Optimal harvesting and complex dynamics in a delayed eco-epidemiological model with weak Allee effects

Abstract

In this article, an eco-epidemiological system with weak Allee effect and harvesting in prey population is discussed by a system of delay differential equations. The delay parameter regarding the time lag corresponds to the predator gestation period. Mathematical features such as uniform persistence, permanence, stability, Hopf bifurcation at the interior equilibrium point of the system is analyzed and verified by numerical simulations. Bistability between different equilibrium points is properly discussed. The chaotic behaviors of the system are recognized through bifurcation diagram, Poincare section and maximum Lyapunov exponent. Our simulation results suggest that for increasing the delay parameter, the system undergoes chaotic oscillation via period doubling. We also observe a quasi-periodicity route to chaos and complex dynamics with respect to Allee parameter; such behavior can be subdued by the strength of the Allee effect and harvesting effort through period-halving bifurcation. To find out the optimal harvesting policy for the time delay model, we consider the profit earned by harvesting of both the prey populations. The effect of Allee and gestation delay on optimal harvesting policy is also discussed.

Appendix

We can write the first equation of the system (4) as

$$\begin{aligned} \frac{\mathrm{d}S}{\mathrm{d}t}= & {} S\left[ \left( 1-S-I\right) \frac{S}{S+\theta } - \beta I -q_1 E\right] . \end{aligned}$$

So,

$$\begin{aligned}&\frac{\mathrm{d}S}{\mathrm{d}t} \le S(1-S),\\&\therefore \lim \sup _{t\rightarrow \infty }S(t)\le 1. \end{aligned}$$

Let $V_1 = S + I$, taking its time derivative along the solution of the system (4), we have

$$\begin{aligned} \dot{V_1}= & {} S(1{-}S{-}I)\frac{S}{S{+}\theta } - aPI - \mu I{-}q_1 E S -q_2 E I, \\\le & {} 1 - S - I.\\&\quad \therefore \dot{V_1} + V_1 \le 1 \Rightarrow \lim _{t\rightarrow \infty } V_1(t) \le 1. \end{aligned}$$

So,

$$\begin{aligned} \lim _{t\rightarrow \infty } I(t) \le 1. \end{aligned}$$

Define a function $V_2 = \frac{1}{a}I(t-\tau ) + \frac{1}{\alpha }P(t)$, taking its time derivative along the solution of the system (4), we have

$$\begin{aligned} \dot{V_2}\le & {} \beta S(t-\tau )I(t-\tau ) - \frac{\mu }{a} I(t-\tau )-\frac{d}{\alpha }P(t), \\\le & {} \beta -\min \{\mu ,d\}V_2.\\&\quad \therefore \lim _{t\rightarrow \infty } P(t) \le M, \end{aligned}$$

where $M=\frac{\beta \alpha }{\min \{\mu ,d\}}.$ Hence, proposition follows.