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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 2386954, 18 pages
https://doi.org/10.1155/2018/2386954
Research Article

Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response

Research Center for Engineering Ecology and Nonlinear Science, North China Electric Power University, Beijing, China

Correspondence should be addressed to Huayong Zhang; nc.ude.upecn@sneecr

Received 29 October 2017; Revised 26 January 2018; Accepted 31 January 2018; Published 29 April 2018

Academic Editor: Hassan A. El-Morshedy

Copyright © 2018 Huayong Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present in this paper an investigation on a discrete predator-prey system with Crowley-Martin type functional response to know its complex dynamics on the routes to chaos which are induced by bifurcations. Via application of the center manifold theorem and bifurcation theorems, occurrence conditions for flip bifurcation and Neimark-Sacker bifurcation are determined, respectively. Numerical simulations are performed, on the one hand, verifying the theoretical results and, on the other hand, revealing new interesting dynamical behaviors of the discrete predator-prey system, including period-doubling cascades, period-2, period-3, period-4, period-5, period-6, period-7, period-8, period-9, period-11, period-13, period-15, period-16, period-20, period-22, period-24, period-30, and period-34 orbits, invariant cycles, chaotic attractors, sub-flip bifurcation, sub-(inverse) Neimark-Sacker bifurcation, chaotic interior crisis, chaotic band, sudden disappearance of chaotic dynamics and abrupt emergence of chaos, and intermittent periodic behaviors. Moreover, three-dimensional bifurcation diagrams are utilized to study the transition between flip bifurcation and Neimark-Sacker bifurcation, and a critical case between the two bifurcations is found. This critical bifurcation case is a combination of flip bifurcation and Neimark-Sacker bifurcation, showing the nonlinear characteristics of both bifurcations.

1. Introduction

Predator-prey interaction shows widespread existence in nature and can take many forms, such as resource-consumer, plant-herbivore, and phytoplankton-zooplankton forms [1]. Due to the ubiquity and importance of the predator-prey interaction between populations, the dynamics of predator-prey systems have attracted the attention of many scholars, and the research on predator-prey systems has become one of the dominant themes in ecology [24]. In recent decades, mathematical models have been established to analyze various complex dynamics of the predator-prey systems in various circumstances [46].

The Lotka-Volterra predator-prey model, proposed in the pioneering works of Lotka and Volterra [7, 8], is probably the simplest and most basic predator-prey model in the field. The development of subsequent predator-prey models is mostly based on the Lotka-Volterra model [3, 9, 10]. The development may be a change of functional response (e.g., Holling type I, II, and III classifications) or numerical response to describe different ecological processes in the predator-prey interaction [11, 12]. Several researchers argued that the nonautonomous model is closer to the realistic predator-prey system than the autonomous model [3]. Therefore, predator-prey models with a time delay were also studied because the predator-prey interaction may have time lag [13]. In order to incorporate the influence of seasonal variation into the predator-prey interaction, predator-prey models with periodically varying parameters were also proposed and investigated [1416].

In earlier approaches, most predator-prey models were time-continuous. However, the continuous models hardly described the dynamics when the populations have nonoverlapping generations or the number of populations is small [17, 18]. For these cases, applying discrete-time models should be more reasonable and appropriate [17]. In recent years, more and more scholars have payed attention to discrete-time predator-prey models. Compared with the continuous models, the discrete dynamic models often have advantages in describing richer nonlinear characteristics and complexity, such as various periodic orbits, quasiperiodic behaviors, and chaotic attractors [1921]. Moreover, a few research works found that the discrete model can lead to more accurate results than the corresponding continuous model in describing predator-prey dynamics [6, 22, 23].

Through the research of He and Lai [2], Liu and Xiao [19], Jing and Yang [20], Agiza et al. [5], Hu et al. [6], Huang and Zhang [21], and others, they found out that flip bifurcation and Neimark-Sacker bifurcation are two basic bifurcations occurring in discrete predator-prey systems. The two bifurcations often start the routes to chaos, revealing the dynamic transition from the nonchaotic state to the chaotic state. Moreover, complex dynamics on the routes to chaos can be observed, including cascades of period doubling, periodic windows, intermittent chaos, and chaotic crisis [1921]. For example, via calculating Lyapunov exponents and fractal dimension, Agiza et al. determined diverse strange attractors of a predator-prey system [5].

When we study the nonlinear characteristics of predator-prey systems, the functional response of the predator to the prey is an essential factor which affects the dynamical properties. From an ecological point of view, functional responses may be determined by the prey escape capability, prey habitat property, and predator hunting capability [24]. Skalski and Gilliam [12] suggested that three classical prey-predator-dependent functional responses—Beddington-DeAngelis, Crowley-Martin, and Hassell-Varley functional responses—can provide a better description of predator feeding over a range of predator-prey abundance presence.

Particularly, Crowley-Martin functional response is much more suitable for the case where the predator feeding rate is decreased by a higher predator density even when the prey density is high [12]. The Crowley-Martin functional response depends on both the predator and the prey and takes into account the interaction between predators, regardless of whether a predator is looking for prey. It is similar to the classical Beddington-DeAngelis functional response; in particular, it shows the interspecies interference with each other. Usually, it is employed to demonstrate the predator-prey stability. And, therefore, it shows ecological significance for researching the predator-prey models with Crowley-Martin functional response. A classical predator-prey system with Crowley-Martin functional response can be described by the following equations [25]: where and represent the densities of the prey and the predator, respectively; parameters , , , , , and are positive constants; the parameter is the intrinsic growth rate of the prey and the parameter is the mortality rate of the predator; and stand for the effect of capture rate and conversion factor denoting the newly born predators for each captured prey; the parameters and are the saturating parameters of Crowley-Martin functional response, measures the magnitude of interference among prey, and expresses the interference among the predators.

In recent years, the research on the predator-prey systems with Crowley-Martin functional response has attracted a lot of attention. For example, Yang studied the stability and instability of positive equilibrium in a predator-prey model with Crowley-Martin functional response and time delay [24]. Asymptotic properties of a stochastic predator-prey model with Crowley-Martin functional response were studied by Liu et al., such as global existence and stochastic boundedness of the positive solution [26]. A stage-structured predator-prey system with Crowley-Martin functional response was investigated, and the persistence of the system and global asymptotic stability of the positive equilibrium were assessed [27]. Sivakumar et. al. analyzed a diffusive density-dependent predator-prey model with Crowley-Martin functional response and revealed Neimark-Sacker bifurcation, Turing bifurcation, and spatiotemporal patterns [28]. Dong et al. investigated the positive solutions of a spatiotemporal predator-prey system with Crowley-Martin functional response and obtained a complete understanding of the uniqueness and nonuniqueness of positive solutions [29, 30]. Li and Wu discussed the extinction and persistence results of time-dependent positive solutions to the system [31]. From previous studies, it can be seen that the predator-prey model with Crowley-Martin functional response often exhibits rich dynamics and attracts the attention of many researchers. However, the complex dynamics of the discrete-time predator-prey model with Crowley-Martin functional response are still in lack of research.

The discretized form of system (1a) and (1b) was never investigated in previous studies. Since the growth, death, feeding, and migration of the predator and prey individuals always occur periodically, we can observe the dynamics of the predator-prey system by a particular time scale. This time scale can be defined by the generation span of the predator and prey populations and it measures the regeneration time of both populations. Its value is mainly determined by the population types; on the other hand, it is influenced by the change of population size or environmental conditions. In this research, we denote the time scale on which predator-prey dynamics are described and observed as parameter . Applying the forward Euler scheme with time interval , system (1a) and (1b) is now transformed to a discrete predator-prey system: where represents the sequence number of iterations. If we give the initial time as , the th iteration represents the time at .

In recent decades, the research on discrete predator-prey systems has become an important topic. Via the research of Li and Wu [31], the dynamical properties of system (1a) and (1b) have been revealed a lot. However, nonlinear characteristics of discrete system (2a) and (2b) are still scarcely known. In the literature, Ren et al. have actually investigated a Leslie-Gower type discrete predator-prey model with Crowley-Martin functional response and found the existence of Marotto chaos [32]. They also controlled chaotic orbits to be a fixed point by a feedback control method. Different from the study of Ren et al. [32], this research focuses on a detailed exploration of the complex dynamics, such as periodic windows, subbifurcations, and chaos crisis, on the routes to chaos induced by the bifurcations of system (2a) and (2b).

In order to facilitate the study of system (2a) and (2b), we rewrite the equations of system (2a) and (2b) as a map; that is,

In this research, the bifurcations and complex dynamics of the discrete predator-prey system with Crowley-Martin functional response are explored through map (3). The exploration is arranged as follows. Section 2 will give the stability analysis on the fixed points of map (3). Section 3 will provide bifurcation analysis and determine occurrence conditions for flip bifurcation and Neimark-Sacker bifurcation. Section 4 will present numerical simulations for complex dynamic behaviors of the discrete predator-prey system. Section 5 will describe the conclusions.

2. Stability Analysis

The fixed points of map (3) can be calculated through the following equations: which directly yield where is a positive root of the following cubic equations: According to previous results in the literature [21], the sufficient conditions for map (3) possessing a unique positive fixed point are determined as follows:

To determine the stability of the fixed points of map (3), the Jacobian matrix associated with the map is applied and can be described as follows:

For the Jacobian matrix of (8), we substitute the values of the three fixed points into it and calculate the two eigenvalues, and . If and , the corresponding fixed point is considered to be stable; if or , an unstable fixed point emerges. For the fixed point , the corresponding Jacobian matrix is the two eigenvalues of which are and . Since is larger than 1, then is unstable. For the fixed point , the Jacobian matrix is obtained as The two eigenvalues of are and . When the following conditions are established,the two eigenvalues are both less than one. Therefore, under conditions (11), is stable. The Jacobian matrix of is presented as the following matrix: Likewise, the two eigenvalues of matrix (3) arewhere We calculate the stability conditions of , that is, and , as follows:

3. Bifurcation Analysis

3.1. Flip Bifurcation

Around the stable fixed point , flip bifurcation of map (3) is studied via the flip bifurcation theorem and center manifold theorem [27]. According to the flip bifurcation theorem, the predator-prey system undergoes flip bifurcation when one of the two eigenvalues of is equal to −1. Therefore, it is explicitly said that ; that is, which directly yields At such critical point , the two eigenvalues of are and . Moreover, the value of should satisfy ; that is,

Consider the bifurcation parameter τ also as a dependent variable. We then translate the fixed point to the origin via the following translation: Consequently, via Taylor expansion near the fixed point, map (3) can be transformed as where is a polynomial with at least four orders with variables , , and . The coefficients can be calculated by where , , and are the orders of the variables , , and , respectively, in the corresponding term of . Here, we define .

Applying a reversible transformation asmap (20) becomes where

The center manifold of map (23) at the fixed point is then determined. According to the center manifold theorem, a central manifold at the fixed point can be approximated by the following: where is assumed to beUtilizing map (23) on both sides of leads to simultaneously balancing the variables in (28); then, we get v , and the expressions of and are

Considering the dynamics of map (23) restricted in the center manifold , hence we obtain a one-dimensional mapping as The coefficients in map (30) are described as follows:

On the basis of map (30), the predator-prey system experiencing flip bifurcation needs the requirement of the following two conditions:which are equivalent to

Hence, the occurrence of flip bifurcation at the fixed point with parameter varying in a small neighborhood of needs the satisfaction of conditions (18) and (33). Furthermore, if , a period-2 orbit bifurcates from and is stable; if , the period-2 orbit bifurcating from is unstable.

3.2. Neimark-Sacker Bifurcation

When map (3) undergoes Neimark-Sacker bifurcation, the two eigenvalues of the fixed point are a pair of conjugate complex numbers whose module is one. Hence, we directly have and , which are further determined as

Under conditions (34), we translate the fixed point to the origin by the following translation: With application of Taylor expansion, map (3) is transformed to The coefficients in the map are also calculated by (21) but . The two eigenvalues at the fixed point of map (36) are also a pair of conjugate complexes with a module of one. These two eigenvalues are written as where and . The modulus of the two eigenvalues satisfies Simultaneously, the two eigenvalues should not be real numbers or imaginary numbers, which means , . Since , we have . In the meantime, . Therefore, , which is equivalent to

Through the transformation the normal form of map (36) is obtained as in which

Calculating the two- and three-order derivatives of and at and , the following results are obtained:

On the basis of map (41), the predator-prey system experiencing Neimark-Sacker bifurcation needs the requirement of the following condition:where Using the above formula, we can get in which

Taking the above calculations together, the occurrence of Neimark-Sacker bifurcation at the fixed point in the discrete predator-prey system needs the satisfaction of conditions (34), (39), and (46). Moreover, if and , an attracting invariant closed curve bifurcates from the fixed point for ; if and , a repelling invariant closed curve bifurcates from the fixed point for .

4. Numerical Results

To show the bifurcations and complex dynamics of the discrete predator-prey system, numerical simulations are carried out. We fix the parameters as , , , , , and and assume that varies. Under such conditions, the fixed point is and the critical point for flip bifurcation is . At the critical bifurcation point, the two eigenvalues are and . Considering the dynamics of map (23) restricted to the center manifold, we then obtain the values of two quantities as