Qualitative Properties Of Wolbachia Spreading Dynamics

The dynamics of biological models have received intensive study and it has been an important aspect in the field of nonlinear partial differential equations.In this paper,we investigate the combined effects of dispersal and spatial variations on the outcome of the Wolbachia spreading.In Chapter 1,the background and history about the related work are introduced.In Chapter 2 and 3,we study Wolbachia Spreading Dynamics with the homogeneous Neu-mann boundary condition in homogeneous environment.We consider the general diffusion and the cross-diffusion,respectively.At first,by using Maximum Principal and Integral Estimates,a priori estimates of positive steady-states of the problem is given;then the nonexistence of positive steady-states is proved by energy analysis;and finally the existence of positive steady-states is obtained by the homotopy invariance of the Leray-Schauder degree,which means Turing Pattern and the cross-diffusion can make Turing Pattern as well.In Chapter 4,we consider the spatial variation on the outcome of the Wolbachia Spreading System.With the total resources being fixed at the exactly the same level,we show that a heterogeneous distribution of resources is usually superior to its homogeneous counterpart in the presence of diffusion.