Stability dynamics of a nonlocal interaction model with memory and Allee effect

Interaction dynamics among species yield to Allee effects, spatial memory, and nonlocal competition. When predators are not subject to the Allee effect, the coexistence equilibrium point remains locally asymptotically stable despite nonlocal competition. However, in the presence of the Allee effect, nonlocal competition leads to the destabilization of the coexistence point. Moreover, the model will undergo stability switches, Hopf bifurcation, and Turing bifurcation, and induced complex dynamics will appear. It is also found that when the memory diffusion is small, it has no effect on the stability switches; as it increases, only the nonlocal model exhibits the spatially inhomogeneous Hopf bifurcation. When it exceeds the maximum threshold, this phenomenon occurs in both models. Furthermore, when the memory diffusion coefficient exceeds the threshold, the stability range of the coexistence equilibrium will decrease in both the local and nonlocal models. Finally, numerical simulations verify the theoretical results.