Stability Of Semi-trivial Solutions For A Predator-prey Model In Heterogeneous Environment

In this paper,we mainly consider the dynamic behavior of the predator-prey model with Holling Type function response in spatially heterogeneous environment,as follows:that is,the effect of diffusion and spatial heterogeneity of the environment on the stability of the semi-trivial steady state of the model is studied.And compared with the homogeneous environment,the stability of the semi-trivial solution in spatially heterogeneous environment is much more complicated.In this paper,we mainly through the analysis of eigenvalues study for certain ranges of death and diffusion rates of the predator,the semi-trivial steady state of the model in spatially heterogeneous environment could change its stability multiple times as the diffusion rates of the prey varies from small to large;however,in spatially homogeneous environment the stability of the semi-trivial steady state of the model is irrelevant with the diffusion rates of the predator and prey.