# Complex Dynamic Behaviors of a Modified Discrete Leslie–Gower Predator–Prey System with Fear Effect on Prey Species

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Existence of Equilibria

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

- (1)
- ${E}_{0}(0,0)$, ${E}_{1}(0,hm)$, if ${r}_{0}\le d$;
- (2)
- ${E}_{0}(0,0)$, ${E}_{1}(0,hm)$, and ${E}_{2}\left(\frac{{r}_{0}-d}{a},0\right)$, if $d<{r}_{0}\le (d+bhm)(1+hkm)$;
- (3)
- ${E}_{0}(0,0)$, ${E}_{1}(0,hm)$, ${E}_{2}\left(\frac{{r}_{0}-d}{a},0\right)$, and ${E}^{*}({x}^{*},{y}^{*})$, if ${r}_{0}>(d+bhm)(1+hkm)$.

**Remark**

**2.**

## 3. The Local Stability of Equilibria

#### 3.1. The Local Stability of Boundary Equilibria ${E}_{0},\phantom{\rule{4pt}{0ex}}{E}_{1},\phantom{\rule{4pt}{0ex}}{E}_{2}$

**Theorem**

**3.**

- (1)
- A source if ${r}_{0}>d$;
- (2)
- A saddle if ${r}_{0}<d$;
- (3)
- Non-hyperbolic if ${r}_{0}=d$.

**Proof.**

**Theorem**

**4.**

- (1)
- A sink if ${r}_{0}<(d+bhm)(1+hkm)$ and $h<2$;
- (2)
- A source if ${r}_{0}>(d+bhm)(1+hkm)$ and $h>2$;
- (3)
- A saddle if one of the following conditions holds:
- (i)
- ${r}_{0}<(d+bhm)(1+hkm)$ and $h>2$;
- (ii)
- ${r}_{0}>(d+bhm)(1+hkm)$ and $h<2$;

- (4)
- Non-hyperbolic if ${r}_{0}=(d+bhm)(1+hkm)$ or $h=2$.

**Proof.**

**Theorem**

**5.**

- (1)
- A source if ${r}_{0}-d>2$;
- (2)
- A saddle if ${r}_{0}-d<2$;
- (3)
- Non-hyperbolic if ${r}_{0}-d=2$.

**Proof.**

#### 3.2. The Local Stability of the Positive Equilibrium ${E}^{*}$

**Theorem**

**6.**

- (1)
- A sink if $P<Q\u2a7d2$ or $0<2Q-4<P<Q$;
- (2)
- A source if one of the following conditions holds:
- (i)
- $Q\u2a7d2$ and $P>Q$;
- (ii)
- $Q>2$ and $P>max(Q,2Q-4)$;

- (3)
- A saddle if $Q>2$ and $P<2Q-4$;
- (4)
- Non-hyperbolic if $P=2Q-4$, where$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& P=h\left({\displaystyle a+\frac{h{r}_{0}k}{{(1+k{y}^{*})}^{2}}+bh}\right){x}^{*},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& Q=a{x}^{*}+h.\hfill \end{array}$$

**Proof.**

**Remark**

**3.**

## 4. The Global Attractivity of Equilibria

#### 4.1. The Global Attractivity of the Positive Equilibrium ${E}^{*}$

**Theorem**

**7.**

**Proof.**

**(I)**- Through the first iteration, we can prove that there exist ${U}_{1}^{x},\phantom{\rule{3.33333pt}{0ex}}{U}_{1}^{y}>0$, such that ${S}_{1}\le {U}_{1}^{x}$ and ${S}_{2}\le {U}_{1}^{y}$.
**(i)**- Given by the first equation of system (11), we get$$\begin{array}{cc}\hfill x(n+1)& =x\left(n\right)exp\left({\displaystyle \frac{{r}_{0}}{1+ky\left(n\right)}-d-ax\left(n\right)-by\left(n\right)}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le x\left(n\right)exp\left({r}_{0}-d-ax\left(n\right)\right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}n=0,\phantom{\rule{3.33333pt}{0ex}}1,\phantom{\rule{3.33333pt}{0ex}}2,\phantom{\rule{3.33333pt}{0ex}}\cdots \hfill \end{array}$$Consider the auxiliary equation$$u(n+1)=u\left(n\right)exp\left({r}_{0}-d-au\left(n\right)\right);$$$${S}_{1}=\underset{n\to \infty}{lim\; sup}x\left(n\right)\le \underset{n\to \infty}{lim}u\left(n\right)={\displaystyle \frac{{r}_{0}-d}{a}}.$$Hence, for sufficiently small $\epsilon >0$ there exists an integer ${N}_{1}>2$ such that if $n\ge {N}_{1}$, then$$x\left(n\right)\le {\displaystyle \frac{{r}_{0}-d}{a}}+\epsilon :={U}_{1}^{x}.$$
**(ii)**- Given by the second equation of system (11), we get$$\begin{array}{cc}\hfill y(n+1)& =y\left(n\right)exp\left({\displaystyle h-\frac{y\left(n\right)}{m+x\left(n\right)}}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le y\left(n\right)exp\left({\displaystyle h-\frac{y\left(n\right)}{m+{U}_{1}^{x}}}\right),\phantom{\rule{3.33333pt}{0ex}}\forall n\ge {N}_{1}.\hfill \end{array}$$Consider the auxiliary equation$$u(n+1)=u\left(n\right)exp\left({\displaystyle h-\frac{y\left(n\right)}{m+{U}_{1}^{x}}}\right).$$Since $h\le 1$, we obtain that $u\left(n\right)\le m+{U}_{1}^{x}$ for all $n\ge {N}_{1}$ from Lemma 3 in [35]. According to Lemma 4 in [35], we have that $f\left(u\right)=uexp\left({\displaystyle h-\frac{y\left(n\right)}{m+{U}_{1}^{x}}}\right)$ is nondecreasing for $u\in (0,m+{U}_{1}^{x}]$. Hence, according to Lemma 5 in [35], we have that $y\left(n\right)\le u\left(n\right)$ for all $n\ge {N}_{1}$. Therefore, as with (I) (i), we get$${S}_{2}=\underset{n\to \infty}{lim\; sup}y\left(n\right)\le \underset{n\to \infty}{lim}u\left(n\right)=h(m+{U}_{1}^{x}).$$Hence, for sufficiently small $\epsilon >0$ there exists an integer ${N}_{2}>{N}_{1}$ such that if $n\ge {N}_{2}$, then$$y\left(n\right)\le h(m+{U}_{1}^{x})+\epsilon :={U}_{1}^{y}.$$

**(II)**- Through the second iteration, we can prove that there exist ${V}_{1}^{x},\phantom{\rule{3.33333pt}{0ex}}{V}_{1}^{y}>0$, such that ${I}_{1}\ge {V}_{1}^{x}$ and ${I}_{2}\ge {V}_{1}^{y}$.
**(i)**- Given by the first equation of system (11), we obtain that$$x(n+1)\ge x\left(n\right)exp\left({\displaystyle \frac{{r}_{0}}{1+k{U}_{1}^{y}}-d-b{U}_{1}^{y}-ax\left(n\right)}\right),\phantom{\rule{3.33333pt}{0ex}}\forall n\ge {N}_{2}.$$Consider the auxiliary equation$$u(n+1)=u\left(n\right)exp\left({\displaystyle \frac{{r}_{0}}{1+k{U}_{1}^{y}}-d-b{U}_{1}^{y}-au\left(n\right)}\right).$$Since $(1+k{U}_{1}^{y})(d+b{U}_{1}^{y})<{r}_{0}\le d+1$, we obtain$$0<\frac{{r}_{0}}{1+k{U}_{1}^{y}}-d-b{U}_{1}^{y}\le {r}_{0}-d\le 1.$$Hence $u\left(n\right)\le 1/a$ for all $n\ge {N}_{2}$ according to Lemma 3 in [35]. From Lemma 4 in [35], we know that $f\left(u\right)=uexp\left({\displaystyle \frac{{r}_{0}}{1+k{U}_{1}^{y}}-d-b{U}_{1}^{y}-au}\right)$ is nondecreasing for $u\in (0,1/a]$. Hence, according to Lemma 5 in [35], we have that $x\left(n\right)\ge u\left(n\right)$ for all $n\ge {N}_{2}$. According to Lemma 3 in [35], we have$${I}_{1}=\underset{n\to \infty}{lim\; inf}x\left(n\right)\ge \underset{n\to \infty}{lim}u\left(n\right)={\displaystyle \frac{1}{a}}\left({\displaystyle \frac{{r}_{0}}{1+k{U}_{1}^{y}}-d-b{U}_{1}^{y}}\right).$$Hence, for sufficiently small $\epsilon >0$, there exists an integer ${N}_{3}>{N}_{2}$ such that if $n\ge {N}_{3}$, then$$x\left(n\right)\ge {\displaystyle \frac{1}{a}}\left({\displaystyle \frac{{r}_{0}}{1+k{U}_{1}^{y}}-d-b{U}_{1}^{y}}\right)-\epsilon :={V}_{1}^{x}.$$
**(ii)**- Given by the second equation of system (11), we obtain$$y(n+1)\ge y\left(n\right)exp\left({\displaystyle h-\frac{y\left(n\right)}{m+{V}_{1}^{x}}}\right),\phantom{\rule{3.33333pt}{0ex}}\forall n\ge {N}_{3}.$$Consider the auxiliary equation$$u(n+1)=u\left(n\right)exp\left(h-\frac{u\left(n\right)}{m+{V}_{1}^{x}}\right).$$Since $h\le 1$, with a similar argument as above, we can obtain$${I}_{2}=\underset{n\to \infty}{lim\; inf}y\left(n\right)\ge \underset{n\to \infty}{lim}u\left(n\right)=h(m+{V}_{1}^{x}).$$Hence, for sufficiently small $\epsilon >0$ there exists an integer ${N}_{4}>{N}_{3}$ such that if $n\ge {N}_{4}$, then$$y\left(n\right)\ge h(m+{V}_{1}^{x})-\epsilon :={V}_{1}^{y}.$$

**(III)**- Through the third iteration, we can prove that there exist ${U}_{2}^{x}\le {U}_{1}^{x},\phantom{\rule{3.33333pt}{0ex}}{U}_{2}^{y}\le {U}_{1}^{y}$, such that ${S}_{1}\le {U}_{2}^{x}$ and ${S}_{2}\le {U}_{2}^{y}$.
**(i)**- Given by the first equation of system (11), we obtain that$$x(n+1)\le x\left(n\right)exp\left({\displaystyle \frac{{r}_{0}}{1+k{V}_{1}^{y}}-d-b{V}_{1}^{y}-ax\left(n\right)}\right),\phantom{\rule{3.33333pt}{0ex}}\forall n\ge {N}_{4}.$$Since ${U}_{1}^{y}>{V}_{1}^{y}$, we obtain $0<\frac{{r}_{0}}{1+k{U}_{1}^{y}}-d-b{U}_{1}^{y}<\frac{{r}_{0}}{1+k{V}_{1}^{y}}-d-b{V}_{1}^{y}\le {r}_{0}-d\le 1$, according to Lemma 5 in [35], we have that$${S}_{1}=\underset{n\to \infty}{lim\; sup}x\left(n\right)\le {\displaystyle \frac{1}{a}}\left(\frac{{r}_{0}}{1+k{V}_{1}^{y}}-d-b{V}_{1}^{y}\right).$$Hence, for sufficiently small $\epsilon >0$, there exists an integer ${N}_{5}>{N}_{4}$ such that if $n\ge {N}_{5}$, then$$x\left(n\right)\le {\displaystyle \frac{1}{a}}\left({\displaystyle \frac{{r}_{0}}{1+k{V}_{1}^{y}}-d-b{V}_{1}^{y}}\right)+{\displaystyle \frac{\epsilon}{2}}:={U}_{2}^{x}\le {U}_{1}^{x}.$$
**(ii)**- Given by the second equation of system (11), we obtain$$y(n+1)\le y\left(n\right)exp\left(h-\frac{y\left(n\right)}{m+{U}_{2}^{x}}\right),\phantom{\rule{3.33333pt}{0ex}}\forall n\ge {N}_{5}.$$Since $h\le 1$, with a similar argument as above, we can obtain$${S}_{2}=\underset{n\to \infty}{lim\; sup}y\left(n\right)\le h(m+{U}_{2}^{x}).$$Hence, for sufficiently small $\epsilon >0$, there exists an integer ${N}_{6}>{N}_{5}$ such that if $n\ge {N}_{6}$, then$$y\left(n\right)\le h(m+{U}_{2}^{x})+\frac{\epsilon}{2}:={U}_{2}^{y}\le {U}_{1}^{y}.$$

**(IV)**- Through the fourth iteration, we can prove that there exist ${V}_{2}^{x}\ge {V}_{1}^{x},\phantom{\rule{3.33333pt}{0ex}}{V}_{2}^{y}\ge {V}_{1}^{y}$, such that ${I}_{1}\ge {V}_{2}^{x}$ and ${I}_{2}\ge {V}_{2}^{y}$.
**(i)**- Similarly, from the first equation of system (11), we obtain$$x(n+1)\ge x\left(n\right)exp\left({\displaystyle \frac{{r}_{0}}{1+k{U}_{2}^{y}}-d-b{U}_{2}^{y}-ax\left(n\right)}\right),\phantom{\rule{3.33333pt}{0ex}}\forall n\ge {N}_{6}.$$Since ${U}_{1}^{y}\ge {U}_{2}^{y}$, we can obtain$$0<\frac{{r}_{0}}{1+k{U}_{1}^{y}}-d-b{U}_{1}^{y}\le \frac{{r}_{0}}{1+k{U}_{2}^{y}}-d-b{U}_{2}^{y}\le {r}_{0}-d\le 1.$$According to Lemma 5 in [35], we have$${I}_{1}=\underset{n\to \infty}{lim\; inf}x\left(n\right)\ge {\displaystyle \frac{1}{a}}\left({\displaystyle \frac{{r}_{0}}{1+k{U}_{2}^{y}}-d-b{U}_{2}^{y}}\right).$$Hence, for sufficiently small $\epsilon >0$ there exists an integer ${N}_{7}>{N}_{6}$ such that if $n\ge {N}_{7}$, then$$x\left(n\right)\ge {\displaystyle \frac{1}{a}}\left({\displaystyle \frac{{r}_{0}}{1+k{U}_{2}^{y}}-d-b{U}_{2}^{y}}\right)-{\displaystyle \frac{\epsilon}{2}}:={V}_{2}^{x}\ge {V}_{1}^{x}.$$
**(ii)**- Similarly, from the second equation of system (11), we obtain$$y(n+1)\ge y\left(n\right)exp\left(h-\frac{y\left(n\right)}{m+{V}_{2}^{x}}\right),\phantom{\rule{3.33333pt}{0ex}}\forall n\ge {N}_{7}.$$Since $h\le 1$, with a similar argument as above, we can obtain$${I}_{2}=\underset{n\to \infty}{lim\; inf}y\left(n\right)\ge h(m+{V}_{2}^{x}).$$Hence, for sufficiently small $\epsilon >0$ there exists an integer ${N}_{8}>{N}_{7}$ such that if $n\ge {N}_{8}$, then$$y\left(n\right)\ge h(m+{V}_{2}^{x})-\frac{\epsilon}{2}:={V}_{2}^{y}\ge {V}_{1}^{y}.$$

#### 4.2. The Global Attractivity of the Boundary Equilibrium ${E}_{1}$

**Proof.**

**Theorem**

**9.**

**Proof.**

## 5. Bifurcation Analysis

#### 5.1. Flip Bifurcation

**Theorem**

**10.**

**Theorem**

**11.**

**Proof.**

#### 5.2. Transcritical Bifurcation

**Theorem**

**12.**

**Proof.**

## 6. Numerical Simulations

**Example**

**1.**

- (1)
- When$$(k,d,a,b,h,m,{r}_{0})=(0.7,0.3,3.0,0.5,0.5,1.0,4.0),$$we take the initial values of system (11) as $(0.6,0.3)$, we have $(1+hkm)(d+bhm)=0.74<{r}_{0}$, then the system (11) admits a unique positive equilibrium ${E}^{*}(0.62,0.81)$ according to Theorem 2. At this point, we have $P=1.18,Q=2.35$, then ${E}^{*}$ is locally asymptotically stable according to Theorem 6, as shown in Figure 2. In fact, if we take the initial values of system (11) as $(0.6,0.3),(0.9,0.2),(0.7,0.4),(0.4,0.5)(0.2,0.2),(0.8,0.8)$, we can see that ${E}^{*}$ is also globally stable in Figure 3, but at this point we have ${r}_{0}=4>1.3=d+1$. It shows that it is possible for ${E}^{*}$ to be globally attractive even if the conditions of Theorem 7 are not satisfied. This means that sufficient conditions to ensure the globally asymptotically stable equilibrium ${E}^{*}$ is too strict, which is probably due to the contraction and expansion of the proof.
- (2)
- When$$(k,d,a,b,h,m,{r}_{0})=(0.0,0.3,3.0,0.5,0.5,1.0,4.0)$$

**Example**

**2.**

- (1)
- When$$(k,d,a,b,h,m,{r}_{0})=(0.7,0.3,3.0,0.5,0.5,1.0,0.4)$$and $\left(x\right(0),y(0\left)\right)=(0.5,0.3)$, we have $(1+hkm)(d+bhm)=0.74>{r}_{0}$ and $h<2$, then ${E}_{1}$ is locally asymptotically stable according to Theorem 4, as shown in Figure 5. Further, we can see that $d+bhm=0.55>{r}_{0}>d$ and $h<ln2+1$, which means that ${E}_{1}$ is globally attractive according to Theorem 9. We take the initial values of the system (11) as$$(0.6,0.3),(0.9,0.2),(0.7,0.4),(0.4,0.5),(0.2,0.2),(0.8,0.8),$$respectively, for numerical simulation, and the results verify the accuracy of the conclusion of Theorem 9, as shown in Figure 6.
- (2)
- When$$(k,d,a,b,h,m,{r}_{0})=(0.7,0.3,3.0,0.5,1.9,1.0,2.8)$$we have$$(1+hkm)(d+bhm)=2.91>{r}_{0}=2.8>1.25=d+bhm$$and $ln2+1<h=1.9<2$, which satisfy the conditions of Theorem 4, but do not meet the requirements of Theorem 9. However, we take the initial values of the system as$$(0.5,1.5),(0.3,2.7),(1.1,1.2),(0.7,2.4)(0.2,0.2),(0.8,1.8),$$we can see that ${E}_{1}(0,1.9)$ is also globally attractive at this time from Figure 7.

**Example**

**3.**

- (1)
- First, we consider the case where there is no fear effect. Set each parameter value of system (11) as $(k,d,a,b,h,m)=(0.0,0.3,3.0,0.5,0.5,1.0)$ and set $\left(x\right(0),y(0\left)\right)=(0.5,0.3)$. Through numerical simulation, we find that when ${r}_{0}<2.67$, system (11) has a stable equilibrium point. When ${r}_{0}$ continues to increase, this equilibrium point first bifurcates into a two-period orbit ($2.67<{r}_{0}<3.25$), then bifurcates into a four-period orbit ($3.25<{r}_{0}<3.33$), and finally chaos occurs (${r}_{0}>3.33$), as shown in Figure 8. We can see that with the increasing of grow rate of prey species, the dynamic behaviors of both predator and prey becomes complicated.
- (2)
- Then, we consider the fear effect. Set each parameter value of system (11) as:$$(d,a,b,m,h,{r}_{0})=(0.3,3.0,0.5,1.0,0.5,4.0),$$set $\left(x\right(0),y(0\left)\right)=(0.5,0.3)$, and plot with k as the abscissa. We can see from Figure 9 that the system (11) changes from chaotic state ($0<k<0.204$) to eight-period orbit ($0.204<k<0.214$), four-period orbit ($0.214<k<0.277$), then to two-period orbit ($0.277<k<0.440$), to stable state ($0.440<k<12.545$), and finally to be the state that the prey is driven to extinction while the predator survive in a stable state ($k>12.545$). This means that the stability of system (11) increases as the fear effect of prey increases within a certain range, which is similar to the result in [35]. In addition, it can be seen from Figure 9 (c)-(d) that the positive equilibrium solution $({x}^{*}\left(k\right),{y}^{*}\left(k\right))$ of the system (11) will decrease with the increasing of k, which is consistent with the conclusion of Remark 2. If k is sufficiently large, the prey will become extinct, while the predator will not become extinct due to the presence of other food sources.

**Example**

**4.**

**Example**

**5.**

- (1)
- First, we consider the case where there is no fear effect and has the variable other food resource. Set each parameter value of system (11) as $(k,d,a,b)=(0.0,0.3,3.0,0.5)$ and set $\left(x\right(0),y(0\left)\right)=(0.5,0.3)$. Depending on ${r}_{0}>2$, ${r}_{0}<2$, $h>2$ and $h<2$, we give the corresponding numerical simulations (Figure 12, Figure 13, Figure 14 and Figure 15). We found that in any cases, too many other food sources can accelerate the demise of prey populations. The possible reason is that with the increasing of other food sources, the number of predator populations has increased. Therefore, although the influence of a single predator on the prey is reduced, the overall predator population will have an excessive impact on the prey population, which will lead to the decreasing of the prey population, and finally the prey species will be driven to extinction.
- (2)
- Then, we consider the influence of fear effect. Set each parameter value of system (11) as:$$(d,a,b)=(0.3,3.0,0.5),$$set $\left(x\right(0),y(0\left)\right)=(0.5,0.3)$, and plot with k as the abscissa. Corresponding to the four cases discussed above, we choose suitable m, such that two species could be coexist, no matter in the stable state or chaos state. Figure 16, Figure 17, Figure 18 and Figure 19 show that in any cases, if the fear effect is too large, the prey species will be driven to extinction, while depending on the intrinsic growth rate of predator species, the predator species will state in stable state for the case $h<2$, and in chaos state for the case $h>2$.

## 7. Discussion

**Conjecture 1.**

**Conjecture 2.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Lin, S.; Chen, F.; Li, Z.; Chen, L.
Complex Dynamic Behaviors of a Modified Discrete Leslie–Gower Predator–Prey System with Fear Effect on Prey Species. *Axioms* **2022**, *11*, 520.
https://doi.org/10.3390/axioms11100520

**AMA Style**

Lin S, Chen F, Li Z, Chen L.
Complex Dynamic Behaviors of a Modified Discrete Leslie–Gower Predator–Prey System with Fear Effect on Prey Species. *Axioms*. 2022; 11(10):520.
https://doi.org/10.3390/axioms11100520

**Chicago/Turabian Style**

Lin, Sijia, Fengde Chen, Zhong Li, and Lijuan Chen.
2022. "Complex Dynamic Behaviors of a Modified Discrete Leslie–Gower Predator–Prey System with Fear Effect on Prey Species" *Axioms* 11, no. 10: 520.
https://doi.org/10.3390/axioms11100520