Dynamical transition and bifurcation of a diffusive predator-prey model with an Allee effect on prey

We study the stability and dynamic transitions of a diffusive predator-prey model with Allee effect on prey in a two-dimensional domain. Our tool includes the recently developed dynamic transition theory by Ma and Wang (2019). We first verify the principle of exchange of stability(PES), then we obtain rigorously the nonlinear transition behavior of the reaction-diffusion system. For single eigenvalue crossing, global transitions where solutions jump away from origin are also discovered along with local pitch-fork and Hopf bifurcations. We also investigates the rarely studied double eigenvalue crossing case which shows more complex bifurcations. All these dynamic behaviors are classified according to a nondimensional transition number and detailed orbital changes are illustrated. Further numerical studies are included to reflect the theoretical results and display intricate relationships between solution patterns and parameters, among which the distinct role of the Allee effect is analyzed in detail.& COPY; 2023 Elsevier B.V. All rights reserved.