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. In this paper, we study the dynamical behavior of a species which inhabits two independent habitat patches. Due to the long range foraging behavior, frequent transfers happen between two patches with an exponentially decaying nonlinear transfer rate. Periodic oscillation is observed as a Hopf bifurcation occurs at some critical values of the delay τ . By applying the center manifold theorem, the Poincaré normal form and the approximate periodic solution near the critical delay values are obtained. The complete synchronization of variations of the population size of species in two patches is analyzed and numerical simulations under various parametric conditions are illustrated. The moment stability of the solution of the stochastic delay equation is also considered by applying the Itô integral.

A two-patch ecological system with nonlinear transfer rate and noise effect