Bifurcation of the diffusive Bazykin’s model with nonlocal competition and delay

In ecological modeling, the interactions between individuals and resource consumption occur not only in the immediate area but also in a wider spatial region; such interactions are often described as nonlocal interactions. The incorporation of delay introduces a temporal element into the interaction. In this paper, we study the dynamics of populations by incorporating nonlocal prey competition and delay effects into the diffusive Bazykin model. Through rigorous mathematical analysis we find the emergence of Hopf bifurcation and Turing bifurcation. When the time delay exceeds the critical delay threshold a Hopf bifurcation occurs in the local competition model and a homogeneous spatially periodic solution appears. Meanwhile, the nonlocal competition can cause Turing bifurcation and Hopf bifurcation, leading to inhomogeneous spatially periodic solutions. The stability of the equilibrium point is affected by time delay and nonlocal effect and induces inhomogeneous spatially periodic oscillations. The induced Hopf bifurcation is also analyzed using the normal form theory and the center manifold theorem. Finally, we use numerical simulations to illustrate the theoretical results.