# Dynamics of a Double-Impulsive Control Model of Integrated Pest Management Using Perturbation Methods and Floquet Theory

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## Abstract

**:**

## 1. Introduction

## 2. Derivation of the Impulsive Control Model

## 3. Dynamics of the Impulsive Model

#### 3.1. Boundedness of the Model Variables

**Theorem**

**1.**

**Lemma**

**1.**

#### 3.2. Existence of the Pest-Free Periodic Orbit

#### 3.3. Stability of the Pest-Free Periodic Solution

**Theorem**

**2.**

- (i)
- Application of biopesticide and chemical pesticide with same time interval $t=n\tau $, provided that$$\begin{array}{ccc}\hfill & & {c}_{1}\alpha -d-\frac{\lambda {v}_{i}{e}^{-\gamma (t-n\tau )}}{1-{e}^{-\gamma \tau}}-\frac{{m}_{1}{s}_{i}{e}^{-\mu (t-n\tau )}}{1-{e}^{-\mu \tau}}<0,\hfill \\ & & {c}_{2}\varphi k\alpha -(d+\delta )-\frac{{m}_{2}{s}_{i}{e}^{-\mu (t-n\tau )}}{1-{e}^{-\mu \tau}}<0;\hfill \end{array}$$
- (ii)
- Application of biopesticide with time interval $t=n{\tau}_{1}$ and chemical pesticide with time interval $t=n{\tau}_{2}$, i.e., for different time intervals, where ${\tau}_{1}\ne {\tau}_{2}$, provided that$$\begin{array}{c}{c}_{1}\alpha -d-\frac{\lambda {v}_{i}{e}^{-\gamma (t-n{\tau}_{1})}}{1-{e}^{-\gamma {\tau}_{1}}}<0,\\ {c}_{2}\varphi k\alpha -(d+\delta )<0,\\ {c}_{1}\alpha -d-\frac{{m}_{1}{s}_{i}{e}^{-\mu (t-n{\tau}_{2})}}{1-{e}^{-\mu {\tau}_{2}}}<0.\end{array}$$

**Proof.**

## 4. Numerical Simulations

`ode45`MATLAB solver.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Impact of biopesticide application in impulsive mode on system (1). Evolution of (

**a**) crop; (

**b**) uninfected pest; (

**c**) infected pest; (

**d**) biopesticide. The set of parameters are $r=0.1$, $k=1$, $\alpha =0.2$, $\beta =0.003$, ${m}_{1}=0.8$, ${m}_{2}=0.6$, ${c}_{1}=0.5$, ${c}_{2}=0.8$, $\gamma =0.15$, $\delta =0.2$, $d=0.05$, $\kappa =100$, $s=0.3$, and $\lambda =0.35$. Here, the time interval is ${\tau}_{1}=5$ days and the rates of biopesticide release are ${v}_{i}=0$ (black line), ${v}_{i}=6$ (red line), and ${v}_{i}=12$ (blue line).

**Figure 2.**Impact of biopesticide on system (1) for different impulse intervals and rates. Evolution of (

**a**) crop; (

**b**) uninfected pest; (

**c**) infected pest; (

**d**) biopesticide. Red line indicates ${v}_{i}=6$ and ${\tau}_{1}=5$, green line indicates ${v}_{i}=12$ and ${\tau}_{1}=5$, and blue line indicates ${v}_{i}=12$ and ${\tau}_{1}=2$.

**Figure 3.**Impact of both biopesticide and chemical pesticide on system (1) with the same impulse interval, ${\tau}_{1}={\tau}_{2}=5$. Evolution of (

**a**) crop; (

**b**) uninfected pest; (

**c**) infected pest; (

**d**) biopesticide; (

**e**) chemical pesticide. The rates of impulses are ${s}_{i}=0.15$ and ${v}_{i}=6$ for black dotted color; ${s}_{i}=0.1$ and ${v}_{i}=6$ for red dashed line; and ${s}_{i}=0.05$ and ${v}_{i}=6$ for blue solid line.

**Figure 4.**Impact of both biopesticide and chemical pesticide on system (1) with same impulse interval, ${\tau}_{1}=3$, ${\tau}_{2}=2$, and where the rates of impulses are ${s}_{i}=0.15$ and ${v}_{i}=6$. Evolution of (

**a**) crop; (

**b**) uninfected pest; (

**c**) infected pest; (

**d**) pesticides (biopesticide in blue, chemical pesticide in red).

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**MDPI and ACS Style**

Al Basir, F.; Chowdhury, J.; Torres, D.F.M.
Dynamics of a Double-Impulsive Control Model of Integrated Pest Management Using Perturbation Methods and Floquet Theory. *Axioms* **2023**, *12*, 391.
https://doi.org/10.3390/axioms12040391

**AMA Style**

Al Basir F, Chowdhury J, Torres DFM.
Dynamics of a Double-Impulsive Control Model of Integrated Pest Management Using Perturbation Methods and Floquet Theory. *Axioms*. 2023; 12(4):391.
https://doi.org/10.3390/axioms12040391

**Chicago/Turabian Style**

Al Basir, Fahad, Jahangir Chowdhury, and Delfim F. M. Torres.
2023. "Dynamics of a Double-Impulsive Control Model of Integrated Pest Management Using Perturbation Methods and Floquet Theory" *Axioms* 12, no. 4: 391.
https://doi.org/10.3390/axioms12040391