Transient and asymptotic dynamics of a prey-predator system with diffusion

In this paper, we study a prey-predator system associated with the classical Lotka-Volterra nonlinearity. We show that the dynamics of the system are controlled by the ODE part. First, we show that the solution becomes spatially homogeneous and is subject to the ODE part as t → ∞. Next, we take the shadow system to approximate the original system as D → ∞. The asymptotics of the shadow system are also controlled by those of the ODE. The transient dynamics of the original system approaches to the dynamics of its ODE part with the initial mean as D → ∞. Although the asymptotic dynamics of the original system are also controlled by the ODE, the time periods of these ODE solutions may be different. Concerning this property, we have that any δ > 0 admits D ₀ > 0 such that if TÌ, the time period of the ODE, satisfies TÌ>δ, then the solution to the original system with D ≥ D ₀ cannot approach the stationary state.