Stability and Bionomic Analysis of Fuzzy Prey–Predator Harvesting Model in Presence of Toxicity: A Dynamic Approach

Abstract

This paper deals with a prey–predator model in which both the species are infected by some toxicants which are released by some other species or source with fuzzy biological parameters. The application of fuzzy differential equation in the modeling of prey–predator populations with the effect of toxicants is presented. The dynamical behavior and harvesting of the fuzzy exploited system are studied by using the utility function method. Sufficient conditions for the local stability of the positive equilibrium are obtained by analyzing the characteristic equation. Furthermore, the possibility of the existence of bionomic equilibrium is studied under imprecise biological parameters. The study of the presence of toxic substance and harvesting in the modeling system can have significant impact on the existence of both the species, which is in line with reality. Numerical simulation results are presented to validate the theoretical analysis.

Appendices

Appendix 1: Proof of Theorem 1

The characteristic equation of the variational matrix at $E_{0}\left( 0,0\right)$ is given by $\left( a_{1}-q_{1}E-\lambda \right) \left( -b_{1}-q_{2}E-\lambda \right) =0$ where $\lambda$ denotes the eigenvalue. The roots of this equation are $\lambda _{1}=a_{1}-q_{1}E$ and $\lambda _{2}=-\left( b_{1}+q_{2}E\right) <0$. Now $\lambda _{1}<0$ if $E>\frac{a_{1}}{q_{1}}$ i.e., $E>w_{1}(\hbox {BTP})_{x_{1l}}+w_{2}(\hbox {BTP})_{x_{1r}}$ hence the steady state $E_{0}\left( 0,0\right)$ is a stable node. Consequently, if $E<w_{1}(\hbox {BTP})_{x_{1l}}+w_{2}(\hbox {BTP})_{x_{1r}}$  then $\lambda _{1}>0$ and $\lambda _{2}<0$; therefore, the steady state $E_{0}\left( 0,0\right)$ is a saddle point.

Appendix 2: Proof of Theorem 2

For the steady state $E_{1}\left( \overline{x}_{1},0\right)$, the characteristic equation is $\left( -a_{2}\overline{x}_{1}-2\gamma _{1} \overline{x}_{1}^{2}-\lambda \right) \left( -b_{1}+b_{2}\overline{x} _{1}-q_{2}E-\lambda \right) =0$. The corresponding eigenvalues are obtained as $\lambda _{1}=-\left( a_{2}\overline{x}_{1}+2\gamma _{1}\overline{x} _{1}^{2}\right) <0$ and $\lambda _{2}=-\left( b_{1}-b_{2}\overline{x} _{1}+q_{2}E\right)$. Here, $\lambda _{1}<0$, hence the steady state $E_{1}\left( \overline{x}_{1},0\right)$ is stable if $\lambda _{2}<0$, i.e., $b_{2}\overline{x}_{1}<b_{1}+q_{2}E\Rightarrow \left( a_{1}-q_{1}E\right) <\left( b_{1}+q_{2}E\right) \left\{ \frac{\gamma _{1}\left( b_{1}+q_{2}E\right) }{b_{2}^{2}}+\frac{a_{2}}{b_{2}}\right\}$.

Appendix 3: Proof of Theorem 3

The characteristic equation of the variational matrix at $\ E_{2}\left( x_{1}^{*},x_{2}^{*}\right)$ is given by

$$\begin{aligned} \lambda ^{2}+A\lambda +B=0 \end{aligned}$$

(20)

where $A=a_{2}x_{1}^{*}+2\gamma _{1}x_{1}^{*2}+\gamma _{2}x_{2}^{*}, B=\left( a_{2}x_{1}^{*}+2\gamma _{1}x_{1}^{*}\right) \gamma _{2}x_{2}^{*}+a_{3}b_{2}x_{1}^{*}x_{2}^{*}$. Sum of the roots of Eq. (20) is $-A<0$  and the product of the roots of the equation is $B>0$ provided $x_{1}^{*}$and $x_{2}^{*}$ are positive. Therefore, the roots of the quadratic Eq. (20) are real negative or complex conjugates with negative real parts. Hence, the steady state $E_{2}\left( x_{1}^{*},x_{2}^{*}\right)$ is a locally stable node or focus if $a_{3}q_{2}>\gamma _{2}q_{1}$ and $\left( b_{1}+q_{2}E\right) \left\{ \gamma _{1}\left( b_{1}+q_{2}E\right) +a_{2}b_{2}\right\} <b_{2}^{2}\left( a_{1}-q_{1}E\right)$.

In the absence of toxicity, i.e., $\gamma _{1}=\gamma _{2}=0$ then $A=a_{2}x_{1}^{*}>0$ and $B=a_{3}b_{2}x_{1}^{*}x_{2}^{*}>0$ provided both $x_{1}^{*}$ and $x_{2}^{*}$ are positive. Hence, the steady state $E_{2}( x_{1}^{*},x_{2}^{*})$ is either a locally stable node or focus. Therefore, the local stability of the system is not directly dependent on the intensities of the toxicant, provided both $x_{1}^{*}$ and $x_{2}^{*}$ are positive, which needs to satisfy $a_{3}q_{2}>\gamma _{2}q_{1}$ and $( b_{1}+q_{2}E) \{ \gamma _{1}( b_{1}+q_{2}E) +a_{2}b_{2}\} <b_{2}^{2}( a_{1}-q_{1}E) ,$ i.e., it depends on the imprecise nature of the biological parameters. Again, if the effect of the toxicity is increased, then both the species will decline and become extinct, and this changes the stability of the system.

Appendix 4: Proof of Theorem 4

Let us consider the Lyapunov function

$$\begin{aligned} V\left( x_{1},x_{2}\right) =x_{1}-x_{1}^{*}-x_{1}^{*}\ln \left( \frac{x_{1}}{x_{1}^{*}}\right) +l\left\{ x_{2}-x_{2}^{*}-x_{2}^{*}\ln \left( \frac{x_{2}}{x_{2}^{*}}\right) \right\} \end{aligned}$$

(21)

where l is the suitable constant determined in the subsequent steps. Also, the function (21) vanishes at the equilibrium point $E_{2}\left( x_{1}^{*},x_{2}^{*}\right)$. The time derivative of V along the trajectories of (6) is $\frac{\hbox {d}V}{\hbox {d}t}=\frac{\left( x_{1}-x_{1}^{*}\right) }{x_{1}}\frac{\hbox {d}x_{1}}{\hbox {d}t}+l\frac{\left( x_{2}-x_{2}^{*}\right) }{x_{2}}\frac{\hbox {d}x_{2}}{\hbox {d}t}=\left( x_{1}-x_{1}^{*}\right) \left\{ a_{1}-a_{2}x_{1}-a_{3}x_{2}-q_{1}E-\gamma _{1}x_{1}^{2}\right\} +l\left( x_{2}-x_{2}^{*}\right) \left( -b_{1}+b_{2}x_{1}-q_{2}E-\gamma _{2}x_{2}\right)$. Now using equilibrium equations we get $\frac{\hbox {d}V}{\hbox {d}t}=-a_{2}\left( x_{1}-x_{1}^{*}\right) ^{2}-a_{2}\left( x_{1}-x_{1}^{*}\right) \left( x_{2}-x_{2}^{*}\right) -\gamma _{1}\left( x_{1}-x_{1}^{*}\right) ^{2}\left( x_{1}+x_{1}^{*}\right) +lb_{2}\left( x_{1}-x_{1}^{*}\right) \left( x_{2}-x_{2}^{*}\right) -l\gamma _{2}\left( x_{2}-x_{2}^{*}\right) ^{2}$. If we choose $l=\frac{a_{2}}{b_{2}}$ we get

$$\begin{aligned} \frac{\hbox {d}V}{\hbox {d}t}=-\left[ \left( x_{1}-x_{1}^{*}\right) ^{2}\left\{ a_{2}+\gamma _{1}\left( x_{1}+x_{1}^{*}\right) \right\} +\frac{ a_{2}\gamma _{2}}{b_{2}}\left( x_{2}-x_{2}^{*}\right) ^{2}\right] <0. \end{aligned}$$

Since $\frac{\hbox {d}V}{\hbox {d}t}$ is negative semidefinite in some neighborhood of $\left( x_{1}^{*},x_{2}^{*}\right)$, the equilibrium point $E_{2}\left( x_{1}^{*},x_{2}^{*}\right)$ is globally and asymptotically stable.