Effect Of Cross-Diffusion On Three Types Of Predator-Prey Models

In this paper,effect of cross-diffusion on three types of predator-prey models are studid.The main works are summarized as follows:(1)In the first chapter,the effect of cross-diffusion on an Ivlev-type predator-prey model with protection zone and the homogeneous Neumann boundary condi-tions is considered,where the cross-diffusion represents the tendency of prey to keep away from its predator.Firstly,the stability of the nonnegative constant steady state solutions is analyzed by the linearization method.Secondly,a priori estimates of positive steady state solutions are given by applying maximum principle and the nonexistence of nonconstant positive steady state solutions is discussed.Finally,The existence of coexistence solutions is obtained by using the bifurcation theory.As a result,it is shown that the cross-diffusion is beneficial for species coexistence.(2)In the second chapter,the effect of cross-diffusion on Beddington-DeAngelis type predator-prey model with protection zone and the homogeneous Neumann boundary conditions is considered,where the cross-diffusion represents the tenden-cy of prey to keep away from its predator.Firstly,the stability of the nonnegative constant steady state solutions is analyzed by the linearization method.Secondly,a priori estimates of positive steady state solutions is given by applying maximum principle.Finally,the existence of coexistence solutions is discussed by using bifur-cation theory.(3)In the third chapter,the effect of cross-diffusion on ratio-dependent type predator-prey model under the homogeneous Dirichlet boundary condition,where two cross-diffusion rates represent the tendency of prey to keep away from its preda-for and the tendency of the predator to chase its prey,respectively.Applying the degree theory and the fixed point index theory,some sufficient conditions for the existence of positive solutions are established.Furthermore,the non-existence of positive steady-state solutions are studied.It is shown that the large cross-diffusion coefficients tend to mean no positive coexistence.