The effect of the defensive strategy taken by the prey on predator-prey interaction

In this paper, we scrutinize a delayed predator-model for the purpose of studying the impact of the strategy considered by the prey population on the evolution of the studied species. The main presumption is to consider the existence of two prey types with different behaviors. The first, has a social behavior and takes the advantage of living in group for the purpose of defending each others, and the second one has a solitary behavior. A mathematical approach is used to study this effect, where the local stability and bifurcation analysis are examined. Indeed, we establish that the system under consideration has a rich dynamics such as Hopf bifurcation in both the absence and the presence of time lags. Further the stability of the periodic solution generated by the presence of the time lags are discussed using the normal form. Some numerical simulations are provided for ensuring the obtained mathematical results.