Qualitative Analysis Of Steady-State Patterns Of Some Prey-Predator Models

It is well known that, due to the crucial significance in theory and extensive applications in reality, pattern formation now becomes an important research aspect of modern science and technology. It can be used to describe the structure changes of interacting species or reactants of ecology, chemical reaction and gene formation in nature.In this dissertation, we are concerned with the pattern formation of some prey-predator models with reaction and diffusion from biology. For the problem of Neumann boundary condition, we mainly study the effects of diffusion and cross-diffusion on the pattern formation (namely, positive non-constant steady-state solutions). When the boundary condition is of Dirichlet type, we mainly investigate the exact multiplicity and stability of positive steady-state solutions (PSS), and determine the asymptotic behavior of PSS.More precisely speaking, we first consider the Holling-Tanner prey-predator model with diffusion and cross-diffusion, establish the fine upper and lower bounds of PSS and then study the existence and non-existence of non-constant PSS. As a consequence, our results show that, under some cases, both diffusion and cross-diffusion can create pattern formation.In the analysis of the non-existence of non-constant PSS of the above model, we adopt a so-called Implicit Function Theorem method, which is quite efficient in obtaining the non-existence of pattern. Moreover, we also apply this method to many biological and chemical systems, such as the well-known Brusselator model, the Noyes-Field model of Belousov-Zhabotinskii reaction and some prey-predator models with ratio-dependent functional response, and greatly improve the previous results of non-existence of pattern.In addition, for the Brusselator model, we obtain some more precise a priori estimates than that by Brown-Davidson and Erneux-Reiss et al. Then, based on this, we use the topology...