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 Open Access
Bifurcation, chaos analysis and control in a discretetime predator–prey system
 Weiyi Liu^{1}Email author and
 Donghan Cai^{1}
https://doi.org/10.1186/s1366201919506
© The Author(s) 2019
 Received: 1 September 2018
 Accepted: 7 January 2019
 Published: 17 January 2019
Abstract
The dynamical behavior of a discretetime predator–prey model with modified Leslie–Gower and Holling’s type II schemes is investigated on the basis of the normal form method as well as bifurcation and chaos theory. The existence and stability of fixed points for the model are discussed. It is showed that under certain conditions, the system undergoes a Neimark–Sacker bifurcation when bifurcation parameter passes a critical value, and a closed invariant curve arises from a fixed point. Chaos in the sense of Marotto is also verified by both analytical and numerical methods. Furthermore, to delay or eliminate the bifurcation and chaos phenomena that exist objectively in this system, two control strategies are designed, respectively. Numerical simulations are presented not only to validate analytical results but also to show the complicated dynamical behavior.
Keywords
 Predator–prey model
 Local stability
 Neimark–Sacker bifurcation
 Marotto’s chaos
 Bifurcation control
 Chaos control
MSC
 37N25
 34H10
 34H15
 34H20
 37M20
 39A28
1 Introduction

\(a_{1}\) and \(a_{2}\) are the natural growth rates of the prey species and the predator species, respectively.

\(b_{1}\) measures the strength of competition among individuals of the prey species.

\(c_{1}\) is the maximum value of the per capita reduction of the prey species due to the predator species, \(c_{2}\) has a similar meaning to \(c_{1}\).

\(e_{1}\) and \(e_{2}\) are the extent to which the environment provides protection to the prey species and the predator species, respectively.
Many differential equations cannot be solved using symbolic computation (“analysis”). For practical purposes, people sometimes construct difference equation to approximate differential equation so that they can write code to solve differential equation numerically. Moreover, while continuous models have been successfully applied in a variety of situations, one fundamental assumption is that the species in question has continuous, overlapping generations. However, it is observed in nature that many species do not possess this quality. For example, some anadromous fishes, such as salmon, have annual spawning seasons, with births taking place at the same time every year. Many insects breed and die before the next generation emerges, often having overwintered as eggs, larvae or pupae. Annual plant species also set seed and die before the next generation germinates. Populations with this characteristic of nonoverlapping generations are much better described by discretetime (difference equations) model than continuous equations (Hu et al., [23]). In addition, the earlier work [24, 25] showed that the discrete dynamics of the onedimensional logistic map can produce a much richer set of patterns than those observed in continuoustime model. Therefore, in this paper, we work under a different perspective, where we will focus on the difference scheme of Eq. (1.1).
In the present paper, we shall consider how the natural growth rates of predator and prey affect the dynamical behavior of model system (1.4). The main purpose of the paper is to show that (1.4) possesses the Neimark–Sacker bifurcation and chaos in the sense of Marotto. Especially, using a hybrid control strategy, the bifurcation threshold value can be raised to a prior setting one so that bifurcation phenomenon be delayed or eliminated in practice. Additionally, if the system is in chaotic state under certain parametric conditions, the chaos orbits can be stabilized to an unstable fixed point by a controller. Numerical simulations are presented to illustrate the analytic results, and to obtain even more dynamical behavior of (1.4), including diagrams for bifurcation, time series plots, phase portraits, strange attractors and the largest Lyapunov exponents.
The paper is organized as follows. In Sect. 2 we discuss the existence and stability of fixed points for model system (1.4). In Sect. 3, we give some details as regards bifurcation analysis of (1.4) as well as accurate control of bifurcation phenomenon. In Sect. 4 conditions on the existence of chaos in the sense of Marotto are given, and some control techniques have been used to stabilize chaos orbits. Finally, some conclusions close the paper in Sect. 5.
2 The existence and stability of fixed points
2.1 Biomass equilibria and their existence
 (i)
The trivial fixed point \(E_{0}=(0,0)\).
 (ii)
The predator free axial fixed point \(E_{1}=(\frac{1}{ \beta },0)\). The biological interpretation of this boundary fixed point is that the prey population reaches in the carrying capacity in the absence of predators.
 (iii)
The steady state of coexistence (interior fixed point) \(E_{2}=(\eta ,\eta +e_{2})\), if
 (H1)
\(me_{2}< e_{1} \)
2.2 Dynamical behavior: stability analysis
In this subsection, we deal with local stability of (1.4). Let \(J_{k}\) denote the Jacobian matrix of (1.4) at the fixed point \(E_{k}\), \(k=0,1,2\), and let \(\lambda _{1}\) and \(\lambda _{2}\) be the two eigenvalues of \(J_{k}\). We first recall some definitions of topological types for a fixed point.
Definition 2.1
 (i)
hyperbolic fixed point, if \(\lambda _{1}\neq 1\) and \(\lambda _{2}\neq 1\);
 (ii)
nonhyperbolic fixed point, if \(\lambda _{1}=1\) or \(\lambda _{2}=1\).
Definition 2.2
 (i)
sink, if \(\lambda _{1}<1\) and \(\lambda _{2}<1\);
 (ii)
source, if \(\lambda _{1}>1\) and \(\lambda _{2}>1\);
 (iii)
saddle, if \(\lambda _{1,2}\) are real with \(\lambda _{1}<1\) and \(\lambda _{2}>1\) (or \(\lambda _{1}>1\) and \(\lambda _{2}<1\)).
2.2.1 The behavior of the model system (1.4) around \(E_{0}\)
2.2.2 The behavior of the model system (1.4) around \(E_{1}\)
2.2.3 The behavior of the model system (1.4) around \(E_{2}\)
In order to discuss the stability of the fixed points, we also need the following lemma, which can be easily proved by the relations between roots and coefficients of a quadratic equation [26].
Lemma 2.1
 (i)
\(\lambda _{1}<1\) and \(\lambda _{2}<1\) iff \(\varPhi (1)>0\) and \(C<1\);
 (ii)
\(\lambda _{1}>1\) and \(\lambda _{2}>1\) iff \(\varPhi (1)>0\) and \(C>1\);
 (iii)
\(\lambda _{1}<1\) and \(\lambda _{2}>1\) (or \(\lambda _{1}>1\) and \(\lambda _{2}<1\)) iff \(\varPhi (1)<0\);
 (iv)
\(\lambda _{1}\) and \(\lambda _{2}\) are complex and \(\lambda _{1}=\lambda _{2}=1\) iff \(B^{2}4C<0\) and \(C=1\);
 (v)
\(\lambda _{1}=1\) and \(\lambda _{2}\neq 1\) iff \(\varPhi (1)=0\) and \(B\neq 0,2\).
Using Definition 2.2 and Lemma 2.1, we obtain the following results.
Theorem 2.1
 (i)a sink if one of the following conditions holds:
 (i.1)
\(0< s_{0}+b<1\) and \(\frac{s_{0}1}{s_{0}+b}< s<\frac{2(1+s _{0})}{s_{0}+b+1}\);
 (i.2)
\(1< s_{0}+b<0\) and \(s<\min \{\frac{2(1+s _{0})}{s_{0}+b+1},\frac{s_{0}1}{s_{0}+b} \}\);
 (i.3)
\(s_{0}+b<1\) and \(\frac{s_{0}1}{s_{0}+b}>s>\frac{2(1+s _{0})}{s_{0}+b+1}\);
 (i.1)
 (ii)a source if one of the following conditions holds:
 (ii.1)
\(0< s_{0}+b<1\) and \(s<\min \{\frac{2(1+s _{0})}{s_{0}+b+1},\frac{s_{0}1}{s_{0}+b} \}\);
 (ii.2)
\(1< s_{0}+b<0\) and \(\frac{s_{0}1}{s_{0}+b}< s<\frac{2(1+s _{0})}{s_{0}+b+1}\);
 (ii.3)
\(s_{0}+b<1\) and \(s>\max \{\frac{2(1+s _{0})}{s_{0}+b+1},\frac{s_{0}1}{s_{0}+b} \}\);
 (ii.1)
 (iii)a saddle if one of the following conditions holds:
 (iii.1)
\(1< s_{0}+b<1\) and \(s>\frac{2(1+s_{0})}{s _{0}+b+1}\);
 (iii.2)
\(s_{0}+b<1\) and \(s< \frac{2(1+s_{0})}{s_{0}+b+1}\);
 (iii.1)
 (iv)nonhyperbolic if one of the following conditions holds:
 (iv.1)
\(s_{0}+b=1\);
 (iv.2)
\(s_{0}+b\neq 1\), and \(s=\frac{2(1+s_{0})}{s _{0}+b+1}\);
 (iv.3)
\(s_{0}+b\neq 0\), \(s=\frac{s_{0}1}{s_{0}+b}\) and \((s_{0}+1s)^{2}<4(s_{0}(1s)bs)\).
 (iv.1)
3 Neimark–Sacker bifurcation analysis and control
3.1 Neimark–Sacker bifurcation analysis
Taking parameters \((m,\beta ,e_{1},e_{2},s)\) arbitrarily from NS. Map (1.4) has an interior fixed point \(E_{2}\), at which eigenvalues \(\lambda _{1}\), \(\lambda _{2}\) satisfy \(\lambda _{1}=\lambda _{2}=1\).
 (H2)
\(s_{1}\neq 1+s_{0}, 2+s_{0}\).
Summarizing, we have established the following result for Neimark–Sacker bifurcation behavior of the model system (1.4):
Theorem 3.1
Assume that conditions (H1) and (H2) hold. Then if \(L\neq 0\), the model system (1.4) undergoes a Neimark–Sacker bifurcation at fixed point \(E_{2}\) when the parameter s varies in a small neighborhood of \(s_{1}\). Moreover, if \(L<0\) (respectively, \(L>0\)), then an attracting (respectively, a repelling) closed invariant curve bifurcates form \(E_{2}\).
Remark 3.1
From the biological point of view, an invariant curve bifurcates from a fixed point, which means that the prey and predator can coexist in a stable way and reproduce their densities. The dynamics on the invariant curve may be either periodic or quasiperiodic.
Example 3.1
Neimark–Sacker bifurcation.
3.2 Controlling Neimark–Sacker bifurcation by using a hybrid control strategy
In [28, 29] the authors control the Neimark–Sacker bifurcation using polynomial functions. In [30] Luo and Chen design a hybrid control strategy to control flip bifurcation, and it is shown that the hybrid control strategy is very effective in controlling bifurcations for onedimensional discrete dynamical systems. In this subsection, we extend the hybrid control strategy to control Neimark–Sacker bifurcation of model system (1.4) and this can be implemented by means of a biological control [31] or some harvesting procedures [32].
Comparing system (3.6) with (3.7), we have the following result.
Theorem 3.2
The controlled system (3.7) and the original system (3.6) have the same κperiodic orbit.
Remark 3.2
 (i)
A continuous control scheme if \(\kappa =1\). Namely, controlling fixed points. Control needs to be added for each iteration.
 (ii)
An impulsive control scheme if \(\kappa >1\). Control is added once only after every κth iteration.
In order to check how the implementation of hybrid control strategy works, we have performed the following numerical simulation.
Example 3.2
Controlling Neimark–Sacker bifurcation.
 (i)
\(1+(0.04975741\gamma 2)+0.0273352331\gamma ^{2}0.04975741 \gamma +1>0\);
 (ii)
\(0.0273352331\gamma ^{2}0.04975741\gamma +1=1\);
 (iii)
\((0.04975741\gamma 2)^{2}4(0.0273352331\gamma ^{2}0.04975741 \gamma +1)<0\).
4 Existence of chaos in the sense of Marotto and chaos control
4.1 Existence of chaos in the sense of Marotto
In this subsection, with the help of Marotto’s theorem [33, 34] we show that map (1.4) exhibits chaotic behavior for specific values of parameters.
Marotto extended LiYork’s theorem on chaos from onedimension to multidimension through introducing the notion of snapback repeller in 1978 [33]. Due to a technical flaw, Marotto redefined a snapback repeller in 2005 [34]. Marotto’s theorem shows that the presence of a snapback repeller is a sufficient criterion for the existence of chaos. Let us describe the notion of snapback repeller and Marotto’s theorem.
Definition 4.1
If z̅ is a fixed point of F and all the eigenvalues of \(\mathrm{D}F(\overline{z})\) exceed one in norm, then z̅ is called a repelling fixed point of F.
Definition 4.2
Let z̅ be a repelling fixed point of F. Suppose that there exist a point \(z_{0}\neq \overline{z}\) in a repelling neighborhood of z̅ and an integer \(M>1\), such that \(z_{M}=\overline{z}\) and \(\operatorname{det}(\mathrm{D}F(z_{k})\neq 0\) for \(1\leqslant k\leqslant M\), where \(z_{k}=F^{k}(z_{0})\). Then z̅ is called a snapback repeller of F.
Theorem 4.1
(Marotto’s theorem)
 (i)
a positive integer N, such that F has a point of period τ, for each integer \(\tau \geqslant N\).
 (ii)a “scrambled set” of F, i.e., an uncountable set S containing no periodic points of F, such that:
 (ii.1)
\(F(S)\subset S\);
 (ii.2)
\(\limsup_{x \to \infty } \parallel F^{k}(p)F ^{k}(q)\parallel >0\), for all \(p,q\in S\), with \(p\neq q\);
 (ii.3)
\(\limsup_{k \to \infty } \parallel F^{k}(p)F ^{k}(q)\parallel >0\), for all \(p\in S\) and periodic point q of F;
 (ii.4)
an uncountable subset \(S_{0}\) of S, such that \(\limsup_{k \to \infty } \parallel F^{k}(p)F^{k}(q)\parallel =0\), for every \(p,q\in S_{0}\).
 (ii.1)
It is straightforward to see that a snapback repeller gives rise to an orbit \(\{z_{k}\}^{\infty }_{k=\infty }\) of F with \(z_{k}= \overline{z}\), for \(k\geqslant M\), and \(z_{k}\rightarrow \overline{z}\) as \(k\rightarrow \infty \). Roughly speaking, the property of this orbit is analogous to the one for homoclinic orbit. In addition, the map F is locally onetoone at each point \(z_{k}\). This leads to the trivial transversality, i.e., the unstable manifold \(\mathbb{R}^{2}\) of full dimension intersects transversally the zerodimensional stable manifold of z̅. Therefore, snapback repeller may be regarded as a special case of a fixed point with a transversal homoclinic orbit [35], which is one of the core concepts in nonlinear dynamics. Especially, homoclinic point, closely related to homoclinic orbit, acts as an organizing center for chaotic motion.
Example 4.1
Marotto’s chaos.
Here, diagrams for bifurcation, chaotic attractors and the largest Lyapunov exponents will be drawn to validate our theoretical result using numerical simulation. As we see in Example 3.1, a Neimark–Sacker bifurcation arises as parameter s varies in a neighborhood of \(s_{1}=0.09096349222\). Now increase the value of s to be 3. By a tedious numerical calculation and simulations with \(\varPhi (1)>0\), \(\varPhi (1)>0\) and \(\varPhi (0)>0\) using MATLAB software, we find a neighborhood \(\mathfrak{B}=\{(u,v)\mid 0< u<0.14, 0.25<v<0.46\}\) of \(E_{2}\), where all eigenvalues of \(\mathrm{D}F(z)\) exceed 1 in norm. There also exists a positive point \(z_{0}=(0.07289604113, 0.2931610206)\) satisfying \(F^{3}(z_{0})=E_{2}\) and \(\operatorname{det}(\mathrm{D}F(z_{k}))=0.6275618341,2.100193893,2.619286018 \neq 0\) for \(k=0,1,2\), respectively. Moreover, \(z_{0}\in \mathfrak{B}\), \(z_{1}=(0.09523173812,0.484637067)\notin \mathfrak{B}\), \(z_{2}=(0.0817510353,0.1557479687) \notin \mathfrak{B}\). Thus, \(E_{2}\) is a snapback repeller.
4.2 Controlling chaotic dynamical system using an improved OGY method
Example 4.2
Controlling chaos.
5 Conclusion
The present paper is concerned with the dynamical behavior and control of a discretetime predator–prey system with modified Leslie–Gower and Holling’s type II schemes. We have seen that our results show far richer dynamics of the discrete model compared with the continuous one, including invariant circle, superstable phenomenon, cascades of perioddoubling bifurcation and chaotic sets. In the meantime, these results demonstrate that the natural growth rates of predator and prey play a vital role for local and global stability of predator–prey system. It indicates that the dynamical behavior of biological model may be very sensitive to bifurcation parameter perturbation. Especially, we provide the method of state feedback and parameter perturbation for bifurcation control, and the improved OGY method for chaos control, which stabilize the chaotic orbit at an unstable fixed point. Numerical simulations are carried out to verify our theoretical analysis and control strategies. Our results can be useful for the specialists in organic agriculture as they need a biological control method. However, it is still a challenging problem to explore the multipleparameter bifurcation in biological systems. But at present the study in this respect is very inadequate. This will be the topic of our future research and we expect to obtain some more analytical results.
Declarations
Acknowledgements
The authors wish to express their gratitude to the editors and reviewers for the helpful comments.
Authors’ information
Dr. Donghan Cai is a professor at the Wuhan University School of Mathematics and Statistics, Bayi Road, Wuhan City, Hubei Province, China. Email: dhcai@whu.edu.cn.
Funding
This research is supported by National Natural Science Foundation of China under grant No. 71271158.
Authors’ contributions
The study presented here was carried out in collaboration between both authors. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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