Dynamical Behaviors of Four-Dimensional Prey-Predator Model

In this paper, we study the dynamic behavior of a four-dimensional prey-predator suggested model of four species. The four species are two prey and two predator species, each of them grows logistically. The two prey live in diverse habitats and have the ability of group defense. In the mentioned model, one predator feeds on the two prey, the top predator feeds on other three species. The existents and, the boundedness of the positive solution, the existence and the local stability of all possible equilibrium points, of the model are investigated. The model has seven equilibrium points at most, four of them always exist and the others exist under certain conditions. Three equilibrium points are not stable while the others are locally asymptotically stable, under given conditions. For the coexistence point, a basin of attraction for it has been found. The steady-state bifurcation relative to the mortality rate of the predators in the neighborhood of three of the equilibrium points and the Hopf-bifurcation relative to the growth rate of the prey in the neighborhood of two of the equilibrium points has been found. Finally, two numerical example has been given to support the theoretical results.