Study On Spatial Pattern Dynamics Of Several Types Of Ecological Models With Self-diusion And Cross-diusion

As one of the most important branches of nonlinear science, the study on the dynamics of the Turing pattern is very extensive and rich. There are a lot of mech-anisms to produce the Turing pattern, the simplest one is reaction diffusion system.It was first proposed by Turing in 1952 in the article of " the chemical basis of mor-phogenesis", which is often referred as Turing instability or instability caused by diffusion. Simply speaking, it is because the diffusion makes the equilibrium become unstable. By constructing the mathematical model with population dynamics, the dynamic morphological analysis can be used to explain the spatial pattern formed by the interaction between populations. At the same time, combined with the results of numerical simulation, it shows that the transition from population to time-space chaos can explain the persistence, extinction and evolution of population in spaceThis paper will use the linear analysis theory, Lyapunov function method and Routh-Hurwitz criterion and multi-scale analysis method, to study three kinds of predator-prey with reaction diffusion model, the following is the main content of the paper:In the first part, Pattern formation and selection in a diffusive predator-prey system with ratio-dependent functional response is studied. The Turing space is obtained by linearization analysis, the amplitude equation of the system is derived by the multi-scale analysis method and the pattern selection is carried out. Selecting the appropriate parameters to get the including the spot, stripe and coexistence patterns in the Turing space.In the second part, based on the hunting of the wild animals in nature, a general two-dimensional model with negative cross diffusion coefficient is studied, moreover,the predator-prey model with ratio-dependent has been studied theoretically and nu-merically, the results show that the negative cross-diffusion coefficient (-d21) affects the formation and selection of Turing pattern. In the case of other parameters fixed,d2i must be less than a critical value, the system will appear Turing instability.In the third part, we consider a strong coupled cross-diffusion system about a three-species food chain model. We first prove that the unique positive equilibrium so-lution is global asymptotic stable for the ODE system and remains global asymptotic stable when the system without cross-diffusion by constructing Lyapunov functions.However, when cross-diffusion is introduce, the positive equilibrium becomes unsta-ble. Then we use the Routh-Hurwitz criterion and Descartes' rule to prove that large cross-diffusion coefficient (k2i or k32 is large enough) can cause the equilibrium to become unstable from the original stability, when the cross-diffusion plays a role in this reaction-diffusion system. Finally, numerical simulations are performed to test our theoretical results by means of Matlab. We can obtain different types of Turing patterns including spotted, striped and coexistence patterns.