Effects of diffusion and delay on the dynamic behavior of a model of two cooperative species

This work considers the dynamic behavior of a diffusive cooperative two species model with a time delay. Unlike those studied previously, the system incorporates the impacts of intrinsic growth rates and diffusion in addition to the time delay. We first apply the Galerkin method to the theoretical DDE equations. Then, we explore theoretically the conditions that promote the existence of Hopf bifurcation curves. In addition, we establish and plot full maps for the stability area and Hopf curves. We consider three different cases: one with two different diffusion rates, one with a single diffusion rate, and one with no diffusion. Two different stability zones are found. Further, the influence of the delay time on all other parameters in the model is examined; this can notably impact the stability zone and bifurcations. Moreover, an asymptotic analysis of the periodic outcome near the Hopf point is explored and compared with the analytical solutions. Lastly, we substantiate the validity of our theoretical findings through an illustrative numerical simulation.