Dynamics and Bifurcations in a Dynamical System of a Predator-Prey Type with Nonmonotonic Response Function and Time-Periodic Variation

Abstract

We study a two dimensional system of ordinary differential equations of a predator-prey type. We use the Holling type IV functional response which models the group defence mechanism. For this system we discuss the number of equilibria in the system and prove it using a geometrical approach. Using the classical Lagrange Multiplier method, we compute fold and cusp bifurcations for equilibrium in the system. As we turn on to numerics, we compute the other bifurcations for equilibrium, namely Hopf bifurcations, and homoclinic bifurcations. As for bifurcation of periodic solution we compute the Fold of Limit Cycle bifurcation. We also include time-periodic variation in the system which translates most of the bifurcation sets for equilibria into bifurcation sets for periodic solutions. Furthermore, we found the swallowtail bifurcation for periodic solution in the system.