Numerical solution of biological reaction diffusion models on fixed domains by the finite element method

Abstract

Several biological phenomena have been described using mathematical models based on reaction diffusion equations. The solution of this type of equations gives rise to formation of spatial-temporal patterns, in agreement with the biological reality of the simulated phenomenon. This article describes the numerical implementation of a set of three well-known reaction diffusion models: the morphogenesis Schnakenberg model, and the GiererMeinhardt and Thomas reaction kinetics models. The aim is to analyze the set of parameters associated with the spatial-temporal pattern formation. The numerical implementation was performed using the finite element method in one dimensional and two dimensional domains. It was concluded that spatialtemporal pattern formation in reaction diffusion models depends on the constant parameters of the model, the initial conditions and the implementation technique. The analysis of these dependences is useful in the formulation and validation of new mathematical models describing biological phenomena.

Keywords

• Reaction diffusion models
• pattern formation
• finite element method
• biomathematics