Combined impact of predatory insects and bio-pesticide over pest population: impulsive model-based study

Abstract

The production with increasing demands maintaining the balance of nature and natural diversity is the most challenging part of the agricultural system. However, pests and other insect populations are significant obstacles to the continuous food supply. This study proposes a crop pest management mathematical model using the predator (pests’ natural enemy) and viral infection through bio-pesticides. Impulsive differential equations have been implemented to study the dynamics between all populations, considering the repetitive release of virus micro-pesticides and predator insects in the crop field. The hypothesized model gives the outlook of complex natural dynamics. Two types of scenarios have been analyzed here using the model: One deals with the complete eradication of the field’s pest population, and another is sounder from a biodiversity conservation perspective, that defines the minimum pest population below the economic injury level, which is, nowadays, the major challenge in the agricultural field. Numerical examples show that pest management is successful when considering the minimum pest level that keeps the economic threshold by optimizing predator and virus levels cost-effectively.

Appendices

Appendix A

Coefficients of equation (11) The coefficients of (11) are given as follows:

$$\begin{aligned} A_1& {} = - F_{11}- F_{22}- F_{33}- F_{44}\\ A_2& {} = - F_{12} F_{21} + F_{11} F_{22} - F_{13} F_{31} - F_{23} F_{32} + F_{11} F_{33}\\&\quad+F_{22} F_{33} - F_{24} F_{42} + F_{11} F_{44} + F_{22} F_{44} + F_{33} F_{44}\\ A_3& {} = F_{13} F_{22} F_{31}- F_{12} F_{23} F_{31}- F_{13} F_{21} F_{32}\\&\quad+ F_{11} F_{23} F_{32}+ F_{12} F_{21} F_{33} - F_{11} F_{22} F_{33} - F_{14} F_{21} F_{42}\\&\quad + F_{11} F_{24} F_{42}+ F_{24} F_{33} F_{42}+ F_{12} F_{21} F_{44}\\&- F_{11} F_{22} F_{44}+ F_{13} F_{31} F_{44}+ F_{23} F_{32} F_{44}\\&\quad- F_{11} F_{33} F_{44}- F_{22} F_{33} F_{44}\\ A_4& {} = -F_{14} F_{23} F_{31} F_{42} + F_{13} F_{24} F_{31} F_{42}\\&\quad+ F_{14} {F_{21}} F_{33} F_{42} -F_{11} F_{24} F_{33} F_{42}\\&- F_{13} F_{22} F_{31} F_{44} + F_{12} F_{23} F_{31} F_{44} + F_{13} F_{21} F_{32} F_{44} \\&\quad- F_{11} F_{23} F_{32} F_{44}\\&\quad- F_{12} F_{21} F_{33} F_{44} + F_{11} F_{22} F_{33} F_{44}. \end{aligned}$$

Appendix B

Proof of Theorem 3:

Proof

Here, we want to check the stability of the system (4) for two cases.

(i) Release predator(P) and Virus(V) at same time interval $\mathbf{t = n\tau }$.

Local Stability: Here, we discuss stability of the system through small amplitude perturbation method at the periodic solution $(0,0 ,P^*,V^*)$. Let,

$$\begin{aligned}&S(t)=\epsilon _1(t), I(t)=\epsilon _2(t),\nonumber \\&P(t)=P^*(t)+\epsilon _3(t),V(t)=V^*(t)+\epsilon _4(t). \end{aligned}$$

(25)

Here, $\epsilon _1, \epsilon _2, \epsilon _3 \text{ and }\epsilon _4$ denote small amplitude perturbation.

First, we consider both predator and virus release at time $t= n\tau$, then the corresponding system of (4) at $(0,0 ,P^*,V^*)$ is given as For $t\ne n\tau$

$$\begin{aligned} \frac{\mathrm{d}\epsilon _1}{\mathrm{d}t}& {} = r\epsilon _1(t)\left\{ 1-\frac{\epsilon _1(t)+\epsilon _2(t)}{K}\right\} -\lambda \epsilon _1(t) (V^*+\epsilon _4(t))-\alpha \epsilon _1(t) (P^*+\epsilon _3(t))\nonumber \\ \frac{\mathrm{d}\epsilon _2}{\mathrm{d}t}& {} = \lambda \epsilon _1(t) (V^*+\epsilon _4(t))-\delta \epsilon _2-\beta \epsilon _2(t) (P^*+\epsilon _3(t))\nonumber \\ \frac{\mathrm{d}\epsilon _3}{\mathrm{d}t}& {} = m \alpha \epsilon _1(t) (P^*+\epsilon _3(t))+m\beta \epsilon _2(t) (P^*+\epsilon _3(t))-d (P^*+\epsilon _3(t)) \nonumber \\ \frac{\mathrm{d}\epsilon _4}{\mathrm{d}t}& {} = n \delta \epsilon _2-\mu (V^*+\epsilon _4(t)). \end{aligned}$$

(26)

At $t=n\tau$, i.e., for impulsive system,

$$\begin{aligned}&\triangle \epsilon _1(t) = 0,\nonumber \\&\triangle \epsilon _2(t)=0,\nonumber \\&\triangle (P^*(t)+\epsilon _3(t))=\rho ,\nonumber \\&\triangle (V^*(t)+\epsilon _4(t))=\nu . \end{aligned}$$

(27)

Since $\epsilon _i(i=1,2,3,4)$ are very small, we can neglect second higher degree terms of $\epsilon _i(i=1,2,3,4)$. Hence, the corresponding linear system of above system is given as, For $t\ne n\tau$

$$\begin{aligned} \frac{\mathrm{d}\epsilon _1}{\mathrm{d}t}& {} = r\epsilon _1(t)-\lambda \epsilon _1(t) V^*-\alpha \epsilon _1(t) P^*\nonumber \\ \frac{\mathrm{d}\epsilon _2}{\mathrm{d}t}& {} = \lambda \epsilon _1(t) V^*-\delta \epsilon _2-\beta \epsilon _2(t)P^*\nonumber \\ \frac{\mathrm{d}\epsilon _3}{\mathrm{d}t}& {} = m \alpha \epsilon _1(t) P^*+m\beta \epsilon _2(t) P^*-d (P^*+\epsilon _3(t)) \nonumber \\ \frac{\mathrm{d}\epsilon _4}{\mathrm{d}t}& {} = n \delta \epsilon _2-\mu (V^*+\epsilon _4(t)). \end{aligned}$$

(28)

At $t=n\tau$, i.e., for impulsive system,

$$\begin{aligned}&\triangle \epsilon _1(t) = 0,\nonumber \\&\triangle \epsilon _2(t)=0,\nonumber \\&\triangle (P^*(t)+\epsilon _3(t))=\rho ,\nonumber \\&\triangle (V^*(t)+\epsilon _4(t))=\nu . \end{aligned}$$

(29)

Fundamental matrix M(t) of (28) will be $\frac{\mathrm{d}M_f(t)}{\mathrm{d}t}=\left[ \begin{array}{ccccc} r(t)-\lambda (t) V^*-\alpha (t) P^* &{} 0 &{} 0 &{} 0 \\ \lambda V^* &{} -\delta -\beta P^*&{} 0 &{} 0 \\ m \alpha P^* &{} m\beta P^* &{} -d &{} 0 \\ 0 &{} n\delta &{} 0 &{} -\mu \\ \end{array}\right] .$

with initial condition $M(0)=I_4$(the identity matrix). Now, fundamental solution matrix is given as $M_f(t)=\left[ \begin{array}{ccccc} \exp \int _0^t{(r(t)-\lambda (t) V^*-\alpha (t) P^*)\mathrm{d}s}&{} 0 &{} 0 &{} 0 \\ \psi _1 &{} \exp\int _0^t{(-\delta -\beta P^*)\mathrm{d}s}&{} 0 &{} 0 \\ \psi _2 &{} \psi _3 &{} \exp(-dt) &{} 0 \\ 0 &{} \psi _4 &{} 0 &{} \exp(-\mu t) \\ \end{array}\right] .$ Here, $\psi _i(t)(i=1,2,3,4)$ is not required to calculate for our further analysis. Now, the monodromy matrix of (28) and (29) is given as $M(t)=I_4 M_f(t)$, where $I_4$ is the identity matrix . According to Bionov et al. 1993[], the periodic solution $E=(0,0,P^*,V^*)$ is local stable if and only if absolute value of all eigenvalue of $M_(t)$ is less than one. The eigenvalue of $M_(t)$ is

$$\begin{aligned}&\lambda _1(t)=\exp\int _0^t{(r(t)-\lambda (t) V^*-\alpha (t) P^*)\mathrm{d}s}\\&\lambda _2(t)=\exp\int _0^t{(-\delta -\beta P^*)\mathrm{d}s},\lambda _3(t)=\exp(-\mathrm{d}t), \\&\lambda _5(t)=\exp(-\mu t) \end{aligned}$$

Now, clearly, $0<\lambda _2<1$, $0<\lambda _3<1$ and $0<\lambda _4<1$ thus the system (4) locally stable around the periodic solution if $0<\lambda _1<1,$ i.e., if

$$\begin{aligned} \exp\int _0^t{(r(t)-\lambda (t) V^*-\alpha (t) P^*)\mathrm{d}s}<1 \end{aligned}$$

(30)

Global stability From Eq. (30), we can choose $\delta _1>0$ such that,

$$\begin{aligned} \eta _1=\exp\int _{n\tau }^{(n+1)\tau }(r(t)-\lambda (t) (V^*-\delta _1)-\alpha (t) (P^*-\delta _1))\mathrm{d}s<1 \end{aligned}$$

Since all state variable are positive,

$$\begin{aligned} \frac{\mathrm{d}P}{\mathrm{d}t} \ge -\mathrm{d}P;~ \frac{\mathrm{d}V}{\mathrm{d}t} \ge -\mu V \end{aligned}$$

Thus, according to comparison theorem and Eqs. (21) and (22), for $\delta _1>0$ there exist $t_0>0$ such that $P(t)\ge P^* - \delta _1$, $V(t)\ge V^* - \delta _1$ for all $t>t_0$.

Now, from the first equation of the system, (4) can be written as

$$\begin{aligned} \dot{S(t)} & {} \le S(t)(r(t)-\lambda (t) (V^*-\delta )-\alpha (t) (P^*-\delta _1)),~ t\ne n\tau \nonumber \\ S(t^+)& {} = S(t),~ t= n\tau \end{aligned}$$

(31)

Integrating (31) on $[n\tau , (n+1)\tau ]$, it can be shown that

$$\begin{aligned} S\{(n+1)\tau \} & {} \le S(n\tau )\exp\int _{n\tau }^{(n+1)\tau }(r(t)-\lambda (t) (V^*-\delta _1)\nonumber \\&-\alpha (t) (P^*-\delta _1)){\rm d}t,\nonumber \\& {} = S(n\tau )\eta _1 \end{aligned}$$

(32)

Similarly,

$$\begin{aligned} S\{n\tau \} & {} \le S\{(n-1)\tau \}\eta _1. \eta _1 \end{aligned}$$

(33)

Hence, from (32) and (33),

$$\begin{aligned} S\{(n+1)\tau \} & {} \le S\{(n-1)\tau \}\eta _1^2 \end{aligned}$$

Proceeding in this way, we get

$$\begin{aligned} S\{(n+1)\tau \} & {} \le S(\tau )\eta _1^n \end{aligned}$$

(34)

Since $\eta _1<1$, hence$\eta _1^n\rightarrow 0$, whenever $n\rightarrow \infty$. Hence, $S\{(n+1)\tau \}\rightarrow 0$ as $n\rightarrow \infty$. Now, we take $n\tau <t~\le ~(n+1)\tau$ then clearly $0<S(t)~\le ~S(n\tau )~\exp~{( r~\tau )}$. Thus, $S(t)\rightarrow 0$ as $t\rightarrow \infty$. Again since

$$\begin{aligned} \eta _2=\exp\int _{n\tau }^{(n+1)\tau }(-\delta _1 -\beta (P^*-\delta _1))\mathrm{d}t<1, \end{aligned}$$

similarly we can prove that $I(t)\rightarrow 0$ as $t\rightarrow \infty$.

Now, we prove that $P(t)\rightarrow P^*(t)$ as $t\rightarrow \infty$. Since $S(t)\rightarrow 0$ and $I(t)\rightarrow 0$ as $t\rightarrow \infty$, then for some $0<\delta _2<\frac{d}{m(\alpha +\beta )}$ there exist $t_1>0$ such that $0<S(t)<\delta _2$ and $0<I(t)<\delta _2$, for all $t>t_1$. Thus, for $t>t_1$, from third equation of the system (4), we can write

$$\begin{aligned} (m \alpha \delta _2+ m \beta \delta _2 -d)P(t)\ge {\dot{P}}(t)\ge (-m \alpha \delta _2- m \beta \delta _2 -d)P(t) \end{aligned}$$

(35)

Let $\acute{P}(t)$ and $\acute{\acute{P}}(t)$ be the solution of the following equation, respectively,

$$\begin{aligned} \dot{ \acute{P}}(t)& {} = (-m \alpha \delta _2- m \beta \delta _2 -d)\acute{\acute{P}}(t), ~t\ne n\tau \\ \acute{P}(t^+)& {} = \acute{P}(t)+\rho ~t=n\tau \end{aligned}$$

and

$$\begin{aligned} \dot{ \acute{P}}(t)& {} = (m \alpha \delta _2+ m \beta \delta _2 -d)\acute{\acute{P}}(t), ~t\ne n\tau \\ \acute{P}(t^+)& {} = \acute{P}(t)+\rho ~t=n\tau \end{aligned}$$

Then, the solution will be

$$\begin{aligned} \acute{P}^*(t)& {} = \frac{\rho e^{-(m \alpha \delta _2+ m \beta \delta _2 +d) (t-n\tau )}}{1-e^{-(m \alpha \delta _2+ m \beta \delta _2 +d) \tau }},\nonumber \\ \acute{\acute{P}}^*(t)& {} = \frac{\rho e^{(m \alpha \delta _2+ m \beta \delta _2 -d) (t-n\tau )}}{1-e^{(m \alpha \delta _2+ m \beta \delta _2 -d) \tau }}, \end{aligned}$$

(36)

From (36), it is clear that when $\delta _2\rightarrow 0,$ then, $\acute{P}^*(t)\rightarrow P^*(t)$ and $\acute{\acute{P}}^*(t)\rightarrow P^*(t)$. Hence, it follows from (35) that $P(t)\rightarrow P^*(t)$ as $t\rightarrow \infty$.

Similarly, we can choose $0<\delta _3<\frac{\mu }{ n \delta }$ and in the same way, we can prove $V(t)\rightarrow V^*(t)$ as $t\rightarrow \infty$.

From Eqs. (21), (22) and (30), we can say that the system (4) is locally, as well as globally, stable if

$$\begin{aligned} r t-\frac{\lambda \nu e^{n\mu \tau }( 1-e^{-\mu t})}{\mu (1-e^{-d \tau })}-\frac{\alpha \rho e^{d n\tau } (1-e^{-d t})}{d(1-e^{-\mu \tau })}<0. \end{aligned}$$

(37)

(ii) Release predator(P) and Virus(V) at different time interval: In this case, two subcases arise.

Subcase I: Release predator(P) at time $t\mathbf{=n\tau _1}$

In this subcase, $\nu =0$, hence, the system (4) is locally stable around the periodic solution if

$$\begin{aligned} r t-\frac{\alpha \rho e^{d n\tau _1} (1-e^{-d t})}{d(1-e^{-\mu \tau _1})}<0. \end{aligned}$$

(38)

Subcase II: Release Virus(V) at time $t\mathbf{=n\tau _2}$ In this subcase, $\rho =0$, hence, the system (4) is locally stable around the periodic solution if

$$\begin{aligned} r t-\frac{\lambda \nu e^{n\mu \tau _2}( 1-e^{-\mu t})}{\mu (1-e^{-d \tau _2})}<0. \end{aligned}$$

(39)

$\square$