# Modelling Predator–Prey Interactions: A Trade-Off between Seasonality and Wind Speed

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Formulation

#### 2.1. Model System 1

- (i)
- $\psi \left(0\right)=1;$ i.e., considering no wind speed, the handling time remains the same as before.
- (ii)
- ${\psi}^{\prime}\left(W\right)>0;$ i.e., considering wind speed and with the acceleration of this speed, the handling time for each prey per predator continuously increases.

#### 2.2. Model System 2

**Remark**

**1.**

#### 2.3. Perturbed Model System 1

#### 2.4. Perturbed Model System 2

## 3. Model Validation

#### 3.1. Positivity

**Theorem**

**1.**

**Proof.**

#### 3.2. Boundedness

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

#### 3.3. Persistence

**Definition**

**1.**

**Lemma**

**1.**

**Theorem**

**3.**

- (i)
- $1+W>K{L}_{2}$;
- (ii)
- ${e}_{2}({L}_{3}-{\u03f5}_{3})>{e}_{1}(2+W+\frac{W}{1+W})$.

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

## 4. General Stability Analysis and Hopf Bifurcation of the Model Systems

#### 4.1. Equilibria, Their Conditions of Existence, and Local Stability Analysis of System (1)

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Remark**

**5.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

#### 4.2. Hopf Bifurcation Analysis

#### 4.3. Equilibria, Their Conditions of Existence, and Local Stability Analysis of System (3)

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

## 5. Global Stability Analysis

#### 5.1. For System (12)

**Theorem**

**11.**

**Proof.**

#### 5.2. For System (19)

**Theorem**

**12.**

- (i)
- $\frac{\left(1+\frac{W}{1+W}\right){y}^{*}}{l\left[1+W+\left(1+\frac{W}{1+W}\right){x}^{*}\right]}+\frac{l+{x}^{*}}{(1+{P}_{1}^{2})({{x}^{*}}^{2}+{P}_{1}^{2})}<1,$
- (ii)
- ${e}_{2}>{e}_{1}l,$

**Proof.**

## 6. Numerical Simulations

#### 6.1. For System (1)

**Remark**

**7.**

#### 6.2. For System (5)

**Remark**

**8.**

**Reason behind the occurrence of chaos and its mitigation:**

#### 6.3. For System (3)

**Remark**

**9.**

#### 6.4. For System (7)

**Remark**

**10.**

**Reason behind the occurrence of period-4 oscillation and its control:**

**Remark**

**11.**

#### 6.5. For Perturbed System 1

#### 6.6. For Perturbed System 2

**Remark**

**12.**

## 7. Conclusions and Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 16.**Time-series plot of the perturbed System (25) considering different values of the wind speed parameter W. The other parameter values have been mentioned in the text.

**Figure 17.**Time-series plot of the perturbed System (27) considering different values of the wind speed parameter W. The other parameter values have been mentioned in the text.

**Table 1.**Units of all non-negative parameters of System (1).

Parameter | Unit |
---|---|

X | $(\mathrm{prey}\mathrm{density})$ |

Y | $(\mathrm{predator}\mathrm{density})$ |

A | ${\left(\mathrm{time}\right)}^{-1}$ |

K | $(\mathrm{prey}\mathrm{density})$ |

C${}_{1}$ | ${(\mathrm{predator}\mathrm{density})}^{-1}$ |

C${}_{2}$ | ${(\mathrm{prey}\mathrm{density})}^{-1}$ |

B${}_{1}={B}_{2}$ | ${(\mathrm{prey}\mathrm{density})}^{-1}$ |

C | ${\left(\mathrm{time}\right)}^{-1}$ |

W | (${\mathrm{time})}^{-1}$ |

**Table 2.**Units of all other non-negative parameters of System (3).

Variable/Parameter | Unit |
---|---|

C${}_{3}$ | (prey density)·(time)${}^{-1}$ |

D${}_{1}$ | (prey density) |

C${}_{4}$ | (time)${}^{-1}$ |

C${}_{5}$ | (predator density)${}^{-1}$·(prey density)·time${}^{-1}$ |

Model System | Parameter Values | Source |
---|---|---|

System 1 | $A=2,\phantom{\rule{3.33333pt}{0ex}}K=40,\phantom{\rule{3.33333pt}{0ex}}{B}_{1}=0.1,\phantom{\rule{3.33333pt}{0ex}}{B}_{2}=0.1,\phantom{\rule{3.33333pt}{0ex}}{C}_{1}=0.1,\phantom{\rule{3.33333pt}{0ex}}{C}_{2}=0.2,\phantom{\rule{3.33333pt}{0ex}}C=1,\phantom{\rule{3.33333pt}{0ex}}W=\ast $ | [23] |

System 2 | $A=2,\phantom{\rule{3.33333pt}{0ex}}K=100,\phantom{\rule{3.33333pt}{0ex}}{B}_{1}=0.1,\phantom{\rule{3.33333pt}{0ex}}{C}_{1}=0.1,\phantom{\rule{3.33333pt}{0ex}}{C}_{3}=1,\phantom{\rule{3.33333pt}{0ex}}W=\ast ,\phantom{\rule{3.33333pt}{0ex}}{D}_{1}=10,\phantom{\rule{3.33333pt}{0ex}}{C}_{4}=0.45,\phantom{\rule{3.33333pt}{0ex}}{C}_{5}=0.2$ | [23] |

Perturbed System 1 | $A=2,\phantom{\rule{3.33333pt}{0ex}}K=75,\phantom{\rule{3.33333pt}{0ex}}{B}_{1}=0.1,\phantom{\rule{3.33333pt}{0ex}}{B}_{2}=0.1,\phantom{\rule{3.33333pt}{0ex}}{C}_{1}=0.1,\phantom{\rule{3.33333pt}{0ex}}{C}_{2}=0.2,\phantom{\rule{3.33333pt}{0ex}}C=1,\theta =0.681,\phantom{\rule{3.33333pt}{0ex}}\u03f5=0.6,\phantom{\rule{3.33333pt}{0ex}}W=\ast $ | [23] |

Perturbed System 2 | $A=2,\phantom{\rule{3.33333pt}{0ex}}K=100,\phantom{\rule{3.33333pt}{0ex}}{B}_{1}=0.1,\phantom{\rule{3.33333pt}{0ex}}{C}_{1}=0.1,\phantom{\rule{3.33333pt}{0ex}}{C}_{3}=1,\phantom{\rule{3.33333pt}{0ex}}W=\ast ,\phantom{\rule{3.33333pt}{0ex}}{D}_{1}=10,\phantom{\rule{3.33333pt}{0ex}}{C}_{4}=0.45,\phantom{\rule{3.33333pt}{0ex}}{C}_{5}=0.2,\theta =0.153,\phantom{\rule{3.33333pt}{0ex}}\u03f5=0.6$ | [23] |

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**MDPI and ACS Style**

Barman, D.; Upadhyay, R.K.
Modelling Predator–Prey Interactions: A Trade-Off between Seasonality and Wind Speed. *Mathematics* **2023**, *11*, 4863.
https://doi.org/10.3390/math11234863

**AMA Style**

Barman D, Upadhyay RK.
Modelling Predator–Prey Interactions: A Trade-Off between Seasonality and Wind Speed. *Mathematics*. 2023; 11(23):4863.
https://doi.org/10.3390/math11234863

**Chicago/Turabian Style**

Barman, Dipesh, and Ranjit Kumar Upadhyay.
2023. "Modelling Predator–Prey Interactions: A Trade-Off between Seasonality and Wind Speed" *Mathematics* 11, no. 23: 4863.
https://doi.org/10.3390/math11234863