Spatiotemporal Dynamics Induced by Additive Allee Effect and Herd Behavior in a Diffusive Leslie-Type Model

This paper investigates a spatial Leslie-type predator–prey model incorporating herd behavior and additive Allee effect. We systematically analyze both local and spatial dynamical behavior, establishing conditions for the existence and stability of equilibria. We demonstrate how the Allee effect regulates dynamical transitions, with the local system generating saddle-node and Hopf bifurcations. For the spatial system, we analyze the existence of solutions and a priori estimates, derive Hopf and Turing bifurcation conditions, and compute normal forms to determine the bifurcation property. Furthermore, we provide numerical examples to validate and expand the theoretical results. Our findings reveal that additive Allee effect of appropriate intensity, particularly weak Allee effect, is more likely to induce periodic oscillations in the system. For positive initial densities, the long-term dynamics either lead to species coexistence or, once the prey density drops below the Allee threshold, trigger a cascading collapse resulting in the joint extinction of both populations.