This article investigates a prey–predator model incorporating a novel refuge proportional to prey and inverse proportion to the predator. We find conditions for the local asymptotic stability of fixed points of the proposed prey–predator model. This article presents Neimark–Sacker bifurcation (NSB) and period-doubling bifurcation (PDB) at particular parameter values for positive equilibrium points of the proposed refuge-based prey–predator system. The system exhibits the chaotic dynamics at increasing values of the bifurcation parameter. The hybrid control methodology will control the chaos of the proposed prey–predator dynamical system and discuss the chaotic situation for different biological parameters through graphical analysis. Numerical simulations support the theoretical outcome and long-term chaotic behavior over a broad range of parameters.

1 INTRODUCTION

In recent years, many new challenging and exciting problems are arising in mathematical biology and ecology. The horizon of this research is widening and deepening day by day due to the exotic usefulness and importance for human beings. There has also been a growing concern about preserving the ecological balance in nature, which plays a vital role in the harmony among different species. As a result, mathematical models for populations with the interaction between species have become popular among biologists and scientists.^1-5 The remarkable exhibition of various dynamical behaviors can be seen in many plant, insects, and animal species. This has stimulated great interest in the study of the dynamical systems of populations or ecosystems. For instance, predator–prey models are widespread and come in many flavors in population ecology.

The stability of the two interacting species may depend on the intrinsic predator and prey growth rates. Some general assumptions are considered in the interaction of prey–predator systems:

(i)
If the two populations inhabit the same area. Their densities are directly proportional to their numbers.^6-12
(ii)
There is no time lag in the responses of either population to changes.
(iii)
There is an abundance of food supply for prey.
(iv)
The prey is the sole source of food for the predator.

The prey–predator interaction is the most fundamental process in population dynamics. Several species, such as monocarpic plants and semelparous animals, have discrete nonoverlapping generations and their births occur in regular breeding seasons. For such species, interactions are described by difference equations or formulated as discrete-time mappings. Discrete-time prey–predator systems can exhibit even more complicated dynamics compare to the corresponding continuous prey–predator models.

Nature can provide some extent of protection to a few quantities of prey populations by providing refuge. Such refugia can help sustain prey–predator interactions by reducing the chance of extinction due to predation and damp prey–predator oscillations. Many researchers in the refuge concept have done extensive studies. The effects of prey refuges on the population dynamics are very complex, but it can be considered for modeling purposes by two components: the first effect on positively the growth of prey and negatively that of predators due to the decrease in predation success, and hence the reduction of prey mortality. The second effect on the trade-offs and by-products of the hiding behavior of prey, which could be advantageous or detrimental for all interacting populations. Almost all the researchers consider the refuge as a constant number of prey or proportional to prey.^13-34 This article presents a novel refuge term which is depending on the quantity of prey population, and the refuge is reducing based on the quantity of proportion to predator. In earlier works, the use of refuges by the fraction of prey or the constant number of prey exerts a stabilizing effect on the dynamics of the interacting populations. In this article, we test the above statement assuming that the quantity of prey in refugia is directly proportional to the prey and inversely proportional to the density of the predator. We analyze such a prey predator system's dynamic properties by modifying the well-known Lotka–Volterra model with prey self-limitation.

The rest of the article is organized as follows: In the second section, a discrete-time prey–predator model with refuge is formulated. In Section 3, we give the existence and stability of fixed points. In Section 4, the conditions on the presence of codimension-one bifurcations, including Neimark–Sacker bifurcation and period-doubling bifurcations are obtained. In Section 5, conditions on the existence of Marottos chaos are presented. Section 6 presents a feedback control method to stabilize chaotic orbits. In Section 7, numerical simulation results are presented to support the theoretical analysis and display the new and rich dynamic behavior. Finally, this article ends with a conclusion in Section 8.

2 DESCRIPTION OF PROPOSED PREY–PREDATOR MODEL

We consider a discrete prey–predator model incorporating the concept of a continuous system of the population densities of prey and predator change continuously as a uniform distribution over space with time and without having stage structure for any species. The model represents a generalized prey–predator model incorporating the evolution of the prey population in the absence of predators with the logistic map

x_{n + 1} = a x_{n} (1 - x_{n})

, where

x_{n} \in (0, 1)

and the positive parameter a represents the constant intrinsic growth rate. The proposed discrete-time prey–predator system incorporating Holling type II functional response can be modeled as follows:

\begin{align} x_{n + 1} & = a x_{n} (1 - x_{n}) - \frac{c x_{n} y_{n}}{e + x_{n}}, \end{align}

()

\begin{align} y_{n + 1} & = \frac{d x_{n} y_{n}}{e + x_{n}}, \end{align}

where

x_{n}

and

y_{n}

denote the density of prey species (zebra) and predator species (tiger), respectively, at a discrete-time step n. c is the maximal per capita consumption rate of predators, d is the efficiency with which predators convert consumed prey into new predators, and e is the half saturated constant.

We modify the model (1) to incorporate prey refuges. A prey obtains protection from predation by hiding in an area where it is inaccessible or cannot easily be found. Constant refuge and proportional refuge are common in the literature. The proposed model considers that the amount of prey in refugia is proportional to the prey and inverse proportion to predator, that is,

\frac{b x_{n}}{y_{n}}

. This is the modified version of the proportional refuge concept. This refuge is an increasing function of prey size but decreasing with predator abundance. Therefore, the proposed prey–predator model (1) can be modified as follows:

\begin{align} x_{n + 1} & = a x_{n} (1 - x_{n}) - \frac{c (y_{n} - b) x_{n} y_{n}}{e y_{n} + (y_{n} - b) x_{n}}, \end{align}

()

\begin{align} y_{n + 1} & = \frac{d (y_{n} - b) x_{n} y_{n}}{e y_{n} + (y_{n} - b) x_{n}}, \end{align}

where

a, b, c, d

, and e are all positive constants consists of the biological meaning of the system in the region

Ω = {(x, y) : x \geq 0, y \geq 0}

3 EQUILIBRIUM POINTS AND THEIR STABILITY ANALYSIS

This section presents the proposed discrete-time prey–predator systems' equilibrium points and discusses their stability.

3.1 Existence of equilibrium points

In this section, we calculate the nonnegative fixed points of the discrete prey–predator system (2).

Theorem 1.Discrete prey–predator system (2) has three nonnegative fixed points.

Proof.The following nonlinear system of equations gives the fixed points of the discrete prey–predator system (2):

x = a x (1 - x) - \frac{c (y - b) x y}{e y + (y - b) x} and y = \frac{d (y - b) x y}{e y + (y - b) x} .

The solution of the above equations gives three nonnegative fixed points as: $(i) P_{0} = (0, 0)$ , $(i i) P_{1} = (\frac{a - 1}{a}, 0)$ for $a > 1$ , $(i i i) P_{2} = (x_{2}, y_{2})$ where $x_{2}$ is the positive root of the equation $x^{2} + A x + B = 0$ where $A = \frac{1 - a}{a} - \frac{e}{d - 1}$ , $B = \frac{c}{d} + \frac{e (a - 1)}{a (d - 1)}$ , and $y_{2} = \frac{b (d - 1) x_{2}}{(d - 1) x_{2} - e} .$

3.2 Local stability analysis

The local behavior of the system (2) for each equilibrium point of the prey–predator system is discussed with the help of the following theorem and lemma.

Lemma 1.Let $F (s) = s^{2} - B s + C$ , suppose that $F (1) > 0, s_{1}$ and $s_{2}$ are the two roots of $F (s) = 0 .$ Then $(i) | s_{1} | < 1$ and $| s_{2} | < 1$ iff $F (- 1) > 0$ and $C < 1; (i i) | s_{1} | < 1$ and $| s_{2} | > 1$ (or $| s_{1} | > 1$ and $| s_{2} | < 1)$ iff $F (- 1) < 0; (i i i) | s_{1} | > 1$ and $| s_{2} | > 1$ iff $F (- 1) > 0$ and $C > 1; (i v) s_{1} = - 1$ and $s_{2} \neq 1$ iff $F (- 1) = 0$ and $B \neq 0, 2; (v) s_{1}$ and $s_{2}$ are complex and $| s_{1} | = | s_{2} |$ iff $B^{2} - 4 C < 0$ and $C = 1 .$

The stability of the system (2) at each fixed point will discuss by constructing the Jacobian matrix. The Jacobian matrix J for the system (2) is $J = [\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}]$ , where $a_{11} = a (1 - 2 x) - \frac{c e (y - b) y^{2}}{{(e y + (y - b) x)}^{2}} = a (1 - 2 x) - c e Q$ , $a_{12} = - \frac{c x [e y^{2} + x {(y - b)}^{2}]}{{(e y + (y - b) x)}^{2}} = - c R$ , $a_{21} = \frac{d e (y - b) y^{2}}{{(e y + (y - b) x)}^{2}} = d e Q$ , $a_{22} = \frac{d x [e y^{2} + x {(y - b)}^{2}]}{{(e y + (y - b) x)}^{2}} = d R .$

The characteristic equation of matrix J is $λ^{2} - T r (J) λ + D e t (J) = 0$ where

$Tr (J) =$ Trace of matrix $= a (1 - 2 x) - c e Q + d R,$

$Det (J) =$ Determinant of matrix $= a d (1 - 2 x) R$

Hence, the model (2) is (i) a dissipative dynamical system if $|a d (1 - 2 x) R| < 1$ , (ii) conservative dynamical one iff $|a d (1 - 2 x) R| = 1$ .

Theorem 2.The fixed point $P_{1} (\frac{a - 1}{a}, 0)$ of prey–predator system (2) is locally asymptotically stable if $1 < a < 3$ and $- 1 < d < 1 .$

Proof.The Jacobian matrix at $P_{1} = (\frac{a - 1}{a}, 0)$ to discuss the dynamical behavior is

$J = [\begin{array}{cc} 2 - a & - c \\ 0 & d \end{array}]$

Therefore, the equilibrium point $P_{1}$ is sink if $|2 - a| < 1$ and $|d| < 1 .$

Lemma 2.Equilibrium point $P_{1} (\frac{a - 1}{a}, 0)$ of prey–predator system (2) is $(i)$ source if $|2 - a| > 1$ and $|d| > 1;$ $(i i)$ saddle if $|2 - a| > 1$ and $|d| < 1$ or $|2 - a| < 1$ and $|d| > 1; (i i i)$ nonhyperbolic if $|2 - a| = 1$ or $|d| = 1$

Theorem 3.The fixed point $P_{2} (x_{2}, y_{2})$ of prey–predator system (2) is locally asymptotically stable.

Proof.The Jacobian matrix at $P_{2} (x_{2}, y_{2})$ to discuss the dynamical behavior is $J = [\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}] a_{11} = a (1 - 2 x_{2}) - \frac{c e (y_{2} - b) y_{2}^{2}}{{(e y_{2} + (y_{2} - b) x_{2})}^{2}} = a (1 - 2 x_{2}) - c e Q_{2}, a_{12} = - \frac{c x_{2} [e y_{2}^{2} + x_{2} {(y_{2} - b)}^{2}]}{{(e y_{2} + (y_{2} - b) x_{2})}^{2}} = - c R_{2}$ , $a_{21} = \frac{d e (y_{2} - b) y_{2}^{2}}{{(e y_{2} + (y_{2} - b) x_{2})}^{2}} = d e Q_{2}, a_{22} = \frac{d x_{2} [e y_{2}^{2} + x_{2} {(y_{2} - b)}^{2}]}{{(e y_{2} + (y_{2} - b) x_{2})}^{2}} = d R_{2}$

The characteristic equation of matrix J is $λ^{2} - T r (J) λ + D e t (J) = 0$ where

$Tr (J) =$ Trace of matrix $= a (1 - 2 x_{2}) - c e Q_{2} + d R_{2}$

$Det (J) =$ Determinant of matrix $= a d (1 - 2 x_{2}) R_{2}$

Therefore, the equilibrium point $P_{2} (x_{2}, y_{2})$ is sink if one of the following conditions holds

(i)
$\max (\frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d}, \frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d}) < R_{2} < \frac{1}{a d (1 - 2 x_{2})}$ and $x_{2} < \min (\frac{1}{2}, \frac{a - 1}{2 a})$ ,
(ii)
$\frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d} < R_{2} < \min (\frac{1}{a d (1 - 2 x_{2})}, \frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d})$ and $\frac{a - 1}{2 a} < x_{2} < \frac{1}{2}$ ,
(iii)
$\max (\frac{1}{a d (1 - 2 x_{2})}, \frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d}) < R_{2} < \frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d}$ and $\frac{1}{2} < x_{2} < \frac{a + 1}{2 a},$
(iv)
$\frac{1}{a d (1 - 2 x_{2})} < R_{2} < \min (\frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d}, \frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d})$ and $x_{2} > \max (\frac{1}{2}, \frac{a + 1}{2 a})$ .

Lemma 3.The fixed point $P_{2} (x_{2}, y_{2})$ of system (2) is source if one of the following conditions holds.

(i)
$\max (\frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d}, \frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d}, \frac{1}{a d (1 - 2 x_{2})}) < R_{2}$ and $x_{2} < \min (\frac{1}{2}, \frac{a - 1}{2 a})$ ,
(ii)
$\max (\frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d}, \frac{1}{a d (1 - 2 x_{2})}) < R_{2} < \frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d}$ and $\frac{a - 1}{2 a} < x_{2} < \frac{1}{2},$
(iii)
$\frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d} < R_{2} < \min (\frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d}, \frac{1}{a d (1 - 2 x_{2})})$ and $\frac{1}{2} < x_{2} < \frac{a + 1}{2 a},$
(iv)
$R_{2} < \min (\frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d}, \frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d}, \frac{1}{a d (1 - 2 x_{2})})$ and $x_{2} > \max (\frac{1}{2}, \frac{a + 1}{2 a})$ .

Lemma 4.The fixed point $P_{2} (x_{2}, y_{2})$ of system (2) is saddle if one of the following conditions holds

(i)
$\frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d} < R_{2} < \frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d}$ and $x_{2} < \frac{a - 1}{2 a}$ ,
(ii)
$R_{2} < \min (\frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d}, \frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d})$ and $\frac{a - 1}{2 a} < x_{2} < \frac{a + 1}{2 a}$ ,
(iii)
$\frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d} < R_{2} < \frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d}$ and $x_{2} > \max (\frac{a - 1}{2 a}, \frac{a + 1}{2 a})$ .

Lemma 5.The fixed point $P_{2} (x_{2}, y_{2})$ of system (2) is nonhyperbolic if one of the following conditions holds

(i)
$\frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d} < R_{2} = \frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d}, x_{2} < \frac{a - 1}{2 a}$ , and $R_{2} \neq \frac{c e Q_{2} - a (1 - 2 x_{2})}{d}, \frac{c e Q_{2} - a (1 - 2 x_{2}) + 2}{d},$
(ii)
$\frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d} = R_{2} < \frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d}, x_{2} > \frac{a - 1}{2 a}$ . and $R_{2} \neq \frac{c e Q_{2} - a (1 - 2 x_{2})}{d}, \frac{c e Q_{2} - a (1 - 2 x_{2}) + 2}{d},$
(iii)
${[a (1 - 2 x_{2}) - c e Q_{2} + d R_{2}]}^{2} - 4 a d (1 - 2 x_{2}) R_{2} < 0$ and $R_{2} = \frac{1}{a d (1 - 2 x_{2})}$ .

At $P_{2} (x_{2}, y_{2})$ if $\frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d} < R_{2} = \frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d}, x_{2} < \frac{a - 1}{2 a}$ , and $R_{2} \neq \frac{c e Q_{2} - a (1 - 2 x_{2})}{d}$ , $\frac{c e Q_{2} - a (1 - 2 x_{2}) + 2}{d}$ or $\frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d} = R_{2} < \frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d}, x_{2} > \frac{a - 1}{2 a}$ , and $R_{2} \neq \frac{c e Q_{2} - a (1 - 2 x_{2})}{d}$ , $\frac{c e Q_{2} - a (1 - 2 x_{2}) + 2}{d}$ then $(x_{2}, y_{2})$ can undergo period-doubling bifurcation.

At $P_{2} (x_{2}, y_{2})$ if ${[a (1 - 2 x_{2}) - c e Q_{2} + d R_{2}]}^{2} - 4 a d (1 - 2 x_{2}) R_{2} < 0$ and $R_{2} = \frac{1}{a d (1 - 2 x_{2})}$ then $(x_{2}, y_{2})$ can undergo Neimark–Sacker bifurcation.

4 BIFURCATION ANALYSIS

In this section, we investigate the parametric conditions for the existence of NSB and PDB at the positive fixed point $P_{2} (x_{2}, y_{2})$ of the prey–predator system (2). When a particular parameter passes through its critical value in the dynamical system, various types of bifurcations emerge from its fixed point. Bifurcation mainly occurs due to changes in the stability of a fixed point, that is, qualitative properties of a dynamical system change. We discuss the emergence of NSB and PDB for the positive fixed point $P_{2} (x_{2}, y_{2})$ of prey–predator system (2), taking b as a bifurcation parameter.

4.1 Period-doubling bifurcation

A PDB in a discrete dynamical system is a bifurcation in which a slight change in a parameter value leads to the system switching to a new behavior with twice the period of the original system.

Here we discuss PDB of the model (2) at $P_{2} (x_{2}, y_{2})$ when parameters vary in a small neighborhood of $A .$ One can see that one of the eigenvalues of the positive fixed point $P_{2} = (x_{2}, y_{2})$ is $λ_{1} = - 1$ and the other $λ_{2}$ is neither 1 nor $- 1$ if parameters of the model are located in the following set

$A_{1} = \{(a, b, c, d, e) : \frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d} < R_{2} = \frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d}, x_{2} < \frac{a - 1}{2 a}, and R_{2} \neq \frac{c e Q_{2} - a (1 - 2 x_{2})}{d}, \frac{c e Q_{2} - a (1 - 2 x_{2}) + 2}{d}\}$ or $A_{2} = \{(a, b, c, d, e) : \frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d} < R_{2} = \frac{c e Q_{2} - a (1 - 2 x_{2}) - 1}{a d (1 - 2 x_{2}) + d}, x_{2} > \frac{a - 1}{2 a} and R_{2} \neq \frac{c e Q_{2} - a (1 - 2 x_{2})}{d}, \frac{c e Q_{2} - a (1 - 2 x_{2}) + 2}{d}\} .$

Theorem 4.The model (2) undergoes PDB at $P_{2} (x_{2}, y_{2})$ for the parameter b varies in a small neighborhood of $A_{1}$ , if $ξ_{2} > 0$ then the period-2 points are stable but unstable for $ξ_{2} < 0$ .

Proof.See the Appendix.

4.2 Neimark–Sacker bifurcation

NSB shows some isolated orbits of periodic behavior along with trajectories that cover the invariant circle densely. The bifurcation can be supercritical or subcritical, resulting in a stable or unstable closed invariant curve, respectively.

This section discusses the NSB of the model (2) at $P_{2} (x_{2}, y_{2})$ when parameters of the model are located in the set $A_{2} = {(a, b, c, d, e) : \frac{a (1 - 2 x_{2}) - c e Q_{2} - 1}{a d (1 - 2 x_{2}) - d} < R_{2} = \frac{1}{a d (1 - 2 x_{2})}$ , ${[a (1 - 2 x_{2}) - c e Q_{2} + d R_{2}]}^{2} - 4 a d (1 - 2 x_{2}) R_{2} < 0}$ .

Theorem 5.The model (2) undergoes NSB at $P_{2} (x_{2}, y_{2})$ when the parameter $b^{*}$ varies in a small neighborhood of the origin. If $Ω < 0 (Ω > 0)$ , then an attracting (repelling) invariant closed curve bifurcates from $P_{2} (x_{2}, y_{2})$ for $b^{*} > 0 (b^{*} < 0) .$

Proof.See the Appendix.

5 EXISTENCE OF MAROTTOS CHAOS

This section presents chaotic nature of the dynamical system (2) in the sense of Marrotto.

Definition 1.Let the function $F : R^{n} \to R^{n}$ be differentiable in $B_{r} (Z)$ . The point $Z \in R^{n}$ is an expanding fixed point of F in $B_{r} (Z)$ , if $F (Z) = Z$ and all eigenvalues of $D F (X)$ exceed 1 in norm for all $X \in B_{r} (Z)$ .

Definition 2.Assume that Z is an expanding fixed point of F in $B_{r} (Z)$ for some $r > 0 .$ Then Z is said to be a snap-back repeller of F if there exists a point $X_{0} \in B_{r} (Z)$ with $X_{0} \neq Z$ , $F^{M} (X_{0}) = Z$ and $D F^{M} (X 0) \neq 0$ for some positive integer M .

Theorem 6. $P_{2}$ is an expanding fixed point of F if $P_{2} \in U_{2} = {(x, y) : 4 a d (1 - 2 x) R > \max (4, {[a (1 - 2 x) - c e Q + d R]}^{2})}$

Proof.For map $F (X_{n}) = (\begin{array}{c} a x_{n} (1 - x_{n}) - \frac{c (y_{n} - b) x_{n} y_{n}}{e y_{n} + (y_{n} - b) x_{n}} \\ \frac{d (y_{n} - b) x_{n} y_{n}}{e y_{n} + (y_{n} - b) x_{n}} \end{array}), X_{n} = {(x_{n} y_{n})}^{T}$ .

The eigenvalues corresponding with the fixed point $P_{2} (x_{2}, y_{2})$ are given by $λ_{1, 2} = \frac{- p (x_{2}, y_{2}) \pm \sqrt{p^{2} (x_{2}, y_{2}) - 4 q (x_{2}, y_{2})}}{2}$ , where $p (x_{2}, y_{2}) = - [a (1 - 2 x_{2}) - c e Q_{2} + d R_{2}], q (x_{2}, y_{2}) = a d (1 - 2 x_{2}) R_{2}$ , $R_{2} = \frac{d x_{2} [e y_{2}^{2} + x_{2} {(y_{2} - b)}^{2}]}{{(e y_{2} + (y_{2} - b) x_{2})}^{2}}$ , $Q_{2} = \frac{(y_{2} - b) y_{2}^{2}}{{(e y_{2} + (y_{2} - b) x_{2})}^{2}}$ .

Therefore the fixed point $P_{2} (x_{2}, y_{2})$ has a pair of complex eigenvalues, and the norm of them exceeds unity if

$p^{2} (x_{2}, y_{2}) - 4 q (x_{2}, y_{2}) < 0$ and $q (x_{2}, y_{2}) - 1 > 0$ , that is, $4 a d (1 - 2 x_{2}) R_{2} > \max (4, {[a (1 - 2 x_{2}) - c e Q_{2} + d R_{2}]}^{2})$

Thus, we can state the following theorem.

Theorem 7. $P_{2} (x_{2}, y_{2})$ is a snap-back repeller in $U_{2}$ .

Proof.According to the definition of a snap-back repeller, one needs to find one point $P_{*} = (x_{0}, y_{0}) \in U_{2}$ such that $P_{*} \neq P_{2}$ , $F^{M} (P_{*}) = P_{2}, | D F^{M} (P_{*}) | \neq 0,$ for some positive integer M, where Map F is defined by (2).

To proceed, notice that

x_{1} = a x_{0} (1 - x_{0}) - \frac{c (y_{0} - b) x_{0} y_{0}}{e y_{0} + (y_{0} - b) x_{0}} and y_{1} = \frac{d (y_{0} - b) x_{0} y_{0}}{e y_{0} + (y_{0} - b) x_{0}}

()

and

x_{2} = a x_{1} (1 - x_{1}) - \frac{c (y_{1} - b) x_{1} y_{1}}{e y_{1} + (y_{1} - b) x_{1}} and y_{2} = \frac{d (y_{1} - b) x_{1} y_{1}}{e y_{1} + (y_{1} - b) x_{1}} .

()

Now, a map $F^{2}$ has been constructed to map the point $P_{*} = (x_{0}, y_{0})$ to the fixed point $P_{2} = (x_{2}, y_{2})$ after two iterations if there are solutions different from $P_{2}$ for Equations (3) and (4). The solutions different from $P_{2}$ for Equation (4) satisfy the following equation

x_{1} = \frac{e y_{1} y_{2}}{(d y_{1} - y_{2}) (y_{1} - b)} and y_{1} = \frac{[e y_{1} + (y_{1} - b) x_{1}] [a x_{1} (1 - x_{1}) - x_{2}]}{c (y_{1} - b) x_{1}} .

()

Substituting $x_{1}$ and $y_{1}$ into Equation (3) and solving $x_{0}, y_{0}$ , we have

x_{0} = \frac{e y_{0} y_{1}}{(d y_{0} - y_{1}) (y_{0} - b)} and y_{0} = \frac{[e y_{0} + (y_{0} - b) x_{0}] [a x_{0} (1 - x_{0}) - x_{1}]}{c (y_{0} - b) x_{0}} .

()

By simple calculations, we get

\begin{align} |D F^{2} (P_{*})| & = [a (B - 2 A B) - \frac{[e D + (D - b) A] [c (B D^{2} + 2 A D E - b B D - b A E)] - c A D (D - b) [e E + B D + A E - b B]}{{[e D + (D - b) A]}^{2}}] \\ \times [\frac{d [e D + A (D - b)] [C D^{2} + 2 A D F - b C D - b A F] - d A D (D - b) [e F + C D + A F - b C]}{{[e D + (D - b) A]}^{2}}] \\ - [a (C - 2 A C) - \frac{c [e D + A (D - b)] [C D^{2} + 2 A D F - b C D - b A F] - c A D (D - b) [e F + A F + C D - b C]}{{[e D + (D - b) A]}^{2}}] \\ \times [\frac{d [e D + A (D - b)] [B D^{2} + 2 A D E - b B D - b A E] - d A D (D - b) [e E + B D + A E - b B]}{{[e D + (D - b) A]}^{2}}], \end{align}

where

A = a x (1 - x) - \frac{c (y - b) x y}{e y + (y - b) x}

B = a (1 - 2 x) - \frac{c e (y - b) y^{2}}{{(e y + (y - b) x)}^{2}}

C = - \frac{c x [e y^{2} + x {(y - b)}^{2}]}{{(e y + (y - b) x)}^{2}}

D = \frac{d (y - b) x y}{e y + (y - b) x}

E = \frac{d e (y - b) y^{2}}{{(e y + (y - b) x)}^{2}}

, and

F = \frac{d x [e y^{2} + x {(y - b)}^{2}]}{{(e y + (y - b) x)}^{2}}

The solutions of Equations (5) and (6) will be farther subject to $(x_{0}, y_{0}), (x_{1}, y_{1}) \neq (x_{2}, y_{2}), (x_{0}, y_{0}) \in U_{2}$ and $| D F^{2} (P_{*}) | \neq 0$ , then $P_{2} (x_{2}, y_{2})$ is a snap-back repeller in $U_{2} .$

Thus, the following theorem is established.

Theorem 8.If $4 a d (1 - 2 x_{2}) R_{2} > \max (4, {[a (1 - 2 x_{2}) - c e Q_{2} + d R_{2}]}^{2})$ the solutions $(x_{1}, y_{1})$ and $(x_{0}, y_{0})$ of Equations (5) and (6) satisfy in addition $(x_{0}, y_{0}), (x_{1}, y_{1}) \neq (x_{2}, y_{2}), (x_{0}, y_{0}) \in U_{2}, (x_{0}, y_{0}) \neq (0, 0)$ and $| D F^{2} (P_{*}) | \neq 0,$ then $P_{2} (x_{2}, y_{2})$ is a snap-back repeller of Map (2), and hence Map (2) is chaotic in the sense of Marotto.

6 CHAOS CONTROL

The study of dynamical systems desires that the system be optimized with respect to some performance criterion and chaos be avoided. The controlling chaos in discrete-time dynamical systems is a topic of great interest for researchers. Many practical methods can be used in many fields such as communications, physics laboratories, cardiology, turbulence, and so on. In discrete-time models, various methods are used to study chaos control; out of them, the state feedback method, pole-placement technique, and hybrid control method are most cited in the literature. This study considers feedback control to stabilize chaotic orbits at an unstable positive fixed point of the model (2). Therefore the controlled form on the prey species of the model (2) is as follows:

x_{n + 1} = a x_{n} (1 - x_{n}) - \frac{c (y_{n} - b) x_{n} y_{n}}{e y_{n} + (y_{n} - b) x_{n}} + S

()

with the following feedback control law as the control force:

S = - q_{1} (x_{n} - x_{2}) - q_{2} (y_{n} - y_{2}),

where

q_{1}

and

q_{2}

are the feedback gain and

(x_{2}, y_{2})

is a positive fixed point of model.

The Jacobian Matrix J for the system (7) at $(x_{2}, y_{2})$ is $J = [\begin{array}{cc} a_{11} - q_{1} & a_{12} - q_{2} \\ a_{21} & a_{22} \end{array}]$ , where $a_{11} = a (1 - 2 x) - \frac{c e (y - b) y^{2}}{{(e y + (y - b) x)}^{2}}$ , $a_{12} = - \frac{c x [e y^{2} + x {(y - b)}^{2}]}{{(e y + (y - b) x)}^{2}}$ , $a_{21} = \frac{d e (y - b) y^{2}}{{(e y + (y - b) x)}^{2}}$ , $a_{22} = \frac{d x [e y^{2} + x {(y - b)}^{2}]}{{(e y + (y - b) x)}^{2}} .$

The corresponding characteristic equation of matrix J is

$λ^{2} - (a_{11} + a_{22} - q_{1}) λ + a_{22} (a_{11} - q_{1}) - a_{21} (a_{12} - q_{2}) .$

Let

λ_{1}

and

λ_{2}

are the eigenvalues

λ_{1} + λ_{2} = a_{11} + a_{22} - q_{1} and λ_{1} λ_{2} = a_{22} (a_{11} - q_{1}) - a_{21} (a_{12} - q_{2}) .

()

The equation $λ_{1} = \pm 1$ and $λ_{1} λ_{2} = 1$ gives the lines of marginal stability. Hence the eigenvalues $λ_{1}$ and $λ_{2}$ have modulus less than 1.

Suppose $λ_{1} λ_{2} = 1$ ; then from (8) we have line $l_{1}$ as follows:

$a_{22} q_{1} - a_{21} q_{2} = a_{22} a_{11} - a_{21} a_{12} - 1$ .

Suppose $λ_{1} = \pm 1;$ from (8) we have line $l_{2}$ and $l_{3}$ as follows:

$(1 - a_{22}) q_{1} + a_{21} q_{2} = a_{11} + a_{22} - 1 - a_{22} a_{11} + a_{21} a_{12}$ , and

$(1 + a_{22}) q_{1} - a_{21} q_{2} = a_{11} + a_{22} + 1 + a_{22} a_{11} - a_{21} a_{12}$ .

A triangular region stream by line $l_{1}, l_{2}$ , and $l_{3}$ contains the stable eigenvalues.

Example: For parameter set $a = 4.5; b = 0.01; c = 3; d = 3.5; e = 0.8$ and the initial population $(0.6, 0.4),$ we observe chaotic behavior of prey–predator show in Figure 1A. In this case fixed point $(0.3, 0.8)$ is unstable. For feedback gain $q_{1} = 0.07$ and $q_{2} = - 0.07,$ we observe chaotic behavior of prey–predator show in Figure 1B. For feedback gain $q_{1} = 0.2$ and $q_{2} = - 0.2$ , we observe limit cycle of prey–predator show in Figure 1C. For feedback gain $q_{1} = 0.3$ and $q_{2} = - 0.3,$ we observe fixed point $(0.3, 0.8)$ is stable show in Figure 1D. Using above parameter set, Figures 2 (red for prey and blue for predator population) and 3 picture the effect of parameters $q_{1}$ and $q_{2}$ on the controlled model (7). Analyzing Figures 2 and 3, we conclude that using the control process system can be changed from chaos to stability and reverse.

Details are in the caption following the image — **FIGURE 1**
Open in figure viewer PowerPoint

Chaos control (A) full developed chaos, (B) chaotic attractor, (C) invariant closed curve, and (D) a stable fixed point

7 NUMERICAL SIMULATIONS

This section reveals the qualitative dynamical behaviors of system (2) by presenting Lyapunov exponent, bifurcation diagram, phase plane for specific parameter values. Bifurcation analyzes systems qualitative and quantitative behavior concerning any parameter value. Bifurcation occurs for the qualitative change for any control parameter, and linear instability produces bifurcation. Bifurcation plotting takes much time as there is a massive number of points against each parameter value. We observe rich dynamics in this bifurcation figure of our model. Parameters are $a = 4.0; b = 0.1; c = 3.0; d = 3.5; e = 0.8$ and initial population $(0.6, 0.4) .$ We take 100,000 iterations for drawing phase portraits and 1000 iterations for the bifurcation diagram. We also take the Wolf algorithm for Lyapunov exponent.

Figure 4 shows that the system trajectory evolves from a fixed point to NSB, finally into a chaotic attractor. There are two visible periodic windows in the bifurcation diagram for both prey and predator. In Figure 5, the largest Lyapunov exponent $L 1$ is greater than zero when $a > 4.47$ , which implies that the system is chaotic.

The phase portraits corresponding to Figure 4 are presented in Figure 6, which clearly depict the process of how a smooth invariant closed curve bifurcates from a stable fixed point and finally into a chaotic attractor.

Figure 7 shows the orbit diagram of prey and predator population for the refuge parameter (b) with other fixed parameter values. The system trajectory evolves into a periodic orbit from the chaotic attractor and finally into a fixed point. There are five obvious periodic windows in all the bifurcation processes for both prey and predator.

In Figure 8, the largest Lyapunov exponent $L 1$ is greater than zero when $b < 0.04$ , which implies that the system is chaotic.

The phase portraits corresponding to Figure 7 are presented in Figure 9, which clearly depict the process of how stable fixed point bifurcates from chaos.

Figure 10 shows the orbit diagram of prey and predator population for the parameter (e) with other fixed parameter values. The system trajectory evolves into a periodic orbit from the chaotic attractor, and finally into a fixed point. There is only two obvious periodic windows in all the bifurcation process for both prey and predator.

In Figure 11, the largest Lyapunov exponent $L 1$ is greater than zero when $e < 0.75$ ; which implies that the system is chaotic.

The phase portraits corresponding to Figure 10 are presented in Figure 12 , which clearly depict the process of how a stable fixed point bifurcates from chaos. The system dynamics described in Figure 12 for four situations for the different values of the half saturated constant, that is, for $e = 0.685$ the system shows a full developed chaos (Figure 12A), but for slight increase for e shows a chaotic attractor for the system (Figure 12B). If we increase the half-saturated constant, furthermore, the system exhibits an invariant closed curve (Figure 12C) for $e = 0.8245$ , and slight increases of e the system dynamics exhibits a stable fixed point (Figure 12D). This analysis shows the different states of the system from chaos, chaotic attractor, invariant closed curve, and finally, the stable fixed point bifurcates.

8 CONCLUSION

Many efforts have been put to manage the subpopulation of the zebra in small reserves. Like zebra, many other species are either driven to extinction or at the verge of extinction due to overexploitation, overpredation, indiscriminate harvesting, environmental pollution, and so on. This motivates us to create biosphere reserve zones/refuges, which may decrease the interaction among species. Because of the above, this article has presented the effects of refuges used by prey on a predator–prey interaction using the analytical and graphical approach. The refuge is considered proportional to prey and inverse proportion to the predator. The study showed that the effects of refuge could stabilize the proposed discrete model. We have evaluated the impact concerning the interior equilibrium point's local stability and the interacting populations' long-term dynamics. The proposed prey–predator model exhibits various bifurcations of codimension 1, including period-doubling bifurcation and Neimark–Sacker bifurcation as the values of the parameters vary, according to the condition given in Theorems 10 and 11. We have observed rich dynamics in the bifurcation figure for the proposed discrete prey–predator model incorporating the prey refuge, where the prey refuge follows proportional to prey species and decreases with proportion to the predator population.

The above study also suggests that the reserved zone's role is crucial in ecology and evolution. The prey refuge stabilizes the prey–predator model with Holling type II functional response. Prey refuge can control the chaotic condition of prey–predator interaction. We can say that the addition of small refuge to the model system does not alter the chaotic behavior of the system, while the addition of considerable values of prey refuge to the system changes the system's chaotic behavior into a stable equilibrium solution, as shown in Figure 5. The modeling of prey–predator incorporating prey refuge with Holling type II functional response can have a better stabilizing effect and can be useful in the case of regions/national parks where the prey and predator species live together wishes to protect the prey species.

APPENDIX

Proof of Theorem 4

Proof.Let b is used as the bifurcation parameter. Further $b^{*} (|b^{*}| ⋘ 1)$ is the perturbation of b, we consider a perturbation of the model as follows:

\begin{align} x_{n + 1} & = a x_{n} (1 - x_{n}) - \frac{c (y_{n} - (b + b^{*})) x_{n} y_{n}}{e y_{n} + (y_{n} - (b + b^{*})) x_{n}} \equiv f (x_{n}, y_{n}, b^{*}), \end{align}

()

\begin{align} y_{n + 1} & = \frac{d (y_{n} - (b + b^{*})) x_{n} y_{n}}{e y_{n} + (y_{n} - (b + b^{*})) x_{n}} \equiv g (x_{n}, y_{n}, b^{*}) . \end{align}

Let $u_{n} = x_{n} - x_{2}, v_{n} = y_{n} - y_{2}$ , to shift the equilibrium point $P_{2} (x_{2}, y_{2})$ into the origin, and then expanding the function f and g as a Taylor series at $(u_{n}, v_{n}, b^{*}) = (0, 0, 0)$ up to the third order, the system (A1) becomes

\begin{align} u_{n + 1} & = α_{1} u_{n} + α_{2} v_{n} + α_{11} u_{n}^{2} + α_{12} u_{n} v_{n} + α_{13} u_{n} b^{*} + α_{23} v_{n} b^{*} + α_{111} u_{n}^{3} \\ + α_{112} u_{n}^{2} v_{n} + α_{113} u_{n}^{2} b^{*} + α_{123} u_{n} v_{n} b^{*} + O ({(|u_{n}| + |v_{n}| + |b^{*}|)}^{4}) \end{align}

()

\begin{align} v_{n + 1} & = β_{1} u_{n} + β_{2} v_{n} + β_{11} u_{n}^{2} + β_{12} u_{n} v_{n} + β_{22} v_{n}^{2} + β_{13} u_{n} b^{*} + β_{23} v_{n} b^{*} + β_{111} u_{n}^{3} \\ + β_{112} u_{n}^{2} v_{n} + β_{113} u_{n}^{2} b^{*} + β_{123} u_{n} v_{n} b^{*} + β_{223} v_{n}^{2} b^{*} + O ({(|u_{n}| + |v_{n}| + |b^{*}|)}^{4}), \end{align}

where

α_{1} = f_{x} (x_{2}, y_{2}, 0), α_{2} = f_{y} (x_{2}, y_{2}, 0), α_{11} = f_{x x} (x_{2}, y_{2}, 0), α_{12} = f_{x y} (x_{2}, y_{2}, 0), α_{13} = f_{x b^{*}} (x_{2}, y_{2}, 0), α_{23} = f_{y b^{*}} (x_{2}, y_{2}, 0), α_{111} = f_{x x x} (x_{2}, y_{2}, 0), α_{112} = f_{x x y} (x_{2}, y_{2}, 0), α_{113} = f_{x x b^{*}} (x_{2}, y_{2}, 0), α_{123} = f_{x y b^{*}} (x_{2}, y_{2}, 0), β_{1} = g_{x} (x_{2}, y_{2}, 0), β_{2} = g_{y} (x_{2}, y_{2}, 0), β_{11} = g_{x x} (x_{2}, y_{2}, 0), β_{12} = g_{x y} (x_{2}, y_{2}, 0), β_{22} = g_{y y} (x_{2}, y_{2}, 0), β_{13} = g_{x b^{*}} (x_{2}, y_{2}, 0), β_{23} = g_{y b^{*}} (x_{2}, y_{2}, 0), β_{111} = g_{x x x} (x_{2}, y_{2}, 0), β_{112} = g_{x x y} (x_{2}, y_{2}, 0), β_{113} = g_{x x b^{*}} (x_{2}, y_{2}, 0), β_{123} = g_{x y b^{*}} (x_{2}, y_{2}, 0), β_{223} = g_{y y b^{*}} (x_{2}, y_{2}, 0)

We define an invertible matrix $T = [\begin{array}{cc} α_{2} & α_{2} \\ - 1 - α_{1} & λ_{2} - α_{1} \end{array}],$ and using the transformation $[\begin{array}{c} u_{n} \\ v_{n} \end{array}] = T [\begin{array}{c} {\overline{x}}_{n} \\ {\overline{y}}_{n} \end{array}]$ , the model (A2) becomes

\begin{align} {\overline{x}}_{n + 1} & = - {\overline{x}}_{n} + f_{1} (u_{n}, v_{n}, b^{*}), \end{align}

()

\begin{align} {\overline{y}}_{n + 1} & = λ_{2} {\overline{y}}_{n} + g_{1} (u_{n}, v_{n}, b^{*}), \end{align}

where the functions

f_{1}

and

g_{1}

denote the terms in the model (A3) in variables

(u_{n}, v_{n}, b^{*})

with order at least two.

Based on the center manifold theorem, a center manifold $W^{c} (0, 0, 0)$ of the model (A3) at $(0, 0)$ in a small neighborhood of $b^{*} = 0$ , approximately described as follows:

$W^{c} (0, 0, 0) = \{({\overline{x}}_{n}, {\overline{y}}_{n}, b^{*}) ϵ R^{3} : {\overline{y}}_{n + 1} = {\overline{α}}_{1} {\overline{x}}_{n}^{2} + {\overline{α}}_{2} {\overline{x}}_{n} b^{*} + O ({(|{\overline{x}}_{n}| + |b^{*}|)}^{3})\},$ where ${\overline{α}}_{1} = \frac{α_{2} [(1 + α_{1}) α_{11} + α_{2} β_{11}]}{1 - λ_{2}^{2}} + \frac{β_{22} {(1 + α_{1})}^{2}}{1 - λ_{2}^{2}} - \frac{(1 + α_{1}) [α_{12} (1 + α_{1}) + α_{2} β_{12}]}{1 - λ_{2}^{2}}$ and ${\overline{α}}_{2} = \frac{(1 + α_{1}) [α_{23} (1 + α_{1}) + α_{2} β_{23}]}{α_{2} {(1 + λ_{2})}^{2}} - \frac{(1 + α_{1}) α_{13} + α_{2} β_{13}]}{{(1 + λ_{2})}^{2}}$ .

The model (A3) is restricted to the center manifold $W^{c} (0, 0, 0)$ , has the following form:

${\overline{x}}_{n + 1} = - {\overline{x}}_{n} + h_{1} {\overline{x}}_{n}^{2} + h_{2} {\overline{x}}_{n} b^{*} + h_{3} {\overline{x}}_{n}^{2} b^{*} + h_{4} {\overline{x}}_{n} b^{* 2} + h_{5} {\overline{x}}_{n}^{3} + O ({(|{\overline{x}}_{n}| + |b^{*}|)}^{3}) \equiv F ({\overline{x}}_{n}, b^{*}),$ where $h_{1} = \frac{{\overline{α}}_{2} [(λ_{2} - {\overline{α}}_{1}) α_{11} - {\overline{α}}_{2} β_{11}]}{1 + λ_{2}} - \frac{β_{22} {(1 + {\overline{α}}_{1})}^{2}}{1 + λ_{2}} - \frac{(1 + {\overline{α}}_{1}) [(λ_{2} - {\overline{α}}_{1}) α_{12} - {\overline{α}}_{2} β_{12}]}{1 + λ_{2}}$ , $h_{2} = \frac{(λ_{2} - {\overline{α}}_{1}) α_{13} - {\overline{α}}_{2} β_{13}}{1 + λ_{2}} - \frac{(1 + {\overline{α}}_{1}) [(λ_{2} - {\overline{α}}_{1}) α_{23} - {\overline{α}}_{2} β_{23}]}{{\overline{α}}_{2} (1 + λ_{2})}$ , $h_{3} = \frac{(λ_{2} - α_{1}) {\overline{α}}_{1} α_{13} - α_{2} β_{13}}{1 + λ_{2}} + \frac{[(λ_{2} - α_{1}) α_{23} - α_{2} β_{23}] (λ_{2} - α_{1}) {\overline{α}}_{1}}{α_{2} (1 + λ_{2})} - \frac{(1 + α_{1}) [(λ_{2} - α_{1}) α_{123} - α_{2} β_{123}]}{1 + λ_{2}} + \frac{α_{2} [(λ_{2} - α_{1}) α_{113} - α_{2} β_{113}]}{1 + λ_{2}} - \frac{β_{223} {(1 + α_{1})}^{2}}{1 + λ_{2}} + \frac{2 α_{2} {\overline{α}}_{2} [(λ_{2} - α_{1}) α_{11} - α_{2} β_{11}]}{1 + λ_{2}} - \frac{2 β_{22} {\overline{α}}_{2} (1 + α_{1}) (λ_{2} - α_{1})}{1 + λ_{2}} + \frac{{\overline{α}}_{2} [(λ_{2} - α_{1}) α_{12} - α_{2} β_{12}] (λ_{2} - 1 - 2 α_{1})}{1 + λ_{2}}$ , $h_{4} = \frac{{\overline{α}}_{2} [(λ_{2} - α_{1}) α_{13} - α_{2} β_{13}]}{1 + λ_{2}} + \frac{[(λ_{2} - α_{1}) α_{23} - α_{2} β_{23}] (λ_{2} - α_{1}) {\overline{α}}_{2}}{α_{2} (1 + λ_{2})} + \frac{2 α_{2} {\overline{α}}_{2} [(λ_{2} - α_{1}) α_{11} - α_{2} β_{11}]}{1 + λ_{2}} + \frac{2 β_{22} {\overline{α}}_{2} (1 + α_{1}) (λ_{2} - α_{1})}{1 + λ_{2}} + \frac{{\overline{α}}_{2} [(λ_{2} - α_{1}) α_{12} - α_{2} β_{12}] (λ_{2} - 1 - 2 α_{1})}{1 + λ_{2}}$ , $h_{5} = \frac{2 α_{2} {\overline{α}}_{1} [(λ_{2} - α_{1}) α_{11} - α_{2} β_{11}]}{1 + λ_{2}} + \frac{2 β_{22} {\overline{α}}_{1} (λ_{2} - α_{1}) (1 + α_{1})}{1 + λ_{2}} + \frac{[(λ_{2} - α_{1}) α_{11} - α_{2} β_{11}] (λ_{2} - 1 - 2 α_{1}) {\overline{α}}_{1}}{1 + λ_{2}} + \frac{{\overline{α}}_{2}^{2} [(λ_{2} - α_{1}) α_{111} - α_{2} β_{111}]}{1 + λ_{2}} - \frac{{\overline{α}}_{2} (1 + α_{1}) [(λ_{2} - α_{1}) α_{112} - α_{2} β_{112}]}{1 + λ_{2}} .$

For flip bifurcation, the two discriminatory quantities $ξ_{1}$ and $ξ_{2}$ be nonzero, $ξ_{1} = (\frac{\partial^{2} F}{\partial \overline{x} \partial b^{*}} + \frac{1}{2} \frac{\partial F}{\partial b^{*}} \frac{\partial^{2} F}{\partial {\overline{x}}^{2}}) |_{(0, 0)}$ and $ξ_{2} = (\frac{1}{6} \frac{\partial^{3} F}{\partial {\overline{x}}^{3}} + {(\frac{1}{2} \frac{\partial^{2} F}{\partial {\overline{x}}^{2}})}^{2}) |_{(0, 0)}$ .

Hence the theorem.

Proof of Theorem 5

Proof.Let b is used as the bifurcation parameter, and $b^{*} (|b^{*}| ⋘ 1)$ is the perturbation of b. Let a perturbation of the system as follows:

\begin{align} x_{n + 1} & = a x_{n} (1 - x_{n}) - \frac{c (y_{n} - (b + b^{*})) x_{n} y_{n}}{e y_{n} + (y_{n} - (b + b^{*})) x_{n}} \equiv f (x_{n}, y_{n}, b^{*}), \end{align}

()

\begin{align} y_{n + 1} & = \frac{d (y_{n} - (b + b^{*})) x_{n} y_{n}}{e y_{n} + (y_{n} - (b + b^{*})) x_{n}} \equiv g (x_{n}, y_{n}, b^{*}) . \end{align}

Let $u_{n} = x_{n} - x_{2}, v_{n} = y_{n} - y_{2}$ to transform the equilibrium $P_{2} (x_{2}, y_{2})$ into the origin, and then expanding f and g as a Taylor series at $(u_{n}, v_{n}) = (0, 0)$ to the third order, the above system (A4) becomes

\begin{align} u_{n + 1} & = α_{1} u_{n} {+ α}_{2} v_{n} {+ α}_{11} u_{n}^{2} {+ α}_{12} u_{n} v_{n} {+ α}_{22} v_{n}^{2} {+ α}_{111} u_{n}^{3} {+ α}_{112} u_{n}^{2} v_{n} {+ α}_{122} u_{n} v_{n}^{2} {+ α}_{222} v_{n}^{3} + O ({(|u_{n}| + |v_{n}|)}^{4}), \end{align}

()

\begin{align} v_{n + 1} & = β_{1} u_{n} {+ β}_{2} v_{n} {+ β}_{11} u_{n}^{2} {+ β}_{12} u_{n} v_{n} {+ β}_{22} v_{n}^{2} {+ β}_{111} u_{n}^{3} {+ β}_{112} u_{n}^{2} v_{n} {+ β}_{122} u_{n} v_{n}^{2} + β_{222} v_{n}^{3} + O ({(|u_{n}| + |v_{n}|)}^{4}), \end{align}

where

α_{1} = f_{x} (x_{2}, y_{2}, 0)

α_{2} = f_{y} (x_{2}, y_{2}, 0)

α_{11} = f_{x x} (x_{2}, y_{2}, 0)

α_{12} = f_{x y} (x_{2}, y_{2}, 0)

α_{22} = f_{y y} (x_{2}, y_{2}, 0)

α_{111} = f_{x x x} (x_{2}, y_{2}, 0)

α_{112} = f_{x x y} (x_{2}, y_{2}, 0), α_{122} = f_{x y y} (x_{2}, y_{2}, 0)

α_{222} = f_{y y y} (x_{2}, y_{2}, 0)

β_{1} = g_{x} (x_{2}, y_{2}, 0)

β_{2} = g_{y} (x_{2}, y_{2}, 0)

β_{11} = g_{x x} (x_{2}, y_{2}, 0)

β_{12} = g_{x y} (x_{2}, y_{2}, 0)

β_{22} = g_{y y} (x_{2}, y_{2}, 0)

β_{111} = g_{x x x} (x_{2}, y_{2}, 0)

β_{112} = g_{x x y} (x_{2}, y_{2}, 0)

β_{122} = g_{x y y} (x_{2}, y_{2}, 0)

β_{222} = g_{y y y} (x_{2}, y_{2}, 0)

Note that the characteristic equation associated with the linearization of the model (A5) at $(u_{n}, v_{n}) = (0, 0)$ is given by $λ^{2} - T r (J_{1} (b^{*})) λ + D e t (J_{1} (b^{*})) = 0$ .

The roots of the characteristic equation are $λ_{1, 2} (b^{*}) = \frac{T r (J_{1} (b^{*})) \pm i \sqrt{4 D e t (J_{1} (b^{*})) - {(T r (J_{1} (b^{*})))}^{2}}}{2} .$

From $|λ_{1, 2} (b^{*})| = 1,$ when $b^{*} = 0$ we have, $|λ_{1, 2} (b^{*})| = {[D e t (J_{1} (b^{*}))]}^{\frac{1}{2}}$ and $l = {[\frac{d |λ_{1, 2} (b^{*})|}{d b^{*}}]}_{h^{*} = 0} \neq 0 .$

Again for $b^{*} = 0$ , $λ_{1, 2}^{i} \neq 1$ , $i = 1, 2, 3, 4$ , which is equivalent to $T r (J_{1} (0)) \neq - 2, - 1, 1, 2 .$

Next we study the normal form. Let $γ = Im (λ_{1, 2})$ and $δ = Re (λ_{1, 2}) .$

We define $T = [\begin{array}{cc} 0 & 1 \\ γ & δ \end{array}]$ and using the transformation $[\begin{array}{c} u_{n} \\ v_{n} \end{array}] = T [\begin{array}{c} {\overline{x}}_{n} \\ {\overline{y}}_{n} \end{array}],$ the model (A5) becomes

\begin{align} {\overline{x}}_{n + 1} & = δ {\overline{x}}_{n} - γ {\overline{y}}_{n} + f_{1} ({\overline{x}}_{n}, {\overline{y}}_{n}), \end{align}

()

\begin{align} {\overline{y}}_{n + 1} & = γ {\overline{x}}_{n} + δ {\overline{y}}_{n} + g_{1} ({\overline{x}}_{n}, {\overline{y}}_{n}), \end{align}

where the functions

f_{1}

and

g_{1}

denote the terms in the model (A6) in variables

({\overline{x}}_{n}, {\overline{y}}_{n})

with the order at least two.

It is required that the discriminatory quantity $Ω$ be nonzero to undergo Hopf Bifurcation:

$Ω = - Re [\frac{(1 - 2 \overline{λ}) {\overline{λ}}^{2}}{1 - λ} ξ_{11} ξ_{20}] - \frac{1}{2} {|ξ_{11}|}^{2} - {|ξ_{02}|}^{2} + Re (\overline{λ} ξ_{21})$ , where $ξ_{20} = \frac{1}{8} δ (2 β_{22} - δ α_{22} - α_{12} + 4 γ α_{22}) + \frac{1}{4} γ α_{12} + \frac{1}{8} δ i (4 γ α_{22} - 2 α_{22} - 2 δ α_{22}) + \frac{1}{8} i (4 γ β_{22} + 2 γ^{2} α_{22} - 2 α_{11}) + \frac{1}{8} β_{12} + \frac{δ α_{11} - 2 β_{11}}{4 γ} + \frac{δ^{3} α_{22} - δ^{2} β_{22}}{4 γ} - \frac{δ^{2} α_{12} - δ β_{12}}{4 γ}$ , $ξ_{11} = \frac{1}{2} γ (β_{22} - δ α_{22}) + \frac{1}{2} i (γ^{2} α_{22} + α_{11} + δ α_{12} + δ^{2} α_{22}) + \frac{β_{11} - δ α_{11}}{2 γ} + \frac{δ β_{12} - δ^{2} α_{12}}{2 γ} - \frac{δ^{2} β_{22} - δ^{3} α_{22}}{γ}$ , $ξ_{02} = \frac{1}{4} γ (2 δ α_{22} + α_{12} + β_{22}) + \frac{1}{4} i (β_{12} + 2 δ β_{22} - 2 δ α_{12} - α_{11}) - \frac{β_{11} - δ α_{11}}{4 γ} - \frac{δ β_{12} - δ^{2} α_{12}}{4 γ} + \frac{1}{4} α_{22} i (γ^{2} - 3 δ^{2}) + \frac{δ^{2} β_{22} - δ^{3} α_{22}}{4 γ}$ , $ξ_{21} = \frac{3}{8} β_{222} (γ^{2} + δ^{2}) + \frac{1}{8} β_{112}$ + $\frac{1}{4} δ α_{112} + \frac{1}{4} δ β_{122} + α_{122} (\frac{1}{8} γ^{2} + \frac{3}{8} δ^{2} - \frac{1}{4} δ) + \frac{3}{8} α_{111} + \frac{3}{8} α_{222} i (γ^{2} + 2 δ^{2}) + \frac{3}{8} α_{122} γ δ i - \frac{1}{8} β_{122} γ i - \frac{3}{8} β_{222} γ δ i - \frac{3 β_{111} - 3 δ α_{111}}{8 γ} i - \frac{3 δ β_{112} - 3 δ^{2} α_{112}}{8 γ} i - \frac{3 δ^{2} β_{122} - 3 δ^{3} α_{122}}{8 γ} i - \frac{3 δ^{3} β_{222} - 3 δ^{4} α_{222}}{8 γ} i$ .

Hence the theorem.

Biographies

Prasun K. Santra holds the degrees of M.Sc. in Applied Mathematics from Bengal Engineering and Science University, Shibpur, presently IIEST Shibpur, India, and Ph.D. from Maulana Abul Kalam Azad University of Technology, West Bengal, India. Dr. Santra has been involved in research for more than 5 years and has published more than 12 research articles in various international journals and proceedings to his credit. His research interest field is Mathematical Biology.
Ghanshaym S. Mahapatra is an Associate Professor at National Institute of Technology Puducherry, India. He obtained M.Sc. and Ph.D. in Applied Mathematics from Bengal Engineering and Science University, Shibpur, presently IIEST Shibpur, India. Dr. Mahapatra has been involved in teaching and research for more than 17 years and has published more than 90 research articles in various International, National journals and proceedings to his credit. His fields of interest in research work are Inventory Management, Reliability, Optimization, Fuzzy Set theory, Soft Computing, and Mathematical Biology.
Ganga R. Phaijoo is an Assistant Professor at Kathmandu University, Dhulikhel, Nepal. His fields of interest in research work are Mathematical Modeling of Infectious Diseases, Prey Predator Models, and Numerics in ODEs and PDEs.

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Citing Literature

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Bifurcation analysis and chaos control of discrete prey–predator model incorporating novel prey–refuge concept

Abstract

1 INTRODUCTION

2 DESCRIPTION OF PROPOSED PREY–PREDATOR MODEL

3 EQUILIBRIUM POINTS AND THEIR STABILITY ANALYSIS

3.1 Existence of equilibrium points

3.2 Local stability analysis

4 BIFURCATION ANALYSIS

4.1 Period-doubling bifurcation

4.2 Neimark–Sacker bifurcation

5 EXISTENCE OF MAROTTOS CHAOS

6 CHAOS CONTROL

7 NUMERICAL SIMULATIONS

8 CONCLUSION

APPENDIX

Proof of Theorem 4

Proof of Theorem 5

Biographies

REFERENCES

Citing Literature

Figures

References

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Bifurcation analysis and chaos control of discrete prey–predator model incorporating novel prey–refuge concept

Abstract

1 INTRODUCTION

2 DESCRIPTION OF PROPOSED PREY–PREDATOR MODEL

3 EQUILIBRIUM POINTS AND THEIR STABILITY ANALYSIS

3.1 Existence of equilibrium points

3.2 Local stability analysis

4 BIFURCATION ANALYSIS

4.1 Period-doubling bifurcation

4.2 Neimark–Sacker bifurcation

5 EXISTENCE OF MAROTTOS CHAOS

6 CHAOS CONTROL

7 NUMERICAL SIMULATIONS

8 CONCLUSION

APPENDIX

Proof of Theorem 4

Proof of Theorem 5

Biographies

REFERENCES

Citing Literature

Figures

References

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