Modeling a Fast-Slow Bitrophic Food Chain with Harvesting

In the present paper, the Rosenzweig-MacArthur predator-prey model (RM), which is a bitrophic food chain model, is considered. We develop the model by adding two assumptions. First, we assume that both species are of economic interest, that is can be harvested. Second, we assume that each specie has its own time scale which range from fast for the prey to slow for the predator. We consider that both the death rate and the catch of the predator are very small which leads to a fast-slow dynamical system. That is, the RM model is transformed into a singular perturbed system with a perturbation parameter epsilon is an element of [0,1]. The existence and stability of equilibria are discussed for epsilon > 0. The model experiences both transcritical and Hopf bifurcations for epsilon > 0. The singular perturbation model at epsilon = 0 is discussed by separating the system into two subsystems; fast and slow and studying them simultaneously. When 0 < epsilon << 1, the model is discussed using geometric singular perturbation techniques. The solution of the model is approximated on the slow manifold and the numerical simulations give very good results for epsilon = 0.005.