# Dynamic Behavior of a Predator–Prey Model with Double Delays and Beddington–DeAngelis Functional Response

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## Abstract

**:**

## 1. Introduction

## 2. Positivity and Boundedness

#### 2.1. Positivity

**Theorem 1.**

**Proof.**

#### 2.2. Boundedness

**Theorem 2.**

**Proof.**

## 3. Stability Analysis

#### 3.1. Equilibrium Points and Existence Criterion

- (1)
- the trivial equilibrium point ${E}_{0}(0,0)$;
- (2)
- the free equilibrium point ${E}_{1}(\frac{k}{\alpha},0)$;
- (3)
- the coexistence equilibrium point ${E}^{*}({x}^{*},{y}^{*})$ satisfying the following equation:$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \frac{k}{1+fy}-\alpha x-\frac{py}{ax+by+c}=0,\\ \hfill \phantom{\rule{1.em}{0ex}}& \frac{\mu px}{ax+by+c}-d-hy=0,\end{array}\right.\end{array}$$

#### 3.2. Local Stability Analysis and Hopf Bifurcation of Equilibria

- (1)
- At ${E}_{0}$:$${J}_{{E}_{0}}=\left[\begin{array}{cc}k& 0\\ 0& -d\end{array}\right].$$

**Theorem 3.**

- (2)
- At ${E}_{1}$:$${J}_{{E}_{1}}=\left[\begin{array}{cc}-k& -\frac{pk}{ak+c\alpha}-\frac{f{k}^{2}}{\alpha}{e}^{-\lambda {\tau}_{1}}\\ 0& -d+\frac{\mu pk}{ak+c\alpha}{e}^{-\lambda {\tau}_{2}}\end{array}\right],$$$$\begin{array}{c}\hfill (-k-\lambda )(-d+\frac{\mu pk}{ak+c\alpha}{e}^{-\lambda {\tau}_{2}}-\lambda )=0.\end{array}$$

**Theorem 4.**

**Remark 1.**

- (3)
- At ${E}^{*}$:$${J}_{{E}^{*}}=\left[\begin{array}{cc}{a}_{11}& {b}_{12}{e}^{-\lambda {\tau}_{1}}+{a}_{12}\\ {c}_{21}{e}^{-\lambda {\tau}_{2}}& {a}_{22}+{c}_{22}{e}^{-\lambda {\tau}_{2}}\end{array}\right],$$

**Theorem 5.**

**Theorem 6.**

**Theorem 7.**

**Theorem 8.**

## 4. Stochastic Delay Model Analysis

#### 4.1. Existence and Uniqueness of Positive Solution

**Theorem 9.**

**Proof.**

#### 4.2. Stochastic Ultimate Boundedness

**Definition 1.**

**Theorem 10.**

**Proof.**

## 5. Numerical Simulations

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Cui, M.; Shao, Y.; Xue, R.; Zhao, J.
Dynamic Behavior of a Predator–Prey Model with Double Delays and Beddington–DeAngelis Functional Response. *Axioms* **2023**, *12*, 73.
https://doi.org/10.3390/axioms12010073

**AMA Style**

Cui M, Shao Y, Xue R, Zhao J.
Dynamic Behavior of a Predator–Prey Model with Double Delays and Beddington–DeAngelis Functional Response. *Axioms*. 2023; 12(1):73.
https://doi.org/10.3390/axioms12010073

**Chicago/Turabian Style**

Cui, Minjuan, Yuanfu Shao, Renxiu Xue, and Jinxing Zhao.
2023. "Dynamic Behavior of a Predator–Prey Model with Double Delays and Beddington–DeAngelis Functional Response" *Axioms* 12, no. 1: 73.
https://doi.org/10.3390/axioms12010073