Stability Switching Dynamics of a Food Chain System Incorporating Gestation Delays

Abstract

In the present study, a three species plant-pest-natural enemy model is proposed incorporating gestation delay for pests and natural enemies. The boundedness and positivity of the model are studied and analyzed. 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 coexisting steady state $$E^*$$ is stable for specific threshold parameters $$\tau _1\in (0,\tau _{10}^+)$$ , $$\tau _2\in (0,\tau _{20}^+)$$ respectively. If both $$\tau _1, \tau _2$$ crosses the threshold parameters i.e., $$\tau _1>\tau _{10}^{++},\tau _2>\tau _{20}^{++}$$ system perceived oscillating behavior and Hopf bifurcation occurs. Furthermore, the coexisting steady state $$E^*$$ is stable for specific threshold parameters $$\tau _1 \tau _{20}^{++}$$ or $$\tau _1>\tau _{10}^{++},\tau _2<\tau _{20}^{++}$$ . The sensitivity analysis of the system at interior steady state is presented and identified the sensitive indices of the variables. Finally, simulations are performed to support our analytic results with a distinct set of parametric values.