Rich dynamics of a food chain model with ratio-dependent type III functional responses

Abstract

This paper deals the dynamics of a tri trophic food chain model with ratio-dependent type III functional response. The investigations that are presented in this paper focus on the computation of food chain with and without time delay. Two types of discrete time delay in top level predator population are considered. In first type time delay may be regarded as a delay due to reaction time or gestation period of the top predator. In second type, delay may introduce in reaction term of top predator population and it assumes that the change rate of predator depends upon the number of prey and of the number of the predators present in some previous time. In absence of delay, the conditions for boundedness of the system are established. Stability analysis of model is carried out by using usual theory of ordinary differential equation. Further, it is proved that the system undergoes Hopf bifurcations, using delay as a bifurcating parameter. We have also shown that Hopf bifurcation may also occur when delay passes its critical value. Finally, our study shows that time delay plays a significant role on the stability of the system. It breaks the stable behaviour of model and drives it to unstable state.

Keywords: Food chain model, Ratio-dependent, Boundedness, Stability, Time delay, Hopf bifurcation