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. • The impact of different cases of diffusion rates with a time delay is discussed. • Theoretical results are examined by applying the Galerkin approach. • The stability region is plotted, and the impacts of delay cases on other free rates are determined. • Bifurcation maps are constructed for both the DDE and DPDE models. • An asymptotic analysis of the periodic outcome close to the Hopf spot is explored.