With The Asymptotic Nature Of The Prey-taxis Of Prey Model

Mathematical ecology is a relatively independent science. Ecology produces lots of problems, mathematics provides models and ways to understand them. There are two kinds of continuous models in mathematical ecology. One is the ordinary differential equation (ODE), and the other is the partial differential equation with diffusion (PDE). Since there exists diffusion in the PDE, a little interesting change happened under certain conditions of the two different models. One of the great examples is Turing instability phenomenon, which was claimed in 1952 in the paper"The chemical basis of morphogenesis"by Alan. M. Turing—one of the most important and influential thinkers of the twentieth century.Turing suggested that, in the absence of the diffusion, the two basic chemicals tend to a linearly stable uniform steady state, while, under certain conditions, the uniform steady state can become unstable, and spatial inhomogeneous patterns can evolve through bifurcations. In the other word, under certain conditions, a constant equilibrium solution can be asymptotically stable with the kinetic equation, but it is unstable with its corresponding reaction-diffusion system. Over the years, Turing's ideas have attracted much more attention and successfully developed on the many scientific backgrounds, such as chemistry, physics, biology, medical,mathematical and so on.This dissertation is devoted to a Lotka-Volterra predator-prey model with prey-taxis. The model is based on the assumption that the populations follow the simple Lotka-Volterra interaction, the spatial and temporal variations of the predator velocity are determined by the prey gradient.The objective of this dissertation is to investigate the long time behaviors of the solution to the coupled reaction-diffusion system. We show that free diffusion and cross-diffusion play important roles in the pattern formation. Moreover, the value of the taxis coefficient can induce instability.In the first part of this dissertation, the background and history about the related work of the Turing instability are introduced. In section 2, a Lotka-Volterra predator-prey model with prey-taxis is first established. Prey-taxis is thus defined as the movement of predators as controlled by prey density.Chapter 2 deals with stability and instability caused by free diffusion, cross-diffusion and also prey-taxis. The local asymptotic stabilities around each of the equilibrium are discussed by using the characteristic decomposition and linearization. Furthermore, the cross-diffusion and prey-taxis are introduced to discuss the stability of the system at the equilibrium point, and the conditions for the instability caused by cross-diffusion coefficient and prey-taxis are given. Our results show that cross-diffusion and prey-taxis can induce the instability of an equilibrium which is stable for the system with free diffusion.Finally the corresponding numerical simulations are given by Matlab to illustrate the main result..