Qualitative Analysis Of A Prey-Predator Model With Stage Structure And Diffusion

In this paper, we consider a nonlinear dynamics near an unstable positive constant equilib-rium for the prey-predator system with cross diffusive and stage structure in predator where Td=(0,π)d (d=1,2,3). The paper is divided into six sections,In Section1, the locally asymptotical stability and the globally asymptotical stability of non-negative constant steady states for ODE system are given by using linearization and Lyapunov functions.In Section2, the locally asymptotical stability and the globally asymptotical stability of non-negative constant steady states for reaction-diffusion system are given by using linearization and Lyapunov functions.In Section3, the locally asymptotical stability of positive constant steady state for cross diffusion system (1) is given by using linearization.In Section4, the instability condition of the corresponding linearized system is found by any general initial perturbation near the positive constant equilibrium, and L2esti-mate of the solution to the linearized system (4) with instability criterion and initial condition is found by using partial differential equation theory, Kramer law and Grownwall inequality.In Section5, the Bootstrap Lemma is proved by using Sobolev embedding theorem, Poincare inequality and Young inequality.In Section6, the main result of this paper is proved. For the prey-predator system with cross diffusive and stage structure in predator, it is proved that given any general perturbation of magnitude δ, its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of the order In1/δ.