Bifurcation and complex dynamics of a discrete-time predator-prey system with simplified Monod-Haldane functional response
- Sarker Md Sohel Rana^{1}Email author
Received: 16 July 2015
Accepted: 27 October 2015
Published: 5 November 2015
Abstract
In this paper, we investigate the dynamics of a discrete-time predator-prey system with simplified Monod-Haldane functional response. The existence and local stability of positive fixed point of the discrete dynamical system is analyzed algebraically. It is shown that the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation in the interior of \(\mathbb {R}^{2}_{+}\) by using bifurcation theory. Numerical simulation results not only show the consistence with the theoretical analysis but also display new and interesting dynamical behaviors, including phase portraits, period-11 orbits, attracting invariant circle, cascade of period-doubling bifurcation from period-11 leading to chaos, quasi-periodic orbits, and the sudden disappearance of the chaotic dynamics and attracting chaotic set. The Lyapunov exponents are numerically computed to characterize the complexity of the dynamical behaviors.
Keywords
discrete-time predator-prey system simplified Monod-Haldane functional response bifurcations chaos Lyapunov exponentsMSC
37D45 37G35 39A30 39A331 Introduction
It is well known the Lotka-Voltera predator-prey model is one of the fundamental population models; a predator-prey interaction has been described first by two pioneers Lotka [1] and Voltera [2] in two independent works. After them, more realistic predator-prey model were introduced by Holling suggesting three types of functional responses for different species to model the phenomena of predation [3]. Qualitative analyses of predator-prey models describe by set of differential equations were studied by many authors [4–8]. Another possible way to understand a predator-prey interaction is by using discrete-time models. In recent years, many authors [4, 5, 9–17] have suggested that discrete-time models governed by difference equations are more appropriate than the continuous ones, especially when the populations have non-overlapping generations. These models are more reasonable, showing that the dynamics of the discrete-time predator-prey models can present a much richer set of patterns than those observed in continuous-time models and lead to unpredictable dynamic behaviors from a biological point of view. However, there are few articles discussing the dynamical behaviors of predator-prey models, which include bifurcations and chaos phenomena for the discrete-time models. Hence, the discrete version has been an important subject of study in diverse phenomenology from the mathematical point of view. The authors [13, 15–17] obtained the flip bifurcation and Hopf bifurcation by using the center manifold theorem and bifurcation theory, while in [9–11], the authors only showed the flip bifurcation and Hopf bifurcation by using numerical simulations. But in [18–21], the authors showed that the system undergoes a flip bifurcation and/or a Neimark-Sacker (NS) bifurcation by using bifurcation theory. Many important and interesting research works on bifurcation theory can be found in [19, 22] and the references cited therein.
This paper is organized as follows. In Section 2, we discuss the existence and local stability of positive fixed point for system (3) in \(\mathbb {R}^{2}_{+}\). In Section 3, we show that there exist some values of the parameters such that (3) undergoes the flip bifurcation and the NS bifurcation in the interior of \(\mathbb {R}^{2}_{+}\). In Section 4, we present the numerical simulations including the bifurcation diagrams, the phase portraits at neighborhood of critical values and the maximum Lyapunov exponents corresponding to the bifurcation diagrams. Finally a short discussion is given in Section 5.
2 Existence and stability of fixed points
In this section, we first determine the existence of the fixed points of system (3), then investigate their stability by the eigenvalues for the Jacobian matrix of (3) at the fixed point.
By a simple algebraic computation, it is straightforward to obtain the following results:
Lemma 2.1
(i) For all permissible parameter values, system (3) has two fixed points, \(E_{0}(0,0)\) and \(E_{1}(K,0)\);
- (iii.a)
When \(K\le x_{1}^{*}\), then system (3) has no positive fixed points;
- (iii.b)
when \(x_{1}^{*}< K< x_{2}^{*}\), then system (3) has one positive fixed points \(E_{2}(x_{1}^{*}, y_{1}^{*})\);
- (iii.c)
when \(x_{1}^{*}< x_{2}^{*}< K\), then system (3) has two positive fixed points \(E_{2}(x_{1}^{*}, y_{1}^{*})\) and \(E_{2}(x_{2}^{*}, y_{2}^{*})\).
In the following we deduce the local dynamics of the positive fixed point \(E_{2}(x^{*}, y^{*})\) only (we left the others). Note that the local stability of the fixed point \((x^{*}, y^{*})\) is determined by the modules of eigenvalues of the characteristic equation at the fixed point.
Using Jury’s criterion [25], we have necessary and sufficient condition for local stability of the fixed point \(E_{2}\), which are given in the following proposition.
Proposition 2.2
- (i)it is a sink if one of the following conditions holds:
- (i.1)
\(\Delta\geq0\) and \(\delta<\frac {-(a_{1}+b_{2})-\sqrt{\Delta}}{a_{1} b_{2}+a_{2} b_{1}}\);
- (i.2)
\(\Delta<0\) and \(\delta<-\frac{a_{1}+b_{2}}{a_{1} b_{2}+a_{2} b_{1}}\);
- (i.1)
- (ii)it is a source if one of the following conditions holds:
- (ii.1)
\(\Delta\geq0\) and \(\delta>\frac {-(a_{1}+b_{2})+\sqrt{\Delta}}{a_{1} b_{2}+a_{2} b_{1}}\);
- (ii.2)
\(\Delta<0\) and \(\delta>-\frac{a_{1}+b_{2}}{a_{1} b_{2}+a_{2} b_{1}}\);
- (ii.1)
- (iii)it is non-hyperbolic if one of the following conditions holds:
- (iii.1)
\(\Delta\geq0\) and \(\delta=\frac {-(a_{1}+b_{2}) \pm\sqrt{\Delta}}{a_{1} b_{2}+a_{2} b_{1}}\);
- (iii.2)
\(\Delta<0\) and \(\delta=-\frac{a_{1}+b_{2}}{a_{1} b_{2}+a_{2} b_{1}}\);
- (iii.1)
- (iv)
it is a saddle for the other values of parameters except those values in (i)-(iii).
The next result is obtained from the above analysis to study the bifurcation of system (3).
Proposition 2.3
- (i)
via a flip point when \(\Delta\geq0\) and \(\delta=\frac{-(a_{1}+b_{2}) \pm\sqrt{\Delta}}{a_{1} b_{2}+a_{2} b_{1}}\);
- (ii)
via a Neimark-Sacker point when \(\Delta<0\) and \(\delta=-\frac{a_{1}+b_{2}}{a_{1} b_{2}+a_{2} b_{1}}\).
3 Bifurcation analysis
In this section, we choose the parameter δ as a bifurcation parameter to study the flip bifurcation and the Neimark-Sacker bifurcation of \(E_{2}\), respectively, by using bifurcation theory in (see Section 4 in [24]; see also [26–28]).
Therefore, \(B(x, y) = \bigl({\scriptsize\begin{matrix}{} B_{1}(x, y) \cr B_{2}(x, y) \end{matrix}} \bigr) \) and \(C(x, y, u) = \bigl({\scriptsize\begin{matrix}{} C_{1}(x, y, u) \cr C_{2}(x, y, u)\end{matrix}} \bigr) \) are symmetric multilinear vector functions of \(x, y, u \in \mathbb {R}^{2}\).
It is easy to see \(\langle p, q \rangle=1\), where \(\langle\cdot , \cdot \rangle\) means the standard scalar product in \(\mathbb {R}^{2}: \langle p, q \rangle=p_{1} q_{2} + p_{2} q_{1}\).
From the above analysis and the theorem in [24, 26–28], we have the following result.
Theorem 3.1
Suppose that \(E_{2}(x^{*}, y^{*})\) is the positive fixed point. If the conditions (13) and (14) hold and \(c(\delta_{F})\ne0\), then system (3) undergoes a flip bifurcation at the fixed point \(E_{2}(x^{*}, y^{*})\) when the parameter δ varies in a small neighborhood of \(\delta_{F}\). Moreover, if \(c(\delta_{F})>0\) (resp., \(c(\delta_{F})<0\)), then the period-2 orbits that bifurcate from \(E_{2}(x^{*}, y^{*})\) are stable (resp., unstable).
We next discuss the existence of a Neimark-Sacker bifurcation by using the NS theorem in [24, 26–28].
It is easy to see \(\langle p, q \rangle=1\), where \(\langle\cdot , \cdot \rangle\) means the standard scalar product in \(\mathbb {C}^{2}: \langle p, q \rangle=\bar{p_{1}} q_{2} + \bar{p_{2}} q_{1}\).
Clearly, (20) and (21) demonstrate that the transversal condition and the nondegenerate condition of system (3) are satisfied. So, summarizing the above discussions, we obtain the following conclusion.
Theorem 3.2
Suppose that \(E_{2}(x^{*}, y^{*})\) is the positive fixed point. If \(a(\delta _{NS}) \ne0\), then system (3) undergoes a Neimark-Sacker bifurcation at the fixed point \(E_{2}\) when the parameter δ varies in the small neighborhood of \(NSB_{E_{2}}\). Moreover, if \(a(\delta_{NS}) < 0\) (resp., >0), then the NS bifurcation of system (3) at \(\delta= \delta_{NS}\) is supercritical (resp., subcritical) and there exists a unique closed invariant curve bifurcation from \(E_{2}\) for \(\delta= \delta_{NS}\), which is attracting (resp., repelling).
Example 3.1
Consider system (3) with \(r = 2\), \(K = 1.2\), \(d = 0.25\), \(\alpha= 0.75\), \(\beta= 2\), \(\delta= \delta_{F} = 1.40356\). Then \((r, K, d, \alpha, \beta, \delta) \in FB1_{E_{2}}\) and there is a unique positive fixed point \((1, 1)\) with multipliers \(\lambda_{1}=-1\), \(\lambda _{2}=0.972639\), and \(c(\delta_{F}) = -23.9313\). Hence, according to Theorem 3.1, the flip bifurcation emerges from the fixed point \((1, 1)\) at \(\delta= \delta_{F}\).
4 Numerical simulations
In this section, by using numeral simulation, we give the bifurcation diagrams, phase portraits and Lyapunov exponents of system (3) to confirm the previous analytic results and show some new interesting complex dynamical behaviors existing in system (3). It is known that maximum Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and frequently are employed to identify chaotic behavior. Since the dynamics of discrete prey-predator model with Holling type I, II, and III functional response has been examined by many researchers, we will now mainly focus our attention on the effect of simplified Monod-Haldane functional response. Based on the previous analysis, we choose the parameter δ as a bifurcation parameter (varied parameter) and the other model parameters are taken as fixed parameters, unless otherwise stated; to study the flip bifurcation and the Neimark-Sacker bifurcation, respectively, for the unique positive fixed point, one can consider the initial condition \((x_{0}, y_{0})\) situated in the basin of attraction of fixed point. Without loss of generality, the bifurcation parameters are considered in the following cases:
Case (i) varying δ in range \(0.5 \le\delta\le1.74\), and fixing \(r = 2\), \(K = 1.2\), \(d = 0.25\), \(\alpha= 0.9\), \(\beta= 1\);
Case (ii) varying β in range \(0.7 \le\beta\le1.25\), and fixing \(r = 2\), \(K = 1.2\), \(d = 0.25\), \(\alpha= 0.9\), \(\delta= 0.816737\).
For case (i): The bifurcation diagrams of system (3) in the \((\delta-x-y)\) space, the \((\delta-x)\) plane and the \((\delta-y)\) plane are given in Figure 2(a)-(b)-(c). After calculation for the fixed point \(E_{2}\) of map (3), the NS bifurcation emerges from the fixed point \((0.303337, 1.63195)\) at \(\delta= \delta_{NS} = 0.816737\) and \((r, K, d, \alpha, \beta, \delta) \in NSB_{E_{2}}\). It shows the correctness of Proposition 2.2. For \(\delta= \delta _{NS}\), we have λ, \(\bar{\lambda} = 0.89639 \pm0.443267 i\), \(|\lambda| = 1\), \(|\bar{\lambda}| = 1\), \(\frac{d|\lambda(\delta)|}{d\delta }|_{\delta= \delta_{NS}} = 0.126859>0\), \(\lambda_{NS} (a_{1} + b_{2}) = -0.20722 \ne-2, -3\), \(g_{20} = 0.288682 + 0.020187 i\), \(g_{11} = 0.24234 - 0.138155 i\), \(g_{02} = -0.503202 + 0.679727 i\), \(g_{21} = 0.058348 - 0.248692 i\), and \(a(\delta_{NS}) = -0.174522\). Therefore, the NS bifurcation is supercritical and it shows the correctness of Theorem 3.2.
From Figure 2(b)-(c), we observe that the fixed point \(E_{2}\) of map (3) is stable for \(\delta< 0.816737\) and loses its stability at \(\delta= 0.816737\) and an invariant circle appears when the parameter δ exceeds 0.816737, we also observe that there are period-doubling phenomenons. The maximum Lyapunov exponents corresponding to Figure 2(b)-(c) are computed and plotted in Figure 2(d), confirming the existence of the chaotic regions and period orbits in the parametric space. From Figure 2(d), we observe that some Lyapunov exponents are bigger than 0, some are smaller than 0, so there exist stable fixed points or stable period windows in the chaotic region. In general the positive Lyapunov exponent is considered to be one of the characteristics implying the existence of chaos. The bifurcation diagrams for x and y together with maximum Lyapunov exponents is presented in Figure 2(e). Figure 2(f) is the local amplification corresponding to Figure 2(b) for \(\delta\in[1.2614, 1.6309]\).
For case (ii): Figure 3 demonstrates the dynamic behavior of the model (3) when the parameter β varies. The bifurcation diagram for system (3) is plotted as a function of the control parameter β. It turns out that the fixed point \(E_{2}\) of map (3) loses its stability through a NS bifurcation, when β varies around 1. As β decreases the behavior of this model becomes very complicated, including the NS bifurcation and many chaotic bands. The sign of the maximal Lyapunov exponent confirms the existence of the strange attractor.
4.1 Sensitive dependence on initial conditions
Also, a sensitive dependence on the initial conditions of two trajectories for y-coordinate of the model (3) is plotted in Figure 6(b). The initial conditions of two trajectories for y-coordinate differ by \(1 \times10^{-3}\), while the other coordinate is kept at the same value. In Figure 6, it is shown that the trajectories of system (3) sensitively depend on the initial conditions, i.e. complex dynamic behavior occurs with initial perturbation.
5 Discussions
In this paper, we investigated the behaviors of the discrete-time predator-prey system (3) involving group defense with simplified Monod-Haldane functional response and showed that it has a complex dynamics in the closed first quadrant \(\mathbb {R}^{2}_{+}\). We showed that the unique positive fixed point of (3) can undergo a flip bifurcation and a NS bifurcation under certain parametric conditions. Some other basic dynamical properties of system (3) have been analyzed by means of bifurcation diagrams, phase portraits, Lyapunov exponents, and the sensitive dependence on the initial conditions. More precisely, as the parameters vary, system (3) exhibits a variety of dynamical behaviors, including period-11 orbits, an invariant cycle, a cascade of period-doubling, quasi-periodic orbits, and chaotic sets, which imply that the predators and prey can coexist in the stable period-n orbits and invariant cycle. Finally, simulation works showed that in certain regions of the parameter space, the model (3) had a great sensitivity to the choice of the initial conditions and parameter values. These results reveal a far richer dynamics of the discrete model compared to the continuous model.
Declarations
Acknowledgements
The author would like to thank the editor and the referees for their valuable comments and suggestions which led to the improvement of the paper.
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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