Dynamics of a Leslie-Gower predator-prey model with advection and free boundaries

This paper investigates a Leslie-Gower predator-prey model with advection and double free boundaries. Firstly, we establish the global existence and uniqueness of the solution. Subsequently, we analyze the long-time behaviors of the solution and demonstrate a spreading-vanishing dichotomy. If the initial habitat size or the expanding capacity of the boundaries is large, the invasive predator will spread throughout the entire space; otherwise, both the prey and predator will eventually vanish. When spreading occurs, two subcases arise, depending on the interaction coefficient $ c $ and the environmental support $ \delta $. If $ c\delta < 1 $, the prey and predator coexist, which we refer to as the weakly hunting case; conversely, if $ c\delta \geq 1 $, the predator survives and spreads throughout the space, while the prey vanishes, which we term the strongly hunting case. In the event of spreading, we also provide an estimate of the spreading speed. Finally, we introduce a front-fixing implicit-explicit finite difference method for the free boundary problem. Numerical simulations validate our theoretical findings and uncover some intriguing new phenomena, prompting further investigation into this and general free boundary problems.