A Predator - Prey Model Analysis Solution

A predator-prey model with Beddington-DeAngelis and modified Leslie-Grower functional responses is studied. Using the bifurcation theory, the maximum principle, the perturbation theory and the stability theory, we discuss the existence of the coexistence, stabil-ity, uniqueness of positive steady and the longtime behavior of species. The main contents in this thesis are as follows.In chapter1, some properties of the positive equilibria of the above system are investigated. Firstly, the global structure of the coexistence solutions is obtained with the help of the bifurcation theory. The results show that the continuum of nontrivial solution is bounded and joins two branches of semi-trivial solutions with the parameter a∈(λ1,λ1+a2/k). For a≥λ1+a2/k, this bifurcation branch of coexistence solutions goes to infinity with the parameter b (see figure1). The local stability of the coexistence is also given. Secondly, the bifurcation solution from a double eigenvalue and its stability are studied by the Liapunov-Schmidt method. Lastly, the existence and uniqueness of positive solutions are obtained in one dimensional case.In chapter2, we determine the longtime behavior of the system. Asymptotic behavior of nonnegative solutions is given with the help of the super-sub solution method and the stability theory. What’s more, several sufficient conditions of the uniform persistence are obtained. It is shown that predator and prey cannot coexist when the parameter a or b is less than λ1.For the parameter a> λ1(a2θb/1+kθb) and b> λ1, the coexistence of the parabolic systems is uniform persistence. Finally, we present some numerical simulations to verify and complement our theoretical analysis with the help of Matlab software.