Stability and Bifurcation Analysis in a Discrete Predator–Prey System of Leslie Type with Radio-Dependent Simplified Holling Type IV Functional Response

Abstract

In this paper, we use a semi-discretization method to consider the predator–prey model of Leslie type with ratio-dependent simplified Holling type IV functional response. First, we discuss the existence and stability of the positive fixed point in total parameter space. Subsequently, through using the central manifold theorem and bifurcation theory, we obtain sufficient conditions for the flip bifurcation and Neimark–Sacker bifurcation of this system to occur. Finally, the numerical simulations illustrate the existence of Neimark–Sacker bifurcation and obtain some new dynamical phenomena of the system—the existence of a limit cycle. Corresponding biological meanings are also formulated.

1. Introduction and Preliminaries

In the past few decades, mathematical models have played a crucial role in the study of biology. According to different application environments, the interaction between populations can be expressed as consumer–resource [1], plant–herbivore [2], parasite–host [3], etc. Predator–prey models are the building blocks of ecosystems, as biomass grows from resource masses. This topic plays an important role in ecology [4,5,6,7,8,9,10,11,12,13,14,15,16].

Two of the most famous predator–prey dynamical models are the Lotka–Volterra model and the Leslie type model. Based on these models, subsequent scholars have considered more influencing factors, such as the fear effect [9], the Allee effect [12], super-predators [13], etc.

In [4], Leslie first proposed that the carrying capacity of the environment for the predator is proportional to the number of prey. So, this form of predator growth is also called Leslie–Gower type and can be represented as

$$\frac{dy}{dt}=sy(1-\frac{y}{hx})$$

Therefore, the dynamical relationship between prey and predator may be expressed as

$$\left\{\begin{array}{c}\frac{dx}{dt}=g\left(x\right)x-f(x,y)y\hfill \\ \frac{dy}{dt}=sy(1-\frac{y}{hx})\hfill \end{array}\right.$$

(1)

where x and y represent prey and predator population sizes or densities, respectively; the function $g\left(x\right)$ characterizes the growth rate of prey in the absence of a predator, and may be represented through logistic growth $g\left(x\right)=r\left(1-\frac{x}{k}\right)$; $f(x,y)$ is a functional response curve and has many different forms, such as Holling types I–IV [17,18], Beddington–DeAngelis type [19,20], Hassell–Varley type [15,21], etc.; the parameter s signifies the intrinsic growth rate of the predator; and k and h denote the carrying capacity of the prey and predator provided by the environment, respectively.

As is well-known, the generalized Holling-IV response function is $f(x,y)=\frac{mx}{a{x}^2+bx+1}$. There is far less research on the Holling-IV response function than on the Holling-I–III response functions. Here, we assume that the prey growth follows logistic growth, and the functional response $f(x,y)$ is taken as a Holling-IV response function. Then, the system (1) can be reformulated as

$$\left\{\begin{array}{c}\frac{dx}{dt}=rx(1-\frac{x}{k})-\frac{mx}{a+x^2}y\hfill \\ \frac{dy}{dt}=sy(1-\frac{y}{hx})\hfill \end{array}\right.$$

(2)

Here, m and a are positive constants, the parameter m is the maximal predator per capita consumption rate, the parameter a is the number of prey necessary to achieve half of the maximum rate m, and the parameter r signifies the intrinsic growth rate of prey.

If one takes the functional response $f(x,y)$ as the ratio-dependent type, i.e., $f(x,y)=\frac{m\frac{x}{y}}{a+{\left(\frac{x}{y}\right)}^2}$., then one has

$$\left\{\begin{array}{c}\frac{dx}{dt}=rx(1-\frac{x}{k})-\frac{mx{y}^2}{a{y}^2+x^2}\hfill \\ \frac{dy}{dt}=sy(1-\frac{y}{hx})\hfill \end{array}\right.$$

(3)

For the sake of the simplicity of mathematical analysis, we now non-dimensionalize the system (3). To accomplish this, let $\frac{x}{k}\to u,\frac{y}{l}\to v,rt\to \tau ,$ $hk\to l,\frac{k^{2}r}{m}\to \alpha ,\frac{a}{k^2}\to \beta ,$ and $\frac{s}{r}\to \delta$. Then, one can derive an equivalent to the system (3) as follows:

$$\left\{\begin{array}{c}\frac{du}{d\tau }=u(1-u)-\frac{\alpha u{v}^2}{\beta v^2+u^2}\hfill \\ \frac{dv}{d\tau }=\delta v(1-\frac{v}{u})\hfill \end{array}\right.$$

(4)

This continuous system has been discussed in [18], whereas its discrete version has not been found to be investigated yet. It is very difficult to solve a complicated continuous equation or system without using a computer. Meanwhile, many models in nature look more reasonable in their discrete forms. So, it is crucial to consider the discrete version corresponding to a continuous model.

Now, we use a semi-discretization method to discretize the system (4). Let $\left[t\right]$ denote the greatest integer not exceeding t. Consider the average change rate of the system (4) at integer number points, namely, the following model:

$$\left\{\begin{array}{c}\frac{1}{u\left(\tau \right)}\frac{du}{d\tau }=1-u\left(\right[\tau \left]\right)-\frac{\alpha v^2\left(\left[\tau \right]\right)}{\beta v^2\left(\left[\tau \right]\right)+u^2\left(\left[\tau \right]\right)}\hfill \\ \frac{1}{v\left(\tau \right)}\frac{dv}{d\tau }=\delta (1-\frac{v\left(\right[\tau \left]\right)}{u\left(\right[\tau \left]\right)})\hfill \end{array}\right.$$

(5)

One can see that a solution $\left(u\right(\tau ),v(\tau \left)\right)$ of the system (5) for $\tau \in [0,+\infty )$ has the following characteristics:

• On the interval $[0,+\infty )$, $u\left(\tau \right)$ and $v\left(\tau \right)$ are continuous;
• $\frac{du\left(\tau \right)}{d\tau }$ and $\frac{dv\left(\tau \right)}{d\tau }$ exist everywhere when $\tau \in [0,+\infty )$, possibly except at the points $\{0,1,2,3,\cdots \}$;
• The system (5) is true on any interval $[n,n+1)$ with $n=0,1,2,\cdots$.

The following system can be obtained by integrating the system (5) over the interval [n,$\tau$] for any $\tau \in [n,n+1)$ and $n=0,1,2,\cdots$

$$\left\{\begin{array}{c}ln\frac{u\left(\tau \right)}{u\left(n\right)}=\left(1-u_n-\frac{\alpha {\left(v_n\right)}^2}{\beta {\left(v_n\right)}^2+{\left(u_n\right)}^2}\right)(\tau -n),\hfill \\ ln\frac{v\left(\tau \right)}{v\left(n\right)}=\delta (1-\frac{v_n}{u_n})(\tau -n)\hfill \end{array}\right.$$

(6)

Subsequently, we simultaneously take the exponent with e as the base for Equation (6):

$$\left\{\begin{array}{c}u\left(\tau \right)=u_n{e}^{\left(1-u_n-\frac{\alpha {\left(v_n\right)}^2}{\beta {\left(v_n\right)}^2+{\left(u_n\right)}^2}\right)(\tau -n)}\hfill \\ v\left(\tau \right)=v_n{e}^{\delta (1-\frac{v_n}{u_n})(\tau -n)}\hfill \end{array}\right.$$

(7)

where $u_n=u\left(n\right)$ and $v_n=v\left(n\right)$. Letting $\tau \to {(n+1)}^{-}$ in the system (7) produces

$$\left\{\begin{array}{c}{u}_{n+1}=u_n{e}^{1-u_n-\frac{\alpha {\left(v_n\right)}^2}{\beta {\left(v_n\right)}^2+{\left(u_n\right)}^2}}\hfill \\ v_{n+1}=v_n{e}^{\delta (1-\frac{v_n}{u_n})}\hfill \end{array}\right.$$

(8)

where the parameters $\alpha ,\beta ,\delta >0$, and they are the same as in (4). The system (8) will be considered in the sequel.

The rest of this paper is organized as follows. In Section 2, we discuss the existence and stability of the positive fixed point because of the biological significance. In Section 3, we provide the sufficient conditions for the existence of the flip bifurcation and Neimark–Sacker bifurcation. In Section 4, we illustrate the theoretical results derived with numerical simulations. Finally, some conclusions and discussions are stated in Section 5.

2. Existence and Stability of Fixed Points

We discuss the existence and stability of non-negative fixed points of the system (8) in this section. The fixed point of the system (8) satisfies the following equation:

$$u-u{e}^{1-u-\frac{\alpha v^2}{\beta v^2+u^2}}=0,v-v{e}^{\delta (1-\frac{v}{u})}=0$$

It is easy to show that the system (8) has two non-negative fixed points $E_1=(1,0)$ and $E_2=(u^{*},v^{*})$ for $\alpha <\beta +1$, where

$$u^{*}=\frac{1+\beta -\alpha }{1+\beta },v^{*}=u^{*}$$

The Jacobian matrix of the system (8) at a fixed point $E(x,y)$ is

$$J\left(E\right)=\left(\begin{array}{cc}\left(1-u+\frac{2\alpha u^2{v}^2}{{(u^2+\beta v^2)}^2}\right){\mathrm{e}}^{1-u-\frac{\alpha v^2}{\beta v^2+u^2}}& -\frac{2\alpha u^{3}v}{{(u^2+\beta v^2)}^2}{\mathrm{e}}^{1-u-\frac{\alpha v^2}{\beta v^2+u^2}}\\ \frac{\delta v^2}{u^2}{\mathrm{e}}^{\delta (1-\frac{v}{u})}& \left(1-\frac{\delta v}{u}\right){\mathrm{e}}^{\delta (1-\frac{v}{u})}\end{array}\right)$$

and its characteristic equation is

$$F\left(\lambda \right)={\lambda }^2-Tr\left(J\left(E\right)\right)\lambda +Det\left(J\left(E\right)\right)=0$$

where

$$Tr\left(J\left(E\right)\right)=\left(1-u+\frac{2\alpha u^2{v}^2}{{(u^2+\beta v^2)}^2}\right){\mathrm{e}}^{1-u-\frac{\alpha v^2}{\beta v^2+u^2}}+\left(1-\frac{\delta v}{u}\right){\mathrm{e}}^{\delta (1-\frac{v}{u})}$$

$$Det\left(J\left(E\right)\right)=\left[\left(1-u+\frac{2\alpha u^2{v}^2}{{(u^2+\beta v^2)}^2}\right)(1-\frac{v}{u})+\frac{2\alpha u{v}^3}{{(u^2+\beta v^2)}^2}\right]{\mathrm{e}}^{1-u-\frac{\alpha v^2}{\beta v^2+u^2}+\delta (1-\frac{v}{u})}$$

Before analyzing the properties of the fixed points of the system (8), we provide the following definition and Lemma [22,23,24,25].

Definition 1.

Let $E(x,y)$ be a fixed point of the system (8) with multipliers ${\lambda }_1$ and ${\lambda }_2$.

$\left(i\right)$ If $|{\lambda }_1| <1$ and $|{\lambda }_2| <1$, $E(x,y)$ is called sink, then a sink is locally asymptotically stable.

$\left(ii\right)$ If $|{\lambda }_1| >1$ and $|{\lambda }_2| >1$, $E(x,y)$ is called source, then a source is locally asymptotically unstable.

$\left(iii\right)$ If $|{\lambda }_1| <1$ and $|{\lambda }_2| >1$ (or $|{\lambda }_1| >1$ and $|{\lambda }_2| <1$), then $E(x,y)$ is called saddle.

$\left(iv\right)$ If either $|{\lambda }_1| =1$ or $|{\lambda }_2| =1$, then $E(x,y)$ is called to be non-hyperbolic.

Lemma 1.

Let $F\left(\lambda \right)={\lambda }^2+B\lambda +C$, where B and C are two real constants. Suppose ${\lambda }_1$ and ${\lambda }_2$ are two roots of $F\left(\lambda \right)=0$. Then, the following statements hold.

(i)

If $F\left(1\right)>0,$ then

(i.1)

$|{\lambda }_1| <1$ and $|{\lambda }_2| <1$ if and only if $F(-1)>0$ and $C<1$;

(i.2)

${\lambda }_1=-1$ and ${\lambda }_2\ne -1$ if and only if $F(-1)=0$ and $B\ne 2$;

(i.3)

$|{\lambda }_1| <1$ and $|{\lambda }_2| >1$ if and only if $F(-1)<0$;

(i.4)

$|{\lambda }_1| >1$ and $|{\lambda }_2| >1$ if and only if $F(-1)>0$ and $C>1$;

(i.5)

${\lambda }_1$ and ${\lambda }_2$ are a pair of conjugate complex roots and $|{\lambda }_1| = |{\lambda }_2| =1$ if and only if $-2<B<2$ and $C=1$;

(i.6)

${\lambda }_1={\lambda }_2=-1$ if and only if $F(-1)=0$ and $B=2$.

(ii)

If $F\left(1\right)=0,$ namely, 1 is one root of $F\left(\lambda \right)=0$, then another root λ satisfies $\left|\lambda \right| =(<,>)1$ if and only if $\left|C\right| =(<,>)1.$

(iii)

If $F\left(1\right)<0,$ then $F\left(\lambda \right)=0$ has one root lying in $(1,\infty )$. Moreover

(iii.1)

the other root λ satisfies $\lambda <(=)-1$ if and only if $F(-1)<(=)0$;

(iii.2)

the other root $-1<\lambda <1$ if and only if $F(-1)>0$.

Due to biological significance, we only consider $E_2$. By using Definition 1 and Lemma 1, the following result can be obtained.

Theorem 1.

For $\alpha <1+\beta$, $E_2=(u^{*},v^{*})=(\frac{1+\beta -\alpha }{1+\beta },\frac{1+\beta -\alpha }{1+\beta })$ is a positive fixed point of the system (8).

Let ${\delta }_1=2+\frac{4\alpha }{(1+\alpha +\beta )(1+\beta )}$ and ${\delta }_2=\frac{3+\beta }{1+\beta }-\frac{1+\beta }{\alpha }$. The following statements are true about the positive fixed point $E_2$.

• Case 1. When $0<\alpha \le \frac{{(1+\beta )}^2}{3+\beta }$, ${\delta }_2\le 0<{\delta }_1$:

$\left(1\right)$ if $0<\delta <{\delta }_1$, then $E_2$ is a sink;

$\left(2\right)$ if $\delta ={\delta }_1$, then $E_2$ is non-hyperbolic;

$\left(3\right)$ if $\delta >{\delta }_1$, then $E_2$ is a saddle.

• Case 2. When $\frac{{(1+\beta )}^2}{3+\beta }<\alpha <1+\beta$, $0<{\delta }_2<{\delta }_1$:

$\left(1\right)$ if $0<\delta <{\delta }_2$, then $E_2$ is a sink;

$\left(2\right)$ if $\delta ={\delta }_2$, then $E_2$ is non-hyperbolic;

$\left(3\right)$ if ${\delta }_2<\delta <{\delta }_1$, then $E_2$ is a source;

$\left(4\right)$ if $\delta ={\delta }_1$, then $E_2$ is non-hyperbolic;

$\left(5\right)$ if $\delta >{\delta }_1$, then $E_2$ is a saddle.

Proof. 

The Jacobian matrix $J\left(E\right)$ of the system (8) at $E_2$ is

$$J\left(E_2\right)=\left(\begin{array}{cc}\frac{\alpha }{1+\beta }+\frac{2\alpha }{{(1+\beta )}^2}& -\frac{2\alpha }{{(1+\beta )}^2}\\ \delta & 1-\delta \end{array}\right)$$

whose characteristic polynomial can be written as

$$F\left(\lambda \right)={\lambda }^2-P\lambda +Q$$

(9)

with

$$P=\frac{\alpha }{1+\beta }+\frac{2\alpha }{{(1+\beta )}^2}+1-\delta$$

$$\begin{array}{cc}\hfill Q& =\left(\frac{\alpha }{1+\beta }+\frac{2\alpha }{{(1+\beta )}^2}\right)(1-\delta )+\frac{2\delta \alpha }{{(1+\beta )}^2}\hfill \\ \hfill & =\frac{\alpha }{1+\beta }\left(\frac{3+\beta }{1+\beta }-\delta \right)\hfill \end{array}$$

□

It is easy to see that $F\left(1\right)=\delta (1-\frac{\alpha }{1+\beta })>0$ always holds for $\alpha <1+\beta .$ Simple calculations display that

$$\begin{array}{cc}\hfill F(-1)& =2[1+\frac{\alpha }{1+\beta }+\frac{2\alpha }{{(1+\beta )}^2}]-\delta (1+\frac{\alpha }{1+\beta })\hfill \\ & =\frac{1+\alpha +\beta }{1+\beta }({\delta }_1-\delta )\hfill \\ \hfill Q-1& =\frac{\alpha }{1+\beta }+\frac{2\alpha }{{(1+\beta )}^2}-\frac{\delta \alpha }{1+\beta }-1\hfill \\ \hfill & =\frac{\alpha }{1+\beta }({\delta }_2-\delta )\hfill \end{array}$$

We can see that when $\delta >(=,<){\delta }_1$, then $F(-1)<(=,>)0$, and when $\delta >(=,<){\delta }_2$, then $Q<(=,>)1$.

• Case 1. If $0<\alpha \le \frac{{(1+\beta )}^2}{3+\beta }$, then ${\delta }_2\le 0<{\delta }_1$.

If $0<\delta <{\delta }_1$, then $F(-1)>0$. But $Q-1=\frac{\alpha }{1+\beta }({\delta }_2-\delta )<0,$ namely, $Q<1.$ Lemma 1 (i.1) states that $E_2$ is a stable node, i.e., a sink. If $\delta ={\delta }_1$, then $F(-1)=0$; hence, $E_2$ is non-hyperbolic. If $\delta >{\delta }_1$, then $F(-1)<0$, then Lemma 1 (i.3) says that $E_2$ is a saddle.

• Case 2. If $\frac{{(1+\beta )}^2}{3+\beta }<\alpha <1+\beta$, then $0<{\delta }_2<{\delta }_1$.

When $0<\delta <{\delta }_2$, then $F(-1)>0$ and $Q>1$. By Lemma 1 (i.4), $\left|{\lambda }_1\right|>1$ and $\left|{\lambda }_2\right|>1$; therefore, $E_2$ is an unstable node, i.e., a source. When $\delta ={\delta }_2$, $F(-1)=0$ and $Q=1$. On one hand, the following applies:

$$\begin{array}{cc}\hfill P+2& =\frac{\alpha }{1+\beta }+\frac{2\alpha }{{(1+\beta )}^2}+1-\left(\frac{3+\beta }{1+\beta }-\frac{1+\beta }{\alpha }\right)+2\hfill \\ \hfill & =\frac{\alpha }{1+\beta }+\frac{2\alpha }{{(1+\beta )}^2}+\frac{2\beta }{1+\beta }+\frac{1+\beta }{\alpha }>0\hfill \end{array}$$

So, $P>-2.$ On the other hand, the following applies:

$$\begin{array}{cc}\hfill P-2& =\frac{\alpha }{1+\beta }+\frac{2\alpha }{{(1+\beta )}^2}-\left(\frac{3+\beta }{1+\beta }-\frac{1+\beta }{\alpha }\right)-1\hfill \\ \hfill & =\frac{\alpha }{1+\beta }+\frac{2\alpha }{{(1+\beta )}^2}-\frac{4+2\beta }{1+\beta }+\frac{1+\beta }{\alpha }\hfill \\ \hfill & =\frac{1}{\alpha (1+\beta )}\left(\frac{3+\beta }{1+\beta }{\alpha }^2-2(1+\beta )\alpha +{(1+\beta )}^2\right)\hfill \\ \hfill & <0\mathrm{because}\frac{{(1+\beta )}^2}{3+\beta }<\alpha <1+\beta \hfill \end{array}$$

Hence, $P<2.$ Accordingly, $-2<P<2$. By Lemma 1 (i.5), Equation (9) has a pair of conjugate complex roots ${\lambda }_1$ and ${\lambda }_1$ with $\left|{\lambda }_1\right|=\left|{\lambda }_2\right|=1$, implying that $E_2$ is non-hyperbolic. When ${\delta }_2<\delta <{\delta }_1$, $Q<1$. Lemma 1 (i.1) tells us that $\left|{\lambda }_1\right|<1$ and $\left|{\lambda }_2\right|<1$, so $E_2$ is a stable node, i.e., a sink.

When $\delta ={\delta }_1$, then $F(-1)=0$ and $E_2$ is non-hyperbolic.

When $\delta >{\delta }_1$, according to Lemma 1 (i.3), $\left|{\lambda }_1\right|<1$ and $\left|{\lambda }_2\right|>1$. Therefore, $E_2$ is a saddle. Summarizing the above discussions, we obtain the following

Table 1. Type of the fixed point $E_2$.

Conditions		Eigenvalues	Properties
$0<\alpha \le \frac{{(1+\beta )}^2}{3+\beta }$	$0<\delta <{\delta }_1$	$|{\lambda }_1| <1$, $\left|{\lambda }_2\right|<1$	sink (stable node)
	$\delta ={\delta }_1$	${\lambda }_1=-1, {\lambda }_2\ne -1$	non-hyperbolic
	$\delta >{\delta }_1$	$\left|{\lambda }_1\right|<1$, $\left|{\lambda }_2\right|>1$	saddle
$\frac{{(1+\beta )}^2}{3+\beta }<\alpha <1+\beta$	$0<\delta <{\delta }_2$	$\left|{\lambda }_1\right|>1$, $\left|{\lambda }_2\right|>1$	source (unstable node)
	$\delta ={\delta }_2$	$\left|{\lambda }_1\right|=\left|{\lambda }_2\right|=1$	non-hyperbolic
	${\delta }_2<\delta <{\delta }_1$	$\left|{\lambda }_1\right|<1$, $\left|{\lambda }_2\right|<1$	sink (stable node)
	$\delta ={\delta }_1$	${\lambda }_1=-1, {\lambda }_2\ne -1$	non-hyperbolic
	$\delta >{\delta }_1$	$\left|{\lambda }_1\right|<1$, $\left|{\lambda }_2\right|>1$	saddle

.

3. Bifurcation Analysis

In this section, we employ the center manifold theorem and bifurcation theory to research the local bifurcation problems of the system (8) at the fixed point $E_2$.

3.1. Main Results

One can see from Theorem 1 that the fixed point $E_2$ is a non-hyperbolic fixed point when $\delta ={\delta }_1$. As soon as the parameter $\delta$ goes through the critical value ${\delta }_1$, a flip bifurcation probably occurs near the fixed point $E_2$. Namely, the bifurcation probably occurs in the space of parameters

$$\left(\alpha ,\beta \right)\in {\Omega }_1=\left\{\left(\alpha ,\beta \right)\in R_{+}^2\mid 0<\alpha <1+\beta ,\beta >0\right\}$$

In fact, a result is obtained as follows.

Theorem 2.

Suppose the paramenters $(\alpha ,\beta )\in {\Omega }_1$. Let ${\delta }_1=2+\frac{4\alpha }{(1+\alpha +\beta )(1+\beta )}$. Assume $c_{200}^2+c_{300}\ne 0$, where $c_{200}$ and $c_{300}$ will be defined in the sequel. If the parameter δ varies in a small neighborhood of the critical value ${\delta }_1$, then the system (8) experiences a flip bifurcation at the fixed point $E_2$.

When $\delta ={\delta }_2$, there is a pair of conjugate imaginary roots with $\left|{\lambda }_1\right|=\left|{\lambda }_2\right|=1$, which ensures the necessary condition for a Neimark–Sacker bifurcation to occur. The following result may be obtained.

Theorem 3.

Suppose we have the following parameters:

$$\left(\alpha ,\beta \right)\in {\Omega }_2=\left\{\left(\alpha ,\beta \right)\in R_{+}^2\mid \frac{{(1+\beta )}^2}{3+\beta }<\alpha <1+\beta ,\beta >0\right\}$$

Let ${\delta }_2=\frac{3+\beta }{1+\beta }-\frac{1+\beta }{\alpha }$, and let L be defined in (18). Then, the system (8) undergoes a Neimark–Sacker bifurcation at the fixed point $E_2$ when the parament δ varies in a small neighborhood of the critical value ${\delta }_2$. Moreover, if $L<(>)0$, then a (an) stable (unstable) invariant closed orbit is bifurcated out from the fixed point $E_2$ of the system (8).

3.2. Proof of Main Results

Proof of Theorem 2.

First, let $X_n=u_n-u^{*}$ and $Y_n=v_n-u^{*}$, which transforms the fixed point $E_2$ to the origin. Then, the system (8) becomes

$$\left\{\begin{array}{c}{X}_{n+1}=(X_n+u^{*})e^{1-X_n-u^{*}-\frac{\alpha {(Y_n+u^{*})}^2}{\beta {(Y_n+u^{*})}^2+{(X_n+u^{*})}^2}}-u^{*}\hfill \\ Y_{n+1}=(Y_n+u^{*})e^{\delta (1-\frac{Y_n+u^{*}}{X_n+u^{*}})}-u^{*}\hfill \end{array}\right.$$

(10)

Second, giving a small perturbation ${\delta }^{*}$ of the parameter $\delta$ around ${\delta }_1$, i.e., ${\delta }^{*}=\delta -{\delta }_1$ with $0<\left|{\delta }^{*}\right|\ll 1$, and letting ${\delta }_{n+1}^{*}={\delta }_n^{*}={\delta }^{*}$, the system (10) is perturbed into

$$\left\{\begin{array}{c}{X}_{n+1}=(X_n+u^{*})e^{1-X_n-u^{*}-\frac{\alpha {(Y_n+u^{*})}^2}{\beta {(Y_n+u^{*})}^2+{(X_n+u^{*})}^2}}-u^{*}\hfill \\ Y_{n+1}=(Y_n+u^{*})e^{({\delta }_1+{\delta }_n^{*})(1-\frac{Y_n+u^{*}}{X_n+u^{*}})}-u^{*}\hfill \\ {\delta }_{n+1}^{*}={\delta }_n^{*}\hfill \end{array}\right.$$

(11)

Using the Taylor expansion of the system (11) at $(X_n,Y_n,{\delta }_n^{*})=(0,0,0)$, one has

$$\left\{\begin{array}{cc}{X}_{n+1}=\hfill & a_{100}{X}_n+a_{010}{Y}_n+a_{200}{X}_n^2+a_{020}{Y}_n^2+a_{110}{X}_n{Y}_n\hfill \\ & +a_{300}{X}_n^3+a_{030}{Y}_n^3+a_{210}{X}_n^2{Y}_n+a_{120}{X}_n{Y}_n^2+o\left({\rho }_1^3\right)\hfill \\ Y_{n+1}=\hfill & b_{100}{X}_n+b_{010}{Y}_n+b_{001}{\delta }_n^{*}+b_{200}{X}_n^2+b_{020}{Y}_n^2+b_{002}{\left({\delta }_n^{*}\right)}^2\hfill \\ & +b_{110}{X}_n{Y}_n+b_{101}{X}_n{\delta }_n^{*}+b_{011}{Y}_n{\delta }_n^{*}+b_{300}{X}_n^3+b_{030}{Y}_n^3\hfill \\ & +b_{003}{\left({\delta }_n^{*}\right)}^3+b_{210}{X}_n^2{Y}_n+b_{201}{X}_n^2{\delta }_n^{*}+b_{120}{X}_n{Y}_n^2+b_{021}{Y}_n^2{\delta }_n^{*}\hfill \\ & +b_{102}{X}_n{\left({\delta }_n^{*}\right)}^2+b_{012}{Y}_n{\left({\delta }_n^{*}\right)}^2+b_{111}{X}_n{Y}_n{\delta }_n^{*}+o\left({\rho }_1^3\right)\hfill \end{array}\right.$$

(12)

where ${\rho }_1=\sqrt{X_n^2+Y_n^2+{\left({\delta }_n^{*}\right)}^2}.$

$$\begin{array}{cc}\hfill a_{100}& =\frac{\alpha }{1+\beta }+\frac{2\alpha }{{(1+\beta )}^2},a_{010}=-\frac{2\alpha }{{(1+\beta )}^2}\hfill \\ \hfill a_{200}& =\frac{2\alpha }{u^{*}{(1+\beta )}^2}\left(1+\frac{\alpha }{{(1+\beta )}^2}-\frac{u^{*}}{2}\right)+\frac{\alpha (\beta -3)}{u^{*}{(1+\beta )}^3}\hfill \\ \hfill a_{020}& =\frac{2{\alpha }^2}{u^{*}{(1+\beta )}^4}+\frac{\alpha (3\beta -1)}{u^{*}{(1+\beta )}^3}\hfill \\ \hfill a_{110}& =-\frac{2\alpha }{{(1+\beta )}^2}\left(\frac{2\alpha }{u^{*}{(1+\beta )}^2}-\frac{\alpha }{1+\beta }\right)+\frac{4\alpha (1-\beta )}{u^{*}{(1+\beta )}^3}\hfill \\ \hfill a_{300}& =\frac{4u^{*}}{3}{\left(\frac{\alpha }{u^{*}{(1+\beta )}^2}-\frac{1}{2}\right)}^3+2{\left(\frac{\alpha }{u^{*}{(1+\beta )}^2}-\frac{1}{2}\right)}^2-\frac{2\alpha }{{\left(u^{*}\right)}^2{(1+\beta )}^3}\hfill \\ \hfill & +\frac{2\alpha (\beta -3)}{{\left(u^{*}\right)}^2{(1+\beta )}^3}\left(\frac{\alpha }{u^{*}{(1+\beta )}^2}-\frac{1}{2}\right)+\frac{\alpha (\beta -3)}{1+\beta }\left(\frac{1}{{\left(u^{*}\right)}^2{(1+\beta )}^2}-\frac{2{\left(u^{*}\right)}^2}{\beta (1-3\beta )}\right)\hfill \\ \hfill a_{030}& =\frac{\alpha (3\beta -1)}{{\left(u^{*}\right)}^2{(1+\beta )}^3}\left(\frac{\alpha }{{(1+\beta )}^2}+\frac{2}{1+\beta }+\alpha \right)+\frac{2\alpha }{{\left(u^{*}\right)}^2{(1+\beta )}^3}\left(\frac{2\alpha }{3{(1+\beta )}^3}-1\right)\hfill \\ \hfill a_{210}& =\frac{4\alpha }{u^{*}{(1+\beta )}^2}\left(\frac{u^{*}-\beta }{u^{*}(1+\beta )}+\frac{1}{2}\right)+\frac{\alpha \beta }{{(1+\beta )}^3}\left(16-\frac{3\beta }{{\left(u^{*}\right)}^2}\right)\hfill \\ \hfill & -\frac{4\alpha }{3{(1+\beta )}^2}{\left(\frac{\alpha }{u^{*}{(1+\beta )}^2}-\frac{1}{2}\right)}^2+\frac{2\alpha (\beta -3)}{{\left(u^{*}\right)}^2{(1+\beta )}^3}\left(1-\frac{\beta }{1+\beta }-\frac{\alpha }{{(1+\beta )}^2}\right)\hfill \\ \hfill & +\frac{4(1-\beta )}{u^{*}{(1+\beta )}^2}\left(\frac{\alpha }{1+\beta }-2\right)\left(\frac{\alpha }{u^{*}{(1+\beta )}^2}-\frac{1}{2}\right)\hfill \\ \hfill a_{120}& =\left(\frac{10{\alpha }^2}{3u^{*}{(1+\beta )}^4}+\frac{(\alpha +1)(3\beta -1)}{u^{*}{(1+\beta )}^3}\right)\left(\frac{\alpha }{u^{*}{(1+\beta )}^2}-\frac{1}{2}\right)\hfill \\ \hfill & +\frac{16\alpha \beta }{{\left(u^{*}\right)}^2{(1+\beta )}^3}\left(\frac{\beta }{{\left(u^{*}\right)}^2(1+\beta )}-1\right)+\frac{\alpha (3\beta -1)}{{\left(u^{*}\right)}^2{(1+\beta )}^3}\left(\frac{2\beta }{1+\beta }+1\right)\hfill \\ \hfill & +\frac{2\alpha }{{\left(u^{*}\right)}^2{(1+\beta )}^2}\left(\frac{\alpha }{{(1+\beta )}^2}-\frac{\beta }{{\left(u^{*}\right)}^2(1+\beta )}+1\right)\hfill \\ \hfill & +\frac{{\alpha }^2}{{\left(u^{*}\right)}^2{(1+\beta )}^5}\left(\frac{(\beta -3)}{3u^{*}{(1+\beta )}^2}-8(1+\beta )\right)\hfill \\ \hfill b_{001}& =b_{002}=b_{003}=b_{102}=b_{012}=0,b_{100}=m{\delta }_1,b_{010}=1-{\delta }_1,b_{200}=-\frac{{\delta }_1(2-{\delta }_1)}{2u^{*}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill b_{020}& =-\frac{{\delta }_1(2-{\delta }_1)}{2u^{*}},b_{110}=\frac{{\delta }_1(2-{\delta }_1)}{u^{*}},b_{101}=1,b_{011}=-1,b_{030}=\frac{{\delta }_1^2(3-{\delta }_1)}{6{\left(u^{*}\right)}^2}\hfill \\ \hfill b_{300}& =\frac{{\delta }_1}{2{\left(u^{*}\right)}^2}\left(2-{\delta }_1-\frac{{\delta }_1(3-{\delta }_1)}{3}\right),b_{210}=\frac{{\delta }_1}{{\left(u^{*}\right)}^2}\left(-\frac{{\delta }_1^2}{2}+\frac{5{\delta }_1}{2}-2\right),b_{201}=\frac{({\delta }_1-1)}{u^{*}}\hfill \\ \hfill b_{120}& =\frac{{\delta }_2}{{\left(u^{*}\right)}^2}\left(\frac{{\delta }_1^2}{2}-2{\delta }_1+1\right),b_{021}=\frac{{\delta }_1-1}{u^{*}},b_{111}=\frac{2(1-{\delta }_1)}{u^{*}}\hfill \end{array}$$

Therefore, we obtain the Jacobian matrix of the system (11) at the fixed point $E_2$

$$J\left(E_2\right)=\left(\begin{array}{ccc}\frac{\alpha }{1+\beta }+\frac{2\alpha }{{(1+\beta )}^2}& -\frac{2\alpha }{{(1+\beta )}^2}& 0\\ {\delta }_1& 1-{\delta }_1& 0\\ 0& 0& 1\end{array}\right)$$

and its eigenvalues

$${\lambda }_1=-1,{\lambda }_2=a_{100}+2-{\delta }_1,{\lambda }_3=1$$

with corresponding eigenvectors

$${\xi }_1=\left(\begin{array}{c}1\\ M\\ 0\end{array}\right),{\xi }_2=\left(\begin{array}{c}1\\ N\\ 0\end{array}\right),{\xi }_3=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)$$

where $M=1+\frac{(1+\beta )(1+\alpha +\beta )}{2\alpha }$ and $N=\frac{2(1+\beta )}{1+\alpha +\beta }.$

Set $T=\left({\xi }_1,{\xi }_2,{\xi }_3\right)$, i.e.

$$T=\left(\begin{array}{ccc}1& 1& 0\\ M& N& 0\\ 0& 0& 1\end{array}\right)$$

then

$$T^{-1}=\left(\begin{array}{ccc}\frac{N}{N-M}& -\frac{1}{N-M}& 0\\ -\frac{M}{N-M}& \frac{1}{N-M}& 0\\ 0& 0& 1\end{array}\right)$$

Taking the transformation

$${(X_n,Y_n,{\delta }_n^{*})}^T=T{(l_n,m_n,{\omega }_n)}^T$$

the system (11) is changed into the following form:

$$\left(\begin{array}{c}{l}_{n+1}\\ m_{n+1}\\ {\omega }_{n+1}\end{array}\right)=\left(\begin{array}{ccc}-1& 0& 0\\ 0& {\lambda }_2& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}{l}_n\\ m_n\\ {\omega }_n\end{array}\right)+\left(\begin{array}{c}{g}_1(l_n,m_n,{\omega }_n)+o\left({\rho }_2^3\right)\\ g_2(l_n,m_n,{\omega }_n)+o\left({\rho }_2^3\right)\\ 0\end{array}\right)$$

(13)

where ${\rho }_2=\sqrt{l_n^2+m_n^2+{\omega }_n^2}$.

$$\begin{array}{cc}\hfill g_1(l_n,m_n,{\omega }_n)& =c_{200}{l}_n^2+c_{020}{m}_n^2+c_{002}{\omega }_n^2+c_{110}{l}_n{m}_n+c_{101}{l}_n{\omega }_n+c_{011}{m}_n{\omega }_n\hfill \\ & +c_{300}{l}_n^3+c_{030}{m}_n^3+c_{003}{\omega }_n^3+c_{210}{l}_n^2{m}_n+c_{201}{l}_n^2{\omega }_n+c_{120}{l}_n{m}_n^2\hfill \\ & +c_{102}{l}_n{\omega }_n^2+c_{012}{m}_n{\omega }_n^2+c_{021}{m}_n^2{\omega }_n+c_{111}{l}_n{m}_n{\omega }_n\hfill \\ \hfill g_2(l_n,m_n,{\omega }_n)& =d_{200}{l}_n^2+d_{020}{m}_n^2+d_{002}{\omega }_n^2+d_{110}{l}_n{m}_n+d_{101}{l}_n{\omega }_n+d_{011}{m}_n{\omega }_n\hfill \\ & +d_{300}{l}_n^3+d_{030}{m}_n^3+d_{003}{\omega }_n^3+d_{210}{l}_n^2{m}_n+d_{201}{l}_n^2{\omega }_n+d_{120}{l}_n{m}_n^2\hfill \\ & +d_{102}{l}_n{\omega }_n^2+d_{012}{m}_n{\omega }_n^2+d_{021}{m}_n^2{\omega }_n+d_{111}{l}_n{m}_n{\omega }_n\hfill \end{array}$$

$$\begin{array}{cc}\hfill c_{102}& =c_{012}=c_{002}=c_{003}=0\hfill \\ \hfill c_{200}& =\gamma a_{200}+\mu b_{200}+M(\gamma a_{110}+\mu b_{110})+M^2(\gamma a_{020}+\mu b_{020})\hfill \\ \hfill c_{110}& =2(\gamma a_{200}+\mu b_{200})+(M+N)(\gamma a_{110}+\mu b_{110})+2MN(\gamma a_{020}+\mu b_{020})\hfill \\ \hfill c_{020}& =\gamma a_{200}+\mu b_{200}+N(\gamma a_{110}+\mu b_{110})+N^2(\gamma a_{020}+\mu b_{020})\hfill \\ \hfill c_{101}& =\mu M{b}_{011}+\mu b_{101},c_{011}=\mu N{b}_{011}+\mu b_{101}\hfill \\ \hfill c_{300}& =\gamma a_{300}+\mu b_{300}+M^3(\gamma a_{030}+\mu b_{030})+M(\gamma a_{210}+\mu b_{210})\hfill \\ \hfill +& M^2(\gamma a_{120}+\mu b_{120}),c_{030}=\gamma a_{300}+\mu b_{300}+N^3(\gamma a_{030}+\mu b_{030})\hfill \\ \hfill +& N(\gamma a_{210}+\mu b_{210})+N^2(\gamma a_{120}+\mu b_{120})\hfill \\ \hfill c_{210}& =3(\gamma a_{300}+\mu b_{300})+3M^{2}N(\gamma a_{030}+\mu b_{030})+(2M+N)(\gamma a_{210}+\mu b_{210})\hfill \\ \hfill +& (M^2+2MN)(\gamma a_{120}+\mu b_{120}),c_{120}=3(\gamma a_{300}+\mu b_{300})+3M{N}^2(\gamma a_{030}+\mu b_{030})\hfill \\ \hfill +& (M+2N)(\gamma a_{210}+\mu b_{210})+(N^2+2MN)(\gamma a_{120}+\mu b_{120})\hfill \\ \hfill c_{201}& =\mu b_{201}+\mu M^2{b}_{021}+\mu M{b}_{111},c_{021}=\mu b_{201}+\mu N^2{b}_{021}+\mu N{b}_{111}\hfill \\ \hfill c_{111}& =\mu b_{201}+\mu MN{b}_{021}+\frac{\mu (M+N)b_{111}}{2}\hfill \\ \hfill d_{102}& =d_{012}=d_{002}=d_{003}=0\hfill \\ \hfill d_{200}& =ϵa_{200}-\mu b_{200}+M(ϵa_{110}-\mu b_{110})+M^2(ϵa_{020}-\mu b_{020})\hfill \\ \hfill d_{110}& =2(ϵa_{200}-\mu b_{200})+(M+N)(ϵa_{110}-\mu b_{110})+2MN(ϵa_{020}-\mu b_{020})\hfill \\ \hfill d_{020}& =ϵa_{200}-\mu b_{200}+N(ϵa_{110}-\mu b_{110})+N^2(ϵa_{020}+\mu b_{020})\hfill \\ \hfill d_{101}& =-\mu M{b}_{011}-\mu b_{101},c_{011}=-\mu N{b}_{011}-\mu b_{101}\hfill \\ \hfill d_{300}& =ϵa_{300}-\mu b_{300}+M^3(ϵa_{030}-\mu b_{030})+M(ϵa_{210}-\mu b_{210})\hfill \\ \hfill +& M^2(ϵa_{120}-\mu b_{120}),d_{030}=ϵa_{300}-\mu b_{300}+N^3(ϵa_{030}-\mu b_{030})\hfill \\ \hfill +& N(ϵa_{210}-\mu b_{210})+N^2(ϵa_{120}-\mu b_{120})\hfill \\ \hfill d_{210}& =3(ϵa_{300}-\mu b_{300})+3M^{2}N(ϵa_{030}-\mu b_{030})+(2M+N)(ϵa_{210}-\mu b_{210})\hfill \\ \hfill +& (M^2+2MN)(ϵa_{120}-\mu b_{120}),d_{120}=3(ϵa_{300}-\mu b_{300})+3M{N}^2(ϵa_{030}-\mu b_{030})\hfill \\ \hfill +& (M+2N)(ϵa_{210}-\mu b_{210})+(M^2+2MN)(ϵa_{120}-\mu b_{120})\hfill \\ \hfill d_{201}& =-\mu b_{201}-\mu M^2{b}_{021}-\mu M{b}_{111},d_{021}=-\mu b_{201}-\mu N^2{b}_{021}-\mu N{b}_{111}\hfill \\ \hfill d_{111}& =-\mu b_{201}-\mu MN{b}_{021}-\frac{\mu (M+N)b_{111}}{2}\hfill \\ \hfill \gamma =& \frac{N}{N-M},ϵ=-\frac{M}{N-M},\mu =-\frac{1}{N-M}\hfill \end{array}$$

Assume on the center manifold that

$$\begin{array}{c}\hfill m_n=h(l_n,{\omega }_n)=h_{20}{l}_n^2+h_{11}{l}_n{\omega }_n+h_{02}{\omega }_n^2+o\left({\rho }_3^2\right)\end{array}$$

where ${\rho }_3=\sqrt{l_n^2+{\omega }_n^2}$. Then, according to (13), we obtain

$$\begin{array}{cc}\hfill m_{n+1}=& h(l_{n+1},{\omega }_{n+1})={\lambda }_{2}h(l_n,{\omega }_n)+g_2(l_n,h(l_n,{\omega }_n),{\omega }_n)+o\left({\rho }_3^3\right)\hfill \\ \hfill h(l_{n+1},{\omega }_{n+1})& =h_{20}{\left(-l_n+g_1(l_n,h(l_n,{\omega }_n),{\omega }_n)\right)}^2\hfill \\ \hfill & +h_{11}\left(-l_n+g_1(l_n,h(l_n,{\omega }_n),{\omega }_n)\right){\omega }_n+h_{02}{\omega }_n^2+o\left({\rho }_3^3\right)\hfill \end{array}$$

Comparing the corresponding coefficients of terms in the above center manifold equation, it is easy to derive that

$$\begin{array}{c}\hfill h_{20}=\frac{d_{200}}{1-{\lambda }_2},h_{11}=\frac{d_{101}}{1-{\lambda }_2},h_{02}=0\end{array}$$

So, the system (13) restricted to the center manifold is given by

$$\begin{array}{cc}\hfill l_{n+1}& =f(l_n,{\omega }_n)=:-l_n+g_1(l_n,h(l_n,{\omega }_n),{\omega }_n)+o\left({\rho }_3^3\right)\hfill \\ \hfill & =-l_n+c_{20}{l}_n^2+c_{11}{l}_n{\omega }_n+c_{30}{l}_n^3+c_{21}{l}_n^2{\omega }_n+c_{12}{l}_n{\omega }_n^2+o\left({\rho }_3^3\right)\hfill \end{array}$$

Accordingly, we have the following:

$$f^2(l_n,{\omega }_n)=l_n-2c_{11}{l}_n{\omega }_n-2(c_{20}^2+c_{30})l_n^3+(c_{11}^2-2c_{12})l_n{\omega }_n^2-c_{11}{c}_{20}{l}_n^2{\omega }_n+o\left({\rho }_3^3\right)$$

with $c_{20}=c_{200},c_{11}=c_{101},c_{30}=c_{300},c_{21}=c_{100}{h}_{11}+c_{011}{h}_{20}+c_{201},$ and $c_{12}=c_{011}.$

It is not difficult to calculate

$$\begin{array}{cc}\hfill f(l_n,{\omega }_n){|}_{(0,0)}& =0,\frac{𝜕f}{𝜕l_n}{|}_{(0,0)}=-1,\frac{𝜕f^2}{𝜕{\omega }_n}{|}_{(0,0)} =0,\frac{{𝜕}^2{f}^2}{𝜕l_n^2}{|}_{(0,0)}=0\hfill \\ \hfill \frac{{𝜕}^2{f}^2}{𝜕l_n𝜕{\omega }_n}{|}_{(0,0)}& =-2c_{11}=-2c_{101}=-2\mu (1-M)\hfill \\ \hfill & =\frac{2(1+\beta ){(1+\alpha +\beta )}^2}{2\alpha (1+\beta -\alpha )-(1+\beta ){(1+\alpha +\beta )}^2}\hfill \\ \hfill & =\frac{2(1+\beta ){(1+\alpha +\beta )}^2}{-2\alpha ({\beta }^2+\beta +1)-(1+\beta )\left({(1+\beta )}^2+{\alpha }^2\right)}<0(\ne 0)\hfill \\ \hfill \frac{{𝜕}^3{f}^2}{𝜕l_n^3}{|}_{(0,0)}& =-12(c_{20}^2+c_{30})=-12(c_{200}^2+c_{300})\ne 0\hfill \end{array}$$

According to (21.1.43)–(21.1.46) in [26], p. 507, all conditions are valid for a flip bifurcation to occur; hence, the system (8) undergoes a flip bifurcation at the fixed point $E_2$. The proof is complete. □

Next, we provide a proof for Theorem 3.

Proof of Theorem 3.

First, give a small perturbation ${\delta }^{**}$ of the parameter $\delta$ around ${\delta }_2$ in the system (10), i.e., ${\delta }^{**}=\delta -{\delta }_2$ with $0<\left|{\delta }^{**}\right|\ll 1$. Under the perturbation, the system (10) is

$$\left\{\begin{array}{c}{X}_{n+1}=(X_n+u^{*})e^{1-X_n-u^{*}-\frac{\alpha {(Y_n+u^{*})}^2}{\beta {(Y_n+u^{*})}^2+{(X_n+u^{*})}^2}}-u^{*}\hfill \\ Y_{n+1}=(Y_n+u^{*})e^{({\delta }_2+{\delta }^{**})(1-\frac{Y_n+u^{*}}{X_n+u^{*}})}-u^{*}\hfill \end{array}\right.$$

(14)

The characteristic equation of the linearized equation of the system (14) at the origin (0,0) is

$$F\left(\lambda \right)={\lambda }^2-p\left({\delta }^{**}\right)\lambda +q\left({\delta }^{**}\right)=0$$

(15)

where

$$\begin{array}{cc}\hfill p\left({\delta }^{**}\right)& =\frac{\alpha (3+\beta )}{1+\beta }+1-{\delta }_2\hfill \\ \hfill q\left({\delta }^{**}\right)& =\frac{\alpha (3+\beta )}{1+\beta }(1-{\delta }_2-{\delta }^{**})+\frac{\alpha ({\delta }_2-{\delta }^{**})}{{(1+\beta )}^2}\hfill \end{array}$$

Notice that $p^2\left(0\right)-4q\left(0\right)={\left(2+\frac{\alpha {\delta }_2}{1+\beta }-{\delta }_2\right)}^2-4<0$; so, for $0<\left|{\delta }^{**}\right|\ll 1$, the two roots of $F\left(\lambda \right)=0$ in (15) are

$${\lambda }_{1,2}\left({\delta }^{**}\right)=\frac{p\left({\delta }^{**}\right)\pm i\sqrt{4q\left({\delta }^{**}\right)-p^2\left({\delta }^{**}\right)}}{2}$$

The occurrence of a Neimark–Sacker bifurcation requires the following two conditions to be satisfied [26]:

• $\left(\frac{d|{\lambda }_{1,2}\left({\delta }^{**}\right)|}{d{\delta }^{**}}\right){|}_{{\delta }^{**}=0}\ne 0;$
• ${\lambda }_{1,2}^i\left(0\right)\ne 1,i=1,2,3,4.$

It is easy to observe that $\left|{\lambda }_{1,2}\right({\delta }^{**}\left)\right|=\sqrt{q\left({\delta }^{**}\right)}$ and ${\left.\left(\right|{\lambda }_{1,2}\left({\delta }^{**}\right)\left|\right)\right|}_{{\delta }^{**}=0}=\sqrt{q\left(0\right)}=1$.

Therefore

$$\left(\frac{d|{\lambda }_{1,2}\left({\delta }^{**}\right)|}{d{\delta }^{**}}\right){|}_{{\delta }^{**}=0}=-\frac{\alpha {(2+\beta )}^2}{{(1+\beta )}^2}<0(\ne 0)$$

Obviously, ${\lambda }_{1,2}^i\left(0\right)\ne 1$ for $i=1,2,3,4,$; so, the two conditions are satisfied.

Second, in order to derive the normal form of the system (14), one expands (14) in a power series up to the third-order term around the origin to obtain

$$\left\{\begin{array}{cc}{X}_{n+1}=\hfill & s_{10}{X}_n+s_{01}{Y}_n+s_{20}{X}_n^2+s_{11}{X}_n{Y}_n+s_{02}{Y}_n^2\hfill \\ & +s_{30}{X}_n^3+s_{21}{X}_n^2{Y}_n+s_{12}{X}_n{Y}_n^2+s_{03}{Y}_n^3+o\left({\rho }_7^3\right)\hfill \\ Y_{n+1}=\hfill & t_{10}{X}_n+t_{01}{Y}_n+t_{20}{X}_n^2+t_{11}{X}_n{Y}_n+t_{02}{Y}_n^2\hfill \\ & +t_{30}{X}_n^3+t_{21}{X}_n^2{Y}_n+t_{12}{X}_n{Y}_n^2+t_{03}{Y}_n^3+o\left({\rho }_7^3\right)\hfill \end{array}\right.$$

(16)

where ${\rho }_7=\sqrt{X_n^2+Y_n^2}.$

$$\begin{array}{cc}\hfill s_{10}& =a_{100},s_{01}=a_{010},s_{20}=a_{200},s_{11}=a_{110}\hfill \\ \hfill s_{02}& =a_{020},s_{30}=a_{300},s_{21}=a_{210},s_{12}=a_{120},s_{03}=a_{030}\hfill \\ \hfill t_{10}& ={\delta }_2,t_{01}=1-{\delta }_2,t_{20}=\frac{{\delta }_2({\delta }_2-2)}{2u^{*}},t_{02}=\frac{{\delta }_2({\delta }_2-2)}{2u^{*}}\hfill \\ \hfill t_{11}& =-\frac{{\delta }_2({\delta }_2-2)}{u^{*}},t_{30}=\frac{m{\delta }_2({\delta }_2^2-6{\delta }_2+6)}{6{\left(u^{*}\right)}^2}\hfill \\ \hfill t_{03}& =-\frac{{\delta }_2^2({\delta }_2-3)}{6{\left(u^{*}\right)}^2},t_{12}=\frac{{\delta }_2({\delta }_2^2-2{\delta }_2+2)}{2{\left(u^{*}\right)}^2}\hfill \\ \hfill b_{21}& =-\frac{{\delta }_2({\delta }_2^2-5{\delta }_2+4)}{2{\left(u^{*}\right)}^2}\hfill \end{array}$$

Take matrix

$$T=\left(\begin{array}{cc}0& s_{01}\\ \eta & 1-\zeta \end{array}\right),\mathrm{then}{T}^{-1}=\left(\begin{array}{cc}\frac{\zeta -1}{{\eta }_{01}}& \frac{1}{\eta }\\ \frac{1}{s_{01}}& 0\end{array}\right)$$

Make a change in variables

$${(X,Y)}^T=T{(M,N)}^T$$

Then, the system (16) is changed to the following form:

$$\left(\begin{array}{c}M\\ N\end{array}\right)\to \left(\begin{array}{cc}\zeta & -\eta \\ \eta & \zeta \end{array}\right)\left(\begin{array}{c}M\\ N\end{array}\right)+\left(\begin{array}{c}{g}_3(M,N)+o\left({\rho }_8^4\right)\\ g_4(M,N)+o\left({\rho }_8^4\right)\end{array}\right)$$

(17)

where ${\rho }_8=\sqrt{M^2+N^2}.$

$$\begin{array}{cc}\hfill & g_3(M,N)=j_{20}{X}^2+j_{11}XY+j_{02}{Y}^2+j_{30}{X}^3+j_{21}{X}^{2}v+j_{12}X{Y}^2+j_{03}{Y}^3\hfill \\ \hfill & g_4(M,N)=k_{20}{X}^2+k_{11}XY+k_{02}{Y}^2+k_{30}{X}^3+k_{21}{X}^{2}Y+k_{12}X{Y}^2+k_{03}{Y}^3\hfill \\ \hfill & X=s_{01}N,Y=\eta M+(1-\zeta )N\hfill \end{array}$$

$$\begin{array}{cc}\hfill & j_{20}=\frac{s_{20}(\zeta -1)}{\eta s_{01}}+\frac{t_{20}}{\eta },j_{02}=\frac{s_{02}(\zeta -1)}{\eta a_{01}}+\frac{t_{02}}{\eta },j_{11}=\frac{s_{11}(\zeta -1)}{\eta s_{01}}+\frac{t_{11}}{\eta }\hfill \\ \hfill & j_{30}=\frac{s_{30}(\zeta -1)}{\eta s_{01}}+\frac{t_{30}}{\eta },j_{03}=\frac{s_{03}(\zeta -1)}{\eta s_{01}}+\frac{t_{03}}{\eta },j_{12}=\frac{s_{12}(\zeta -1)}{\eta a_{01}}+\frac{t_{12}}{\eta }\hfill \\ \hfill & j_{21}=\frac{s_{21}(\zeta -1)}{\eta s_{01}}+\frac{t_{21}}{\eta },k_{20}=\frac{s_{20}}{s_{01}},k_{02}=\frac{s_{02}}{s_{01}},k_{11}=\frac{s_{11}}{s_{01}},k_{30}=\frac{s_{30}}{s_{01}}\hfill \\ \hfill & k_{03}=\frac{s_{03}}{s_{01}},k_{12}=\frac{s_{12}}{s_{01}},k_{21}=\frac{s_{21}}{s_{01}}\hfill \end{array}$$

Furthermore

$$\begin{array}{cc}\hfill & F_{XX}{|}_{(0,0)}=2j_{02}{\eta }^3,F_{XY}{|}_{(0,0)}=j_{11}{s}_{01}\eta +2j_{02}\eta (1-\zeta )\hfill \\ \hfill & F_{YY}{|}_{(0,0)}=2j_{02}{s}_{01}^2+2j_{11}{s}_{01}(1-\zeta ),F_{XXX}{|}_{(0,0)}=6j_{03}{\eta }^3\hfill \\ \hfill & F_{XXY}{|}_{(0,0)}=2j_{21}{s}_{01}{\eta }^2+6j_{03}{\eta }^2(1-\zeta )\hfill \\ \hfill & F_{XYY}{|}_{(0,0)}=2j_{21}{s}_{01}^2\eta +4j_{12}{s}_{01}\eta (1-\zeta )+6j_{03}\eta {(1-\zeta )}^2\hfill \\ \hfill & F_{YYY}{|}_{(0,0)}=4{(1-\zeta )}^3+6j_{30}{s}_{01}^3+4j_{21}{s}_{01}^2(1-\zeta )+6j_{12}{s}_{01}{(1-\zeta )}^2\hfill \\ \hfill & G_{XX}{|}_{(0,0)}=2k_{02}{\eta }^3,G_{XY}{|}_{(0,0)}=k_{11}{s}_{01}\eta +2k_{02}\eta (1-\zeta )\hfill \\ \hfill & G_{YY}{|}_{(0,0)}=2k_{02}{s}_{01}^2+2k_{11}{s}_{01}(1-\zeta ),G_{XXX}{|}_{(0,0)}=6j_{03}{\eta }^3\hfill \\ \hfill & G_{XXY}{|}_{(0,0)}=2k_{21}{s}_{01}{\eta }^2+6k_{03}{\eta }^2(1-\zeta )\hfill \\ \hfill & G_{XYY}{|}_{(0,0)}=2k_{21}{s}_{01}^2\eta +4k_{12}{s}_{01}\eta (1-\zeta )+6k_{03}\eta {(1-\zeta )}^2\hfill \\ \hfill & G_{YYY}{|}_{(0,0)}=4{(1-\zeta )}^3+6k_{30}{s}_{01}^3+4k_{21}{s}_{01}^2(1-\zeta )+6k_{12}{s}_{01}{(1-\zeta )}^2\hfill \end{array}$$

To determine the stability and direction of the bifurcation curve (closed orbit) for the system (8), the discriminating quantity L should be calculated and not be zero, where

$$L=-Re\left(\frac{(1-2{\lambda }_1){\lambda }_2^2}{1-{\lambda }_1}{\theta }_{20}{\theta }_{11}\right)-\frac{1}{2}{\left|{\theta }_{11}\right|}^2-{\left|{\theta }_{02}\right|}^2+Re\left({\lambda }_2{\theta }_{21}\right)$$

(18)

$$\begin{array}{cc}\hfill {\theta }_{20}& =\frac{1}{8}[F_{XX}-F_{YY}+2G_{XY}+i(G_{XX}-G_{YY}-2F_{XY})]{|}_{(0,0)}\hfill \\ \hfill {\theta }_{11}& =\frac{1}{4}[F_{XX}+F_{YY}+i(G_{XX}+G_{YY})]{|}_{(0,0)}\hfill \\ \hfill {\theta }_{02}& =\frac{1}{8}[F_{XX}-F_{YY}-2G_{XY}+i(G_{XX}-G_{YY}+2F_{XY})]{|}_{(0,0)}\hfill \\ \hfill {\theta }_{21}& =\frac{1}{16}[F_{XXX}+F_{XYY}+G_{XXY}+G_{YYY}\hfill \\ \hfill & +i(G_{XXX}+G_{XYY}-F_{XXY}-F_{YYY}){\left]\right|}_{(0,0)}\hfill \end{array}$$

Based on [26,27,28], we see that if $L<(>)0,$ then an attracting (a repelling) invariant closed curve bifurcates from the fixed point.

The proof is then complete. □

4. Numerical Simulation

In this section, by using the software Matlab, we obtain the bifurcation diagrams and phase portraits of the system (8) at the fixed point $E_2$, which illustrate our theoretical results previously derived and reveal some new dynamical behaviors.

First, vary $\delta$ in the range $(2.7,3)$ and $(0.35,0.6)$, respectively, and fix $\alpha =0.8,\beta =0.5$ with the initial value $(x_0,y_0)=(0.4667,0.4667)$. Figure 1a shows the existence of a flip bifurcation at the fixed point $E_2=(0.4667,0.4667)$ when $\delta ={\delta }_1=2+\frac{4\alpha }{(1+\alpha +\beta )(1+\beta )}\approx 2.93$ and indicates the periodic orbits and chaos in the system (8) as $\delta$ increases. Meanwhile, we can calculate that $c_{200}^2+c_{300}\approx -10.92<0$ and $-\frac{{𝜕}^2{f}^2}{𝜕l_n𝜕{\omega }_n}/\frac{{𝜕}^3{f}^2}{𝜕l_n^3}{|}_{(0,0)}>0$, which means that the direction of the flip bifurcation is on the right side of the critical value. Furthermore, according to Case 2, we can clearly see that the nature of the system (8) changes from unstable to stable near ${\delta }_2$. This change is shown in Figure 1b, and the periodic orbit is simulated in Figure 2b. This agrees with the conclusion in Theorem 2.

Then, we choose different values of the parameter $\delta$. The corresponding phase portraits are plotted in Figure 3 and Figure 4, respectively. Figure 3 implies that the closed curve is stable inside, while Figure 4 indicates that the closed curve is stable outside. That is to say, there occurs a stable invariant closed curve around the fixed point $E_2$. This agrees with the conclusion in Theorem 3.

Finally, take initial values $(x_0,y_0)=(0.43,0.43)$ in Figure 2a and $(0.4667,$ $0.4667)$ in Figure 2b. One finds a new dynamical phenomenon—the existence of a limit cycle. This means that the system produces periodic oscillations here.

5. Conclusions and Discussion

In this paper, we analyze a predator–prey model of Leslie type with ratio-dependent simplified Holling type IV functional response. By using the semi-discretization method, the system (4) is transformed to the discrete system (8). At first, one considers the existence and stability of the positive fixed point $E_2=(\frac{1+\beta -\alpha }{1+\beta },\frac{1+\beta -\alpha }{1+\beta })$. Subsequently, one studies the existence conditions of the flip bifurcation and Neimark–Sacker bifurcation of the system (8) at the fixed point $E_2$ by using the center manifold theorem and bifurcation theory. In the end, we confirm the correctness of the theoretical results previously derived through numerical simulations. In the process of simulation, the existence of a limit cycle is also found.

As for the biological significance, our results indicate that a limit cycle will occur when the parameter $\delta$ is small. This means that the interaction between prey and predator leads to periodic oscillations, indicating the rich dynamic properties of the system. When appropriately adding the value of the parameter $\delta$, the prey and predator populations will coexist and the limit cycle will be eliminated. Our studies provide a theoretical basis for the stable coexistence of predator and prey.

However, there still are some questions worth investigating. For example, we only know of the existence and bifurcation of the system (3) at the fixed point $E_2$ when $0<\alpha <1+\beta$. How about the case when $\alpha \ge 1+\beta$? Are there more interesting dynamical properties if we discuss the impact of seasonality on the system’s behavior? How about using discrete methods other than the semi-discretization method that we use in this paper? We hope that interested readers consider these questions.