Dynamical Analysis On Some Predator-Prey Systems With Reaction-Diffusion

Dynamical analysis on predator-prey systems has been an important branch of biomathematics, and in which systems with reaction-diffusion have received more attention by mathematicians and biologists. In particular, there is difference in energy conversion among different organisms. Hence, dynamics on predator-prey systems with diffusion and different functional responses is another important subject in biomathematics. In this paper, we analyze four predator-prey systems with diffusion and different functional responses.Firstly, we discuss a diffusive predator-prey system in Holling Type III with a prey refuge. By constructing the Lyapunov functional, we give a sufficient condition to ensure that the positive equilibrium is globally asymptotically stable. By regarding prey refuge as parameter, we study the existence of Hopf bifurcation. By the theory of normal form and center manifold, an algorithm for determining the direction and stability of Hopf bifurcation is derived. Finally, we summarize the results and discuss the effect of prey refuge on dynamic properties of the system.Secondly, we discuss a diffusive predator-prey system with nonconstant death rate and Holling III functional response. This model takes into account the internal pressure of the predator, that is the mortality rate is an increasing function about the number of predator. After eigenvalue analysis, we give a sufficient condition to ensure that the positive equilibrium is locally asymptotically stabile. We also discuss Turing instability of the positive equilibrium caused by diffusion. Finally, we discuss the existence and property of Hopf bifurcation.In addition, we consider a diffusive predator-prey system with Beddington-De Angelis and modified Leslie-Gower functional response. Beddington-De Angelis functional response can reflect the effect of the interaction among predators on pery. This model also assumes that predators will switch to other resources if their favorable food source is scarce, that is environmental carrying capacity depends upon their favorable food and other resources. We discuss this model from the view of bifurcation, including existence of Turing and Hopf bifurcation, and properties of Hopf bifurcation.We finally consider the effect of delay on a diffusive predator-prey system. By eigenvalue analysis, we gives conditions to ensure that the positive equilibrium is locally asymptotically stable. By using delay as bifurcation parameter, we give a condition of existence of Hopf bifurcation at the positive equilibrium. Some explicit formulas for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived by applying the normal form and center manifold theories.