Dynamics of a discrete-time predator-prey system
- Ming Zhao^{1},
- Zuxing Xuan^{2}Email author and
- Cuiping Li^{1}
https://doi.org/10.1186/s13662-016-0903-6
© Zhao et al. 2016
Received: 22 March 2016
Accepted: 19 June 2016
Published: 13 July 2016
Abstract
We investigate the dynamics of a discrete-time predator-prey 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 Neimark-Sacker 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 period-6, -11, -16, -18, -20, -21, -24, -27, and -37 orbits, attracting invariant cycles, quasi-periodic orbits, nice chaotic behaviors, which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discrete-time predator-prey system. Finally, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.
Keywords
MSC
1 Introduction
It is well known that the Lotka-Volterra predator-prey model [1, 2] is one of the most important population models. There are also many other predator-prey 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 predator-prey models with two variables. However, discrete-time models described are more reasonable than the continuous-time models when populations have nonoverlapping generations. Moreover, using discrete-time 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 discrete-time predator-prey system and prove that the system undergoes flip bifurcation and Neimark-Sacker bifurcation. Recently, the complex dynamics of a discrete-time predator-prey system is investigated in [11]. By analysis it is proved that the discrete-time model has different properties and structures compared with the continuous one. Such systems discussed as discrete-time models can also be found in [12–21] and references therein.
Motivation of this paper is to investigate system (1) in detail. Here we derive the conditions of existence for flip bifurcation and Neimark-Sacker 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 period-6, -11, -16, -18, -20, -21, -24, -27, -37 orbits, attracting invariant cycles, quasi-periodic 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 discrete-time predator-prey 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 codimension-one bifurcations, including flip bifurcation and Neimark-Sacker 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.
2 Existence and stability of fixed points
It is easy to see that system (3) has one extinction fixed point \((0,0)\), one exclusion fixed point \((\frac{a-1}{a},0)\) for \(a>1\), and one coexistence fixed point \((x^{*},y^{*})=(\frac{1+c}{d},\frac{d(a-1)-a(1+c)}{d})\) for \(d>\frac {a(1+c)}{a-1}\) 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
- (i)
\((0,0)\) is asymptotically stable if \(0< a,c<1\);
- (ii)
\((\frac{a-1}{a},0)\) is asymptotically stable if \(1< a\leq3\) and \(\max\{0,\frac{a(c-1)}{a-1}\}< d<\frac{a(1+c)}{a-1}\);
- (iii)\((\frac{1+c}{d},\frac{d(a-1)-a(1+c)}{d})\) is asymptotically stable if and only if one of the following conditions holds:
- (a)
\(1< a\leq3\), \(c>0\) and \(\frac{a(1+c)}{a-1}< d<\frac{a(2+c)}{a-1}\);
- (b)
\(3< a\leq5\), \(c>0\) and \(\frac{a(1+c)(3+c)}{3+a-c+ac}< d<\frac {a(2+c)}{a-1}\);
- (c)
\(5< a<9\), \(0< c<\frac{9-a}{a-5}\) and \(\frac {a(1+c)(3+c)}{3+a-c+ac}< d<\frac{a(2+c)}{a-1}\).
- (a)
Proof
(i) For the fixed point \((0,0)\), the corresponding characteristic equation is \(\lambda^{2}-(a-c)\lambda-ac=0\), 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 \(c>1\).
Clearly, \(J_{0}\) has characteristic roots \(\lambda_{1}=2-a\), \(\lambda_{2}=-c+\frac{d(a-1)}{a}\). Then \(|\lambda_{i}|<1\) (\(i=1,2\)) if and only if \(1< a<3\) and \(\max\{0,\frac{a(c-1)}{a-1}\}< d<\frac{a(1+c)}{a-1}\).
We further will prove that when \(a=3\), the exclusion fixed point \((\frac {a-1}{a},0)\) is asymptotically stable and when \(d=\frac{a(1+c)}{a-1}\), it is unstable by using center manifold theory.
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 \((\frac{a-1}{a},0)\) is asymptotically stable.
Computations show that \(\hat{f}_{1}'(0)=1\) and \(\hat{f}_{1}''(0)=-\frac {2(1+c)}{a-1}<0\). Hence, by [25] the exclusion fixed point \((\frac {a-1}{a},0)\) is unstable. More precisely, it is a semistable fixed point from the right.
Therefore, \((\frac{a-1}{a},0)\) is asymptotically stable when \(1< a\leq3\) and \(\max\{0,\frac{a(c-1)}{a-1}\}< d<\frac{a(1+c)}{a-1}\).
3 Bifurcations
In this section, we mainly focus on the flip bifurcation and Neimark-Sacker bifurcation of the positive fixed point \((x^{*}, y^{*})\). We choose the parameter d as a bifurcation parameter for analyzing the flip bifurcation and Neimark-Sacker 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 \(d=\frac{a(1+c)(3+c)}{3+a-c+ac}\). Moreover, if \(3< a<\frac{9+5c}{1+c}\), then period-2 points that bifurcate from this fixed point are unstable.
Proof
If \(d^{*}=\frac{a(1+c)(3+c)}{3+a-c+ac}\), then the eigenvalues of the fixed point \((x^{*},y^{*})\) are \(\lambda_{1} = -1\) and \(\lambda_{2} = \frac{6-a+4c-ac}{3+c}\). 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 \(d>\frac{a(1+c)}{a-1}\) (\(a>1\)), so we get \(a>3\). Hence, we further assume that \(a>3\) and \(a\neq\frac{9+5c}{1+c}\).
It is easy to check that if \(3< a<\frac{9+5c}{1+c}\), then \(|\lambda _{2}|<1\) and \(\alpha_{2}<0\). Thus, period-2 points that bifurcate from this fixed point are unstable.
This completes the proof of Theorem 3.1. □
For Neimark-Sacker bifurcation, we have the following theorem.
Theorem 3.2
System (3) undergoes a Neimark-Sacker 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 \(d=\bar{d}^{*}=\frac{a(2+c)}{a-1}\). Moreover, \(k < 0\), and thus an attracting invariant closed curve bifurcates from the fixed point for \(d>\bar{d}^{*}\).
Proof
Thus the fixed point \((X,Y)=(0,0)\) is a Neimark-Sacker bifurcation point for the map (16). This completes the proof. □
4 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).
4.1 Numerical simulations for stability and bifurcations of fixed points
We consider the following two cases.
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 Neimark-Sacker 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.
4.2 Further numerical simulations for system (3)
In this subsection, new and interesting dynamical behaviors are investigated as the parameters vary.
- (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 \(d= 3.5\), \(c=0.2\).
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 quasi-period solutions or coexistence of chaos and quasi-period 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.
5 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).
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 \((0.315789, 1.32632)\).
6 Conclusions
In this paper, we have investigated the complex dynamic behaviors of the predator-prey system (3). By using the center manifold theorem and the bifurcation theory we proved that the discrete-time system (3) can undergo a flip bifurcation and a Neimark-Sacker bifurcation. Moreover, system (3) displays much more interesting dynamical behaviors, which include orbits of period-6, -11, -16, -18, -20, -21, -24, -27, and -37, invariant cycles, quasi-periodic orbits, and chaotic sets. They all imply that the predator and prey can coexist at period-n oscillatory balance behaviors or a oscillatory balance behavior, but the predator-prey 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 discrete-time model. Finally, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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