#
Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Existence of Equilibria

**Lemma**

**1.**

- (i)
- Suppose $a\in (0,1)$. Then the existence of boundary equilibria in addition to ${E}_{0}$ is summarized in Table 1.
- (ii)
- Suppose $a=1$. Then besides ${E}_{0}$, there is also another boundary equilibrium ${E}_{4}=({x}_{4},0)=(\sqrt{1-m},0)$ only when $0<m<1$.
- (iii)
- Suppose $a>1$. Then besides ${E}_{0}$, there is also another boundary equilibrium ${E}_{5}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}({x}_{5},0)\phantom{\rule{3.33333pt}{0ex}}\left(\frac{1-a+\sqrt{\mathrm{\Delta}(m)}}{2},0\right)$ only when $0<m<a<{m}^{*}$.

**Lemma**

**2.**

## 3. Local Stability of Equilibria

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

**Proof.**

## 4. Global Asymptotical Stability of the Positive Equilibrium

**Theorem**

**6.**

- (i)
- $a<1$, $m\le a<{m}^{*}$, and $\alpha b{x}_{3}-n>0$;
- (ii)
- $a=1$, $m<a={m}^{*}$, and $\alpha b{x}_{4}-n>0$;
- (iii)
- $a>1$, $m<a<{m}^{*}$, and $\alpha b{x}_{5}-n>0$.

**Proof.**

## 5. Bifurcation Analysis

**Theorem**

**7**

- (1)
- Suppose$$\begin{array}{ccc}{W}^{T}{f}_{\mu}({x}_{0},{\mu}_{0})\hfill & \ne \hfill & 0,\hfill \\ {W}^{T}\left[{D}^{2}{f}_{\mu}({x}_{0},{\mu}_{0})(V,V)\right]\hfill & \ne \hfill & 0.\hfill \end{array}\phantom{\rule{2.em}{0ex}}$$
- (2)
- Suppose$$\begin{array}{ccc}{W}^{T}{f}_{\mu}({x}_{0},{\mu}_{0})\hfill & =\hfill & 0,\hfill \\ {W}^{T}\left[D{f}_{\mu}({x}_{0},{\mu}_{0})V\right]\hfill & \ne \hfill & 0,\hfill \\ \hfill {W}^{T}\left[{D}^{2}{f}_{\mu}({x}_{0},{\mu}_{0})(V,V)\right]& \hfill \ne & 0.\hfill \end{array}\phantom{\rule{2.em}{0ex}}$$

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

## 6. Numerical Simulations

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Two distinct boundary equilibria and one trivial equilibrium when $m<{m}_{SN}$: there are two stable nodes ${E}_{0}$ and ${E}_{3}$, and a saddle ${E}_{2}$. (

**b**) A boundary equilibrium and a trivial equilibrium when $m={m}_{SN}$: ${E}_{3}$ is a saddle-node and ${E}_{0}$ is a stable node. (

**c**) A trivial equilibrium when $m>{m}_{SN}$: ${E}_{0}$ is a stable node.

**Figure 2.**(

**a**) Two distinct boundary equilibria and one trivial equilibrium when $a<{a}_{TC}$: two stable nodes ${E}_{2}$ and ${E}_{3}$, and a saddle ${E}_{0}$. (

**b**) A boundary equilibrium and a trivial equilibrium when $a={a}_{TC}$: ${E}_{3}$ is a stable node and ${E}_{0}$ is a saddle-node. (

**c**) Two boundary equilibria and a trivial equilibrium when $a>{a}_{TC}$: ${E}_{2}$ is a saddle, both ${E}_{3}$ and ${E}_{0}$ are stable nodes.

**Figure 3.**(

**a**) When $a<{a}_{H}$, ${E}^{*}$ is unstable. (

**b**) When $a={a}_{H}$, a stable periodic orbit bifurcated from ${E}^{*}$. (

**c**) When $a>{a}_{H}$, ${E}^{*}$ is stable.

**Figure 4.**When $a={a}_{H}=0.26$, a stable periodic orbit bifurcated form ${E}^{*}$ which is locally stable.

**Figure 5.**(

**a**) With $m=0.2<0.3=a$ and $n=0.2$, ${E}_{0}$ and ${E}_{3}$ are saddle points. (

**b**) With $m=0.2<0.3=a$ and $n=0.5$, ${E}^{*}$ is a stable node.

**Figure 6.**(

**a**) With $m=a=0.3$ and $n=0.2$, ${E}_{0}$ is a saddle-node, ${E}_{3}$ is a saddle point, and ${E}^{*}$ is a stable node. (

**b**) With $m=a=0.3$ and $n=0.5$, ${E}_{0}$ is a saddle node and ${E}_{3}$ is a stable node.

**Figure 7.**(

**a**) With $m=0.4>0.3=a$ and $n=0.1$, there is a stable node ${E}_{0}$, a saddle-node ${E}_{2}$, and a saddle ${E}_{3}$. (

**b**) With $m=0.4>0.3=a$ and $n=0.2$, there is a stable node ${E}_{0}$, two saddles ${E}_{2}$ and ${E}_{3}$, and a stable node ${E}^{*}$. (

**c**) With $m=0.4>0.3=a$ and $n=0.25$, there is a stable node ${E}_{0}$, a saddle ${E}_{2}$, and a saddle-node ${E}_{3}$. (

**d**) With $m=0.4>0.3=a$ and $n=0.5$, there is a stable node ${E}_{0}$, a saddle ${E}_{2}$, and a stable node ${E}_{3}$.

**Figure 8.**(

**a**) When $m={m}^{*}=0.4225>0.3=a$ and $n=0.1$, ${E}_{0}$ is a stable node and ${E}_{1}$ is a saddle-node. (

**b**) When $m={m}^{*}=0.4225>0.3=a$ and $n=0.2$, ${E}_{0}$ is a stable node and ${E}_{1}$ is a saddle-node. (

**c**) When $m={m}^{*}=0.4225>0.3=a$ and $n=0.175$, ${E}_{0}$ is a stable node and ${E}_{1}$ is a saddle.

**Figure 9.**(

**a**) When $m=0.5<1=a={m}^{*}$ and $n=0.1$, there are two saddle points ${E}_{0}$ and ${E}_{4}$, and a stable node ${E}^{*}$. (

**b**) When $m=0.5<1=a={m}^{*}$ and $n=0.2$, ${E}_{0}$ is a saddle and ${E}_{4}$ is a saddle-node. (

**c**) When $m=0.5<1=a={m}^{*}$ and $n=0.3$, ${E}_{0}$ is a saddle and ${E}_{4}$ is a stable node.

**Figure 10.**(

**a**) When $m=0.4<1.3=a<{m}^{*}=1.3225$ and $n=0.2$, both ${E}_{0}$ and ${E}_{5}$ are saddle points and ${E}^{*}$ is a stable node. (

**b**) When $m=0.4<1.3=a<{m}^{*}=1.3225$ and $n=0.25$, ${E}_{0}$ is a saddle point and ${E}_{5}$ is a saddle-node. (

**c**) When $m=0.4<1.3=a<{m}^{*}=1.3225$ and $n=0.3$, ${E}_{0}$ is a saddle point and ${E}_{5}$ is a stable node.

Condition | Boundary Equilibria |
---|---|

$a<{m}^{*}<m$ | No |

$a<m={m}^{*}$ | ${E}_{1}=({x}_{1},0)=(\frac{1-a}{2},0)$ |

$a<m<{m}^{*}$ | ${E}_{2}=({x}_{2},0)=\left(\frac{1-a-\sqrt{\mathrm{\Delta}(m)}}{2},0\right)$ and ${E}_{3}=({x}_{3},0)=\left(\frac{1-a+\sqrt{\mathrm{\Delta}(m)}}{2},0\right)$ |

$0<m=a<{m}^{*}$ | ${E}_{3}$ (${E}_{2}$ and ${E}_{0}$ coincide) |

$0<m<a<{m}^{*}$ | ${E}_{2}$ (${x}_{2}<0$) and ${E}_{3}$ |

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## Share and Cite

**MDPI and ACS Style**

Lai, L.; Zhu, Z.; Chen, F. Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect. *Mathematics* **2020**, *8*, 1280.
https://doi.org/10.3390/math8081280

**AMA Style**

Lai L, Zhu Z, Chen F. Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect. *Mathematics*. 2020; 8(8):1280.
https://doi.org/10.3390/math8081280

**Chicago/Turabian Style**

Lai, Liyun, Zhenliang Zhu, and Fengde Chen. 2020. "Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect" *Mathematics* 8, no. 8: 1280.
https://doi.org/10.3390/math8081280