Persistence of a discrete reaction-diffusion predator-prey system

The T-periodic reaction-diffusion predator-prey model is considered: ut = d1(t)Δu+u[a(t,x)-b(t,x)u-c(t,x)v], vt = d2(t)Δv+v[-e(t,x)+g(t,x)u-f(t,x)v], ∂u/∂v = 0 on [, ∞]×∂D, where D is the bounded region in Rn with smooth boundary, d1,d2 are smooth strictly positive T-periodic functions, a,b,c,e,f,g are smooth functions on R×D which are T-periodic in t,b,c,e,f,g are positive and ∂/∂v denotes differentiation in the direction of the outward normal. The model has a positive T-periodic solution if the principal eigenvalue of the operator M(v) = vt-d2(t)Δv-[g(t,x)U(t,x)-e(t,x)]v is negative.