Perturbation problem for the indefinite nonlocal periodic-parabolic equation

In this paper, we study the perturbation problem for the periodic logistic equation with indefinite weight functions and nonlocal dispersal. Our aim is to find the precisely asymptotic behavior of positive solutions when the perturbation (dispersal rate) is large or small. We establish that the positive solution uniformly tends to the maximum nonnegative solution of the kinetic equation as the dispersal rate goes to zero. However, the positive solution uniformly converges to the maximum nonnegative solution of kinetic equation, which is independent of spatial locations when the dispersal rate goes to infinity. The main results reveal that large dispersal rate corresponding to pure kinetic equation, whereas small dispersal rate corresponding to kinetic equation with parameters.