Bivariate spline solution to a class of reaction-diffusion equations
Slavov, George Petrov
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This work presents a method of solving a time dependent partial differential equation, which arises from classic models in ecology concerned with a species’ population density over two dimensional domains. The species experiences population growth and diffuses over time due to overcrowding. Population growth is modeled using logistic growth with Allee effect. This work introduces the concept of discrete weak solution and establish theory for the existence, uniqueness and stability of the solution. Bivariate splines of arbitrary degree and smoothness across elements are used to approximate the discrete weak solution. More recent efforts focus on modeling the interaction of multiple species, which either compete for a common resource or one predates on the other. Some simulations of population development over some irregular domains are presented at the end.