarzian derivative of this map at \(X=0\) is \(S(\hat{f}(0))=54<0\). Hence, by [25] the exclusion fixed point \((\frac{a1}{a},0)\) is asymptotically stable.
Next we consider the second case, that is, \(1< a<3\) and \(d=\frac {a(1+c)}{a1}\). System (4) becomes
We construct the invertible matrix
and use the translation \(\bigl ( {\scriptsize\begin{matrix}{} u\cr v \end{matrix}} \bigr )=T \bigl ( {\scriptsize\begin{matrix}{} X\cr Y \end{matrix}} \bigr )\). Then the map (7) becomes
where \(\tilde{f}_{1}(X,Y)=aX^{2}+\frac{a+c}{a1}XY\frac{1+c}{a(a1)}Y^{2}\) and \(\tilde{g}_{1}(X,Y)=\frac{a(1+c)}{a1}XY\frac{1+c}{a1}Y^{2}\).
Consider a center manifold with the form \(X=h(Y)=\tilde{\alpha}_{1} Y^{2}+\tilde{\beta}_{1} Y^{3}+O(Y^{4})\). Then it must satisfy
By approximate computation, for the center manifold, we obtain \(\tilde{\alpha}_{1}=\frac{1+c}{a(a1)^{2}}\) and \(\tilde{\beta}_{1}=\frac {(1+c)(2+a+3c)}{a(a1)^{4}}\). Hence, \(h(Y)=\frac{1+c}{a(a1)^{2}} Y^{2}\frac {(1+c)(2+a+3c)}{a(a1)^{4}} Y^{3}+O(Y^{4})\), and on the center manifold \(X=h(Y)\), the new map \(\hat{f}_{1}\) is given by
Computations show that \(\hat{f}_{1}'(0)=1\) and \(\hat{f}_{1}''(0)=\frac {2(1+c)}{a1}<0\). Hence, by [25] the exclusion fixed point \((\frac {a1}{a},0)\) is unstable. More precisely, it is a semistable fixed point from the right.
Therefore, \((\frac{a1}{a},0)\) is asymptotically stable when \(1< a\leq3\) and \(\max\{0,\frac{a(c1)}{a1}\}< d<\frac{a(1+c)}{a1}\).
(iii) Finally, we consider the positive fixed point \((x^{*},y^{*})=(\frac {1+c}{d},\frac{d(a1)a(1+c)}{d})\) for \(d>\frac{a(1+c)}{a1}\) (\(a>1\)). The Jacobian matrix evaluated at the positive fixed point \((x^{*},y^{*})\) is given by
and the characteristic equation of the Jacobian matrix \(J^{*}\) can be written as
According to the Jury conditions [9], in order to find the asymptotically stable region of \((x^{*},y^{*})\), we need to find the region that satisfies the following conditions:
Since
from the relations \(P^{*}(1)>0\), \(P^{*}(1)>0\), and \(\operatorname{det}J^{*}<1\) we have that
This completes the proof of Lemma 2.1. □
Bifurcations
In this section, we mainly focus on the flip bifurcation and NeimarkSacker bifurcation of the positive fixed point \((x^{*}, y^{*})\). We choose the parameter d as a bifurcation parameter for analyzing the flip bifurcation and NeimarkSacker bifurcation of \((x^{*}, y^{*})\) by using the center manifold theorem and bifurcation theory of [23, 24].
First, we have the following result on the flip bifurcation of system (3).
Theorem 3.1
System (3) undergoes a flip bifurcation at \((x^{*},y^{*})\) if the following conditions are satisfied: \(c>0\), \(a>3\), \(a\neq\frac{9+5c}{1+c}\), and \(d=\frac{a(1+c)(3+c)}{3+ac+ac}\). Moreover, if \(3< a<\frac{9+5c}{1+c}\), then period2 points that bifurcate from this fixed point are unstable.
Proof
If \(d^{*}=\frac{a(1+c)(3+c)}{3+ac+ac}\), then the eigenvalues of the fixed point \((x^{*},y^{*})\) are \(\lambda_{1} = 1\) and \(\lambda_{2} = \frac{6a+4cac}{3+c}\). The condition \(\lambda_{2}\neq1\) leads to \(a\neq3,\frac{9+5c}{1+c}\). In addition, note that the existence of the positive fixed point is assured by the relation \(d>\frac{a(1+c)}{a1}\) (\(a>1\)), so we get \(a>3\). Hence, we further assume that \(a>3\) and \(a\neq\frac{9+5c}{1+c}\).
Let \(u=xx^{*}\), \(v=yy^{*}\), and \(\bar{d}=dd^{*}\). We consider the parameter d̄ as a new and dependent variable. Then the map (3) becomes
where
Let
and use the translation \(\Bigl ( {\scriptsize\begin{matrix}{} u\cr \bar{d}\cr v \end{matrix}} \Bigr )=T \Bigl ( {\scriptsize\begin{matrix}{} X\cr \mu\cr Y \end{matrix}} \Bigr )\). Then the map (10) becomes
where
By the center manifold theorem we know that the stability of \((X,Y)=(0,0)\) near \(\mu=0\) can be determined by studying a oneparameter family of maps on a center manifold, which can be represented as follows:
Assume that
By approximate computation for the center manifold, we obtain
Thus, the map restricted to the center manifold is given by
where
If the map (11) undergoes a flip bifurcation, then it must satisfy the following conditions:
and
By a simple calculation we obtain
and
It is easy to check that if \(3< a<\frac{9+5c}{1+c}\), then \(\lambda _{2}<1\) and \(\alpha_{2}<0\). Thus, period2 points that bifurcate from this fixed point are unstable.
This completes the proof of Theorem 3.1. □
For NeimarkSacker bifurcation, we have the following theorem.
Theorem 3.2
System (3) undergoes a NeimarkSacker bifurcation at the fixed point \((x^{*}, y^{*})\) if the following conditions are satisfied: \(c>0\), \(1< a<9\), \(a\neq\frac{5+3c}{1+c}, \frac {7+4c}{1+c}\), and \(d=\bar{d}^{*}=\frac{a(2+c)}{a1}\). Moreover, \(k < 0\), and thus an attracting invariant closed curve bifurcates from the fixed point for \(d>\bar{d}^{*}\).
Proof
The characteristic equation associated with the linearized system (3) at the fixed point \((x^{*}(d),y^{*}(d))\) is given by
The eigenvalues of the characteristic equation (12) are given as
where \(p(d)= ca+(2ad)x^{*}+y^{*}\) and \(q(d)= ac+a(2c+d)x^{*}2ad{x^{*}}^{2}+cy^{*}\).
The eigenvalues \(\lambda_{1,2}\) are complex conjugates for \(p(d)^{2}4q(d)< 0\), which leads to
Let
We get \(q(d)=1\) and \(\lambda_{1,2}=\frac{5a+3cac}{2(2+c)}\pm\frac {i\sqrt{(1a)(1+c)(a95c+ac)}}{2(2+c)}=\rho\pm i\omega \). Under condition (14), we have
In addition, if \(p(\bar{d}^{*})\neq0, 1\), which leads to
then we obtain that \(\lambda_{1,2}^{n}(\bar{d}^{*})\neq1\) (\(n = 1, 2, 3, 4\)).
Letting \(u=xx^{*}\) and \(v=yy^{*}\), the map (3) becomes
where \(f_{1}(u,v)=au^{2}uv\) and \(f_{2}(u,v)=\frac{a(2+c)}{a1}uv\).
Let
and use the translation \(\bigl ( {\scriptsize\begin{matrix}{} u\cr v \end{matrix}} \bigr )=T \bigl ( {\scriptsize\begin{matrix}{} X\cr Y \end{matrix}} \bigr )\). Then the map (15) becomes
where
Notice that (16) is exactly in the form on the center manifold in which the coefficient k [23] is given by
where
Thus, a complex calculation gives
where \(A=3+14c+14c^{2}+4c^{3}+a^{4}(1+c)^{3}+a^{3}(1+c)^{2}(c^{2}3c6)3a^{2}(1+c)^{2}(c^{2}c4)+a(6+19c+35c^{2}+31c^{3}+11c^{4}+c^{5})\).
Thus the fixed point \((X,Y)=(0,0)\) is a NeimarkSacker bifurcation point for the map (16). This completes the proof. □
Numerical simulations
In this section, numerical simulations are given, including bifurcation diagrams, Lyapunov exponents, and fractal dimension and phase portraits, to illustrate the above theoretical analysis and to show new and more complex dynamic behaviors in system (3).
The fractal dimension [26–29] is defined by using Lyapunov exponents as follows:
with \(L_{1}, L_{2}, \ldots, L_{n}\) being Lyapunov exponents, where j is the largest integer such that \(\sum_{i=1}^{i=j}L_{i}\geq0\) and \(\sum_{i=1}^{i=j+1}L_{i}<0\).
Our model is a twodimensional map that has the fractal dimension of the form
Numerical simulations for stability and bifurcations of fixed points
We consider the following two cases.
Case 1. A bifurcation diagram of system (3) in \((d, x)\) plane for \(1.4\leq d\leq2.4\) and \(a = 3.5\) with initial value \((0.6, 0.3)\) is given in Figure 1(a). It shows that there is a flip bifurcation (labeled ‘PD’) emerging from the fixed point \((0.625, 0.3125)\) with \(d= 1.92\), \(\alpha_{1} = 3.53103\), and \(\alpha_{2} = 78.7813<0\).
Case 2. A bifurcation diagram of system (3) in \((d, x)\) plane is displayed in Figure 1(b) for \(3\leq d\leq5\) and \(a = 2.5\) with initial value \((0.3, 0.6)\). Figure 1(b) exhibits a NeimarkSacker bifurcation (labeled ‘NS’), which occurs at fixed point \((0.32727, 0.68182)\) and \(d = 3.66667\) with \(d_{1} = 0.24546 > 0\) and \(k = 0.52652< 0\). Figures 1(a) and 1(b) show the correctness of Theorems 3.1 and 3.2.
Further numerical simulations for system (3)
In this subsection, new and interesting dynamical behaviors are investigated as the parameters vary.
The bifurcation diagrams in the twodimensional plane are considered in the following four cases:

(i)
Varying d in the range \(0\leq d\leq4.5\) and fixing \(a= 3.4\), \(c=0.2\);

(ii)
Varying d in the range \(0 \leq d\leq4.5\) and fixing \(a= 3.6\), \(c=0.2\);

(iii)
Varying d in the range \(2.7\leq d\leq3.5\) and fixing \(a = 4.1\), \(c=0.2\);

(iv)
Varying a in the range \(0 \leq a\leq4.3\) and fixing \(d= 3.5\), \(c=0.2\).
Case (i). The bifurcation diagrams of system (3) in \((d,x)\) plane and in \((d,y)\) plane for \(a=3.4\) and \(c=0.2\) with initial value \((0.4, 1.0)\) are given in Figures 2(a) and 2(c), respectively, which show the dynamical changes of the prey and predator as d varies. From Figures 2(a) and 2(c) we can see that a NeimarkSacker bifurcation emerges at \(d\sim3.1\) and an attracting invariant cycle bifurcates from the fixed point since \(k=0.69494\) and \(d_{1}=0.462032\) by Theorem 3.2. We also observe the period6, 11, 24, 27 windows within the chaotic regions and boundary crisis at \(d =4.19\). The phase portraits corresponding to Figure 2(a) are shown in Figures 2(d)(f) for showing eleven and sixcoexisting chaotic attractors at \(d = 3.58\) and 3.72 and a chaotic attractor at \(d=3.8\). The maximum Lyapunov exponents corresponding to Figure 2(a) are computed in Figure 2(b). When \(d=3.72\), the maximum Lyapunov exponent is \(0.015>0\), which confirms the existence of the chaotic sets.
Case (ii). The bifurcation diagram of system (3) in \((d,x)\) plane for \(a=3.6\) and \(c=0.2\) with initial value \((0.4, 1.1)\) is disposed in Figure 3(a). The maximum Lyapunov exponents corresponding to Figure 3(a) are calculated in Figure 3(b), confirming the existences of chaotic regions and period orbits as the parameter d varying. Figures 3(a) and 3(b) clearly depict two onsets of chaos at \(d=0\) and \(d\sim3.195\), respectively, which are the crisis. The nonattracting chaotic set at \(d=1.9\) and chaotic attractor at \(d=3.7\) are shown in Figures 3(e) and 3(f), respectively. Figure 3(c) is the bifurcation diagram in \((d,y)\) for \(a=3.6\) and \(c=0.2\), which shows the dynamical changes of the predator as d varies. Comparing Figures 3(a) and 3(c), we see that there are similar dynamics for \(d\in(2, 4.1)\), but when the prey is in chaotic dynamic for \(d\in(0, 2)\), the predator tends to extinct (we also can see phase portrait Figure 3(d)).
Case (iii). The bifurcation diagram of system (3) in \((d,x)\) plane for \(a= 4.1\) and \(c=0.2\) with initial value \((0.4, 1.4)\) is shown in Figure 4(a). Figures 4(d) and 4(e) show the local amplifications for \(d\in(3.08, 3.24)\) and \(d\in(3.2, 3.23)\) in (a), respectively. The diagrams show that there is a stable fixed point for \(d\in(2.7, 2.9097)\) and the fixed point loses its stability as d increases. NeimarkSacker bifurcation occurs at \(d\sim2.9097\), and invariant circle appears as d increases, and the invariant circle suddenly becomes to period9 orbits at \(d\sim3.085\). Then, as d grows, chaos appears at \(d\sim3.16\), the chaotic behavior disappears suddenly and transforms to period18 orbits at \(d\sim3.203\). Particularly, the chaotic behavior disappears suddenly and becomes to period11 orbits at \(d\sim3.3741\). The phase portraits for various values of d are shown in Figures 5(a)(i). From Figure 5 we observe that there are period9 and 11 orbits, ninecoexisting chaotic attractors, and attracting chaotic sets.
The maximum Lyapunov exponents and fractal dimension corresponding to (a) are given in Figures 4(b) and 4(c), respectively. The maximum Lyapunov exponents are negative for the parameter \(d\in (2.7, 2.91)\), whereas the fixed point is stable. For \(d\in(2.91, 3.17)\), the maximum Lyapunov exponents are in the neighborhood of zero, which corresponds to quasiperiod solutions or coexistence of chaos and quasiperiod solutions. For \(d\in(3.17, 3.5)\), the maximum Lyapunov exponents are positive with a few negative, which shows that a period window occurs in the chaotic region.
Case (iv). The bifurcation diagram of system (3) in \((a,x)\) plane for \(d= 3.5\) and \(c=0.2\) with initial value \((0.3, 0.7)\) is shown in Figure 6(a). Figures 6(d) and 6(e) are the local amplifications for \(a\in (3.4,3.72)\) and \(a\in(3.5, 3.7)\) in (a). The maximum Lyapunov exponents and fractal dimension corresponding to (a) are plotted in Figures 6(b) and 6(c), respectively. From Figure 6(b) we see that some Lyapunov exponents are greater than 0, some are smaller than 0,and thus there exist stable fixed points or period windows in the chaotic region. The diagrams show that there is a stable fixed point for \(a\in(1.8, 2.69)\), and the fixed point loses its stability as a increases. NeimarkSacker bifurcation occurs at \(a\sim2.69\), and invariant circle appears as a increases, and the invariant circle suddenly becomes to period6 orbits at \(a\sim2.93\) and period11 orbits at \(a\sim3.35\). Furthermore, as a grows, we can observe the period6, 16, 20, 21, 27, and 37 windows within the chaotic regions and boundary crisis at \(a=4.09\). The phase portraits for various values of a are shown in Figure 7, which clearly depicts how a smooth invariant circle bifurcates from the stable fixed point and an invariant circle to chaotic attractors.
Chaos control
In this section, we apply the state feedback control method [30–32] to stabilize chaotic orbits at an unstable fixed point of system (3).
Consider the following controlled form of system (3):
with the following feedback control law as the control force:
where \(k_{1}\) and \(k_{2}\) are the feedback gains, and \((x^{*}, y^{*})\) is the positive fixed point of system (3).
The Jacobian matrix J of the controlled system (17) evaluated at the fixed point \((x^{*}, y^{*})\) is given by
and the characteristic equation of the Jacobian matrix \(J(x^{*}, y^{*})\) is
Assume that the eigenvalues are \(\lambda_{1}\) and \(\lambda_{2}\). Then
and
The lines of marginal stability are determined by the equations \(\lambda _{1} = \pm1\) and \(\lambda_{1}\lambda_{2}= 1\). These conditions guarantee that the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) have moduli equal to 1.
Assume that \(\lambda_{1}\lambda_{2}= 1\). Then from (19) we have
Assume that \(\lambda_{1} = 1\). Then from (18) and (19) we get
Assume that \(\lambda_{1} = 1\). Then from (18) and (19) we obtain
The stable eigenvalues lie within a triangular region by lines \(l_{1}\), \(l_{2}\), and \(l_{3}\) (see Figure 8(a)).
Some numerical simulations have been performed to see how the state feedback method controls the unstable fixed point. The parameter values are fixed as \(a=3.4\), \(d=3.8\), \(c=0.02\). The initial value is \((0.3, 1.25)\), and the feedback gains are \(k_{1} = 0.5\) and \(k_{2} =0.04\). Figures 8(b) and 8(c) show that a chaotic trajectory is stabilized at the fixed point \((0.315789, 1.32632)\).
Conclusions
In this paper, we have investigated the complex dynamic behaviors of the predatorprey system (3). By using the center manifold theorem and the bifurcation theory we proved that the discretetime system (3) can undergo a flip bifurcation and a NeimarkSacker bifurcation. Moreover, system (3) displays much more interesting dynamical behaviors, which include orbits of period6, 11, 16, 18, 20, 21, 24, 27, and 37, invariant cycles, quasiperiodic orbits, and chaotic sets. They all imply that the predator and prey can coexist at periodn oscillatory balance behaviors or a oscillatory balance behavior, but the predatorprey system is unstable if a chaotic behavior occurs. In particular, we observe that when the prey is chaotic, the predator will ultimately tend to extinct or tend to a stable fixed point. In comparison with system (1) for \(c=0\) in [22], system (3) exhibits different dynamical behaviors in the stability properties and the bifurcation structures. These results show far richer dynamics of the discretetime model. Finally, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.
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Acknowledgements
The authors would like to thank the reviewers and the editor for very helpful suggestions and comments, which led to improvements of our original paper. ZX is the corresponding author and is supported in part by NNSFC (No. 91420202) and the Project of Construction of Innovative Teams and Teacher Career Development for Universities and Colleges Under Beijing Municipality (CIT and TCD201504041, IDHT20140508). CL is supported by the National Natural Science Foundation of China (Nos. 61134005, 11272024).
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The main idea of this paper was proposed by MZ, ZX, and CL. MZ, ZX, and CL prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
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Zhao, M., Xuan, Z. & Li, C. Dynamics of a discretetime predatorprey system. Adv Differ Equ 2016, 191 (2016). https://doi.org/10.1186/s1366201609036
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MSC
 39A28
 39A30
Keywords
 predatorprey system
 flip bifurcation
 NeimarkSacker bifurcation
 feedback control