Hopf and Turing bifurcation for a competition and cooperation system with spatial diffusion effect

Populations diffuse in the spatial domain for better habitat and compete for enriching resources for survival and evolution. Some behaviors or chemicals from the species’ competitive interactions will occur and have feedback to affect their growth. In this work, we construct a competitive system with diffusion and stimulatory chemicals to investigate the dynamics of competitive species. We first study the existence of the positive steady-state and obtain the related conditions. Furthermore, we analyze the effect of diffusion and stimulating substances on the dynamics of the species concentration. Without diffusion effect, the stimulatory chemicals will contribute to the oscillation of the species concentration if the production rate of the chemicals is greater than the critical value. The diffusion phenomenon in the system causes Turing instability, leading to the non-homogeneous spatial distribution of the species concentration. Finally, several numerical examples are provided to validate the theoretical conclusions in this work.