Bifurcation dynamics of a plant-pest-natural enemy system in polluted environment incorporating gestation delays

Abstract

In this study, a three species plant-pest-natural enemy compartmental model incorporating gestation delays for both pests and natural enemies in a polluted environment is proposed. The boundedness and positivity properties of the model are established. Equilibria and their stability analysis are carried out for all possible steady states. The existence of Hopf bifurcation in the system is analyzed. It is established that the natural enemy free steady state $$E_2$$ is stable for specific threshold parameter values $$\tau _1\in (0,\tau _{10}^+)$$, i.e., gestation delay for pest species belongs to zero and it’s own critical value, $$\tau _{10}^+$$ and the coexisting steady state $$E^*$$ is stable for specific threshold parameter values $$\tau _1\in (0,\tau _{10}^+)$$ and $$\tau _2\in (0,\tau _{20}^+)$$, i.e., gestation delay for pest species belongs to zero and it’s own critical value, $$\tau _{10}^+$$ and gestation delay for natural enemies belongs to zero and it’s own critical value, $$\tau _{20}^+$$. If both gestation delays for pest and natural enemies, i.e., $$\tau _1, \tau _2$$ respectively cross their threshold parameter values, i.e., $$\tau _1>\tau _{10}^{+},\tau _2>\tau _{20}^{+}$$, then the system perceived oscillating behavior and Hopf bifurcation occurs in the system. The sensitivity analysis of the system at interior steady state is presented and the sensitive indices of the variables are identified. Finally, simulations are performed to support our analytic results with a distinct set of parametric values.