The Hopf bifurcation in prey-predator model by considering logistical carrying capacity of prey.

This paper concerns with the dynamic behavior of a prey-predator model with Holling type II functional responses and the environmental carrying capacity of prey. The carrying capacity of prey grows logistically with the parameter η being the maximum carrying capacity of prey. We assume that predators only eat prey, where predation follows a type II Holling functional response. The local stability analysis shows that there are three equilibrium points. One equilibrium point represents both species that do not persist always unstable, the other points prey persists but the predator cannot defend and the coexistence point are stable under some certain conditions. Several numerical results are presented to illustrate the analytical findings. The dynamic behavior of the system changes quantitatively at the bifurcation point when the parameter (η) is varied, a stable equilibrium becomes unstable and vice versa. The result of numerical continuity of the parameters of maximum prey carrying capacity causes changes in the instability of the system solution so that a Hopf bifurcation occurs at the value of η = 4.13068. [ABSTRACT FROM AUTHOR]