In this paper, we propose a spatiotemporal prey-predator model with fear and Allee effects. We first establish the global existence of solution in time and provide some sufficient conditions for the existence of non-negative spatially homogeneous equilibria. Then, we study the stability and bifurcation for the non-negative equilibria and explore the bifurcation diagram, which revealed that the Allee effect and fear factor can induce complex bifurcation scenario. We discuss that large Allee effect-driven Turing instability and pattern transition for the considered system with the Holling-Ⅰ type functional response, and how small Allee effect stabilizes the system in nature. Finally, numerical simulations illustrate the effectiveness of theoretical results. The main contribution of this work is to discover that the Allee effect can induce both codimension-one bifurcations (transcritical, saddle-node, Hopf, Turing) and codimension-two bifurcations (cusp, Bogdanov-Takens and Turing-Hopf) in a spatiotemporal predator-prey model with a fear factor. In addition, we observe that the circular rings pattern loses its stability, and transitions to the coldspot and stripe pattern in Hopf region or the Turing-Hopf region for a special choice of initial condition.
Keywords: bifurcation theory; center manifold theory; eaction-diffusion system; fear and Allee effect; normal form; spatiotemporal pattern transition.