Bivariate spline solution of time dependent nonlinear PDE for a population density over irregular domains

• We study a bivariate spline numerical solution for a problem of logistic dispersal with Allee effect.
• We demonstrate the uniqueness of the bivariate spline solution.
• The results can be used to study vector dispersal in a vector-borne disease model.
• We conduct computational simulations as a proof of concept.

We study a time dependent partial differential equation (PDE) which arises from classic models in ecology involving logistic growth with Allee effect by introducing a discrete weak solution. Existence, uniqueness and stability of the discrete weak solutions are discussed. We use bivariate splines to approximate the discrete weak solution of the nonlinear PDE. A computational algorithm is designed to solve this PDE. A convergence analysis of the algorithm is presented. We present some simulations of population development over some irregular domains. Finally, we discuss applications in epidemiology and other ecological problems.