Qualitative Analysis of Turing Bifurcation in a Diffusion Population Model With Environmental Toxin Threat and Allee Effect

ABSTRACT In this paper, the spatiotemporal pattern dynamics of a diffusion population model with Allee effect and environmental toxins is considered. Firstly, the boundedness and positivity of the system are proved, the existence and stability of the equilibrium are analyzed, and the conditions of the occurrence of Turing bifurcation are deduced. Subsequently, the mechanism for the selection of patterns around the positive equilibrium was provided through weakly nonlinear analysis. One finds that weak Allee effect can restore the reduction of population density and undergoes the bistable phenomena. Environmental toxins can lead to Hopf bifurcation and saddle‐node bifurcation, and change the topology of the system. Self‐diffusion can induce Turing bifurcation, and the hexagonal hole patterns, mixed patterns of holes and stripes, and striped patterns are shown. Finally, theoretical results are validated through numerical simulations. These findings provide a scientific basis for population conservation strategies under the threat of environmental toxins.