Dynamics of a two-patch logistic model with diffusion and time delay

In this paper, we proposed a two-patch logistic model connected by diffusion, where one patch includes the Gamma type distribution time delay while the other patch does not include the time delay. In general, Routh-Hurwitz criterion is applied to the derivation for the conditions of Hopf bifurcation, but the more the order of the time delay increases the more the difficulty rises. Hence we adopt the polar form method for the characteristic equation to study the stability of coexistence equilibrium. Our findings show that the diffusion prevents the instabilization of the coexistence equilibrium. Besides, we found that the coexistence equilibrium is stable when time delay is small, and becomes unstable as the delay increases. But it can be restabilized for further increasing of time delay and continues to be stable afterwards. In other words, the diffusion and the time delay are beneficial to the stability of the coexistence equilibrium.