# Pattern Dynamics of Cross Diffusion Predator–Prey System with Strong Allee Effect and Hunting Cooperation

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## Abstract

**:**

## 1. Introduction

## 2. Analysis of Self Diffusion Model

#### 2.1. Existence of Equilibrium Point

**Theorem**

**1.**

- (1)
- The model (5) has a semi trivial equilibrium point ${E}_{01}(b,0)$.
- (2)
- The model (5) has a semi trivial equilibrium point ${E}_{02}(1,0)$.
- (3)
- When $c<1+b-2\sqrt{b}$ and $a=\frac{{c}^{2}-2(1+b)c+{(1+b)}^{2}}{4b{c}^{2}}$, the model (5) has a unique positive equilibrium point ${E}^{*}=({u}^{*},{v}^{*})=(\frac{2b}{1+b-c},\frac{2bc}{1+b-c})$.
- (4)
- When $c<1+b-2\sqrt{b}$ and $a<\frac{{c}^{2}-2(1+b)c+{(1+b)}^{2}}{4b{c}^{2}}$, the model (5) has has two positive equilibrium points ${E}_{1}({u}_{1},c{u}_{1})$ and ${E}_{2}({u}_{2},c{u}_{2})$. Where ${u}_{1}=\frac{(1+b-c)+\sqrt{\Delta}}{2(1+a{c}^{2})}$, ${u}_{2}=\frac{(1+b-c)-\sqrt{\Delta}}{2(1+a{c}^{2})}$, $\Delta ={(1+b-c)}^{2}-4b-4ab{c}^{2}$.
- (5)
- When $c<1+b-2\sqrt{b}$ and $a>\frac{{c}^{2}-2(1+b)c+{(1+b)}^{2}}{4b{c}^{2}}$ or $c>1+b$, the model (5) has no positive equilibrium point.

**Proof.**

- (a)
- When $\Delta =0$, that is $a=\frac{{c}^{2}-2(1+b)c+{(1-b)}^{2}}{4b{c}^{2}}$, Equation (8) has a positive root:$$\begin{array}{c}\hfill u=\frac{2b}{1+b-c}.\end{array}$$
- (b)
- When $\Delta >0$, that is $0<a<\frac{{c}^{2}-2(1+b)c+{(1-b)}^{2}}{4b{c}^{2}}$, Equation (8) has two positive roots:$$\begin{array}{c}\hfill {u}_{1}=\frac{(1+b-c)+\sqrt{\Delta}}{2(1+a{c}^{2})},{u}_{2}=\frac{(1+b-c)-\sqrt{\Delta}}{2(1+a{c}^{2})}.\end{array}$$
- (c)
- When $a>\frac{{c}^{2}-2(1+b)c+{(1-b)}^{2}}{4b{c}^{2}}$, Equation (8) has no roots.

#### 2.2. Stability of Equilibrium Point

**Theorem**

**2.**

**Proof.**

#### 2.3. Turing Instability

#### 2.4. Diffusion Induced Pattern

**Figure 2.**The process of formation of the prey patterns at $c=0.55$, the other parameters are fixed as in (23). Times: (

**a**) $t=0$, (

**b**) $t=150$, (

**c**) $t=300$, and (

**d**) $t=1000$.

**Figure 3.**The patterns of u at $c=0.62$, the other parameters are fixed as in (23). Times: (

**a**) $t=0$, (

**b**) $t=50$, (

**c**) $t=200$, and (

**d**) $t=500$.

**Figure 4.**The patterns of u at $c=0.7$, the other parameters are fixed as in (23). Times: (

**a**) $t=0$, (

**b**) $t=50$, (

**c**) $t=200$, (

**d**) $t=500$.

**Figure 5.**The patterns of prey evolution. Parameters: $a=0.14$, $b=0.02$, $c=0.7$, $s=0.23$, ${d}_{1}=0.04$, ${d}_{2}=0.24$. Times: (

**a**) $t=0$, (

**b**) $t=150$, (

**c**) $t=300$, and (

**d**) $t=1000$.

## 3. Analysis of Cross Diffusion Model

#### 3.1. Turing Instability Induced by Cross Diffusion

#### 3.2. Pattern Formation Induced by Self Diffusion and Cross Diffusion

## 4. Conclusions and Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Bifurcation diagram for model (2) in the $c-{d}_{1}$ parameter space. The parameters are taken as $a=0.14$, $b=0.02$, $s=0.23$, and ${d}_{2}=0.24$. The blue curve represents the Turing bifurcation curve, which divides the plane into two regions. The upper part of the blue curve is the stable region and the lower part is the unstable region. ${P}_{1}$, ${P}_{2}$, ${P}_{3}$, and ${P}_{4}$ correspond to the parameter spaces of Figure 2, Figure 3, Figure 4, and Figure 5, respectively.

**Figure 6.**The patterns of the prey with different values at ${d}_{1}$, the other parameters are fixed as in (24). Types: (

**a**) ${d}_{1}=0.12$, (

**b**) ${d}_{1}=0.04$, (

**c**) ${d}_{1}=0.01$, (

**d**) ${d}_{1}=0.002$.

**Figure 7.**The patterns formation process of the prey population at $(a,c,{d}_{12},{d}_{21})=(0.1,0.56,0,0)$, the other parameters are fixed as in (36). Times: (

**a**) $t=0$, (

**b**) $t=200$, (

**c**) $t=500$, and (

**d**) $t=1000$.

**Figure 8.**The patterns of u at $(a,c,{d}_{12},{d}_{21})=(0.1,0.56,0.2,0.02)$, the other parameters are fixed as in (36). Times: (

**a**) $t=0$, (

**b**) $t=1000$, (

**c**) $t=3000$, and (

**d**) $t=5000$.

**Figure 9.**The patterns of u at $(a,c,{d}_{12},{d}_{21})=(0.4,0.6,0.5,0.02)$, the other parameters are fixed as in (36). Times: (

**a**) $t=0$, (

**b**) $t=200$, (

**c**) $t=500$, and (

**d**) $t=2000$.

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**MDPI and ACS Style**

Zhu, M.; Li, J.; Lian, X.
Pattern Dynamics of Cross Diffusion Predator–Prey System with Strong Allee Effect and Hunting Cooperation. *Mathematics* **2022**, *10*, 3171.
https://doi.org/10.3390/math10173171

**AMA Style**

Zhu M, Li J, Lian X.
Pattern Dynamics of Cross Diffusion Predator–Prey System with Strong Allee Effect and Hunting Cooperation. *Mathematics*. 2022; 10(17):3171.
https://doi.org/10.3390/math10173171

**Chicago/Turabian Style**

Zhu, Meng, Jing Li, and Xinze Lian.
2022. "Pattern Dynamics of Cross Diffusion Predator–Prey System with Strong Allee Effect and Hunting Cooperation" *Mathematics* 10, no. 17: 3171.
https://doi.org/10.3390/math10173171