With Diffusion Of Predator - Prey Model Coexistence Solution

The theory of differential equation developed at the end of 17th century, almost at the same time when the calculation and integal generated. Now differential equation becomes strongly tool of the study natural phenomenon.In this paper, we use the theory of differential equation to explain the number variety of creature population.The two-species predator-prey system is discussed,and the study of content,way and result is as follow:In the frist chapter, the condition that the two-species predator-prey system has no non-constant solution, and the HARNACK inequality of the corresponding elliptic system are given. In section 2, the effects of reaction are discussed.In section 3, effects of diffusion are discussed. Our results show that the model hao no non-constant solution if the intrinsic contend are active or the diffusion coffecient are strong.In the second chapter, the two-species predator-prey system is discussed. The dynamics of coupled systems of semilinear parabolic equations are investigated using the method of upper and lower solutions. The asymptotic behavior of the solutio is given. It is shown that the unique and positive solution is the local stability if the natural growth rate a is big and bs-da<0.In the third chapter, the two-species predator-prey system is discussed. The global stability of ODE and PDE systems are investigated using the method of charateristic subspace decomposition and Lyapunov function.lt is shown that the global stability is not relation to diffusion coefficient of species.