A Free Boundary Problem on 2-D Spatial Domain Exhibiting Fingering

Abstract.

We study the population dynamics of a single species modeled by a Fisher-KPP equation with an integral weighted free boundary condition in a two dimensional region. The spatial domain of the model is a rectangle with one side having a moving boundary and the remaining three sides fixed. We prove the global existence and uniqueness of solution, and show the vanishing-balancing-spreading trichotomy dynamics of the model. By numerical simulation, the steady states exhibit a finger-shaped front pattern, a long-lasting topic of interest in spatial ecology. For certain sets of parameters, the finger-shaped free boundary could vary over time.