Stability Analysis Of A Class Of Stochastic Diffusive Predator - Prey System

The application of differential equation of population ecology has a very long history, which takes into account the temporal and spatial environment of life, the application of partial different equation model to study the evolution of species in space cannot be ignored as part of the research project. This paper considers the predator-prey model with cross diffusion; investigate its stability and spatial pattern. The main content of this paper consists of two parts.In the first part, we consider a three species predator-prey model which has two teams of preys and one predator. We first prove that in the absence of diffusion the positive equilibrium solution is globally asymptotically stable, and in the absence of cross diffusion reaction diffusion system is still linear stability. Further we prove that when the system has no cross diffusion the positive equilibrium solution is globally asymptotically stable, but when the cross diffusion in reaction diffusion system play a role in the system it becomes unstable, which are caused by the cross-diffusion, also known as the Turing instability. After the corresponding numerical simulation and also for us to verify the conclusions and we also get the spatial pattern of population.In the second part, we consider the prey to help each other team u1u2u3 for the original system, and discuss the stability of the system, we first discuss the situation without diffusion team. We found that the system has seven equilibrium solutions, and when the coefficient a,b,c,d,e of reaction terms satisfy certain conditions there have two positive equilibrium,E1 and E2. It was found that when the coefficients satisfy certain conditions the two solutions is locally asymptotically stable. Then we discussed the stability of the positive equilibriums in the case of only exists self diffusion term. We found that only self diffusion will not affect the stability of the system, then we discussed the situation which has cross diffusion term, we selected k31,k32 as parameters, then we found that when the coefficients satisfy certain conditions, however, E1 is always the linear stability. For E2, when k31,k32 become very large, itwill lead to linear instability. Finally we through the numerical simulation to verify our conclusion and the corresponding the spatial distribution of population.