Bivariate spline solution to a class of reaction-diffusion equations

This work presents a method of solving a time dependent partial differential equation,which arises from classic models in ecology concerned with a species populationdensity over two dimensional domains. The species experiences population growthand diffuses over time due to overcrowding. Population growth is modeled using logisticgrowth with Allee effect. This work introduces the concept of discrete weaksolution and establish theory for the existence, uniqueness and stability of the solution.Bivariate splines of arbitrary degree and smoothness across elements are usedto approximate the discrete weak solution. More recent efforts focus on modelingthe interaction of multiple species, which either compete for a common resource orone predates on the other. Some simulations of population development over someirregular domains are presented at the end.