This website presents the research work I did during the summer of 2005, from May 16 to August 12. There were two main components of the problem:

1. Solving a system of reaction-diffusion equations with Allee effect for the prey on a growing domain
2. Determining whether the spatial or temporal dynamics of the domain change if the domain grows.

 

 

Using a system of two reaction-diffusion equations for the predator and prey populations

u_t = u_xx + &gamma u (u - &beta) (1 - u) - uv
v_t = D v_xx + uv - &delta v

Morozov et al. show that Allee effect exhibits spatially regular, but temporally chaotic behavior for a certain range of parameters of the system. While they focus on domains with fixed size, we are interested in the interaction of predator and prey populations on growing domains. We explore the range of parameters in the system that lead to chaos in growing domains using different numerical approaches. The Fourier coefficients in a Galerkin expansion demonstrate the regular and chaotic behavior of predator and prey populations for different parameters. Using a Petrov-Galerkin finite element method we dynamically change the mesh size of the growing domain and thus avoid long computation time and numerical error, but predict accurate behavior of the predator and prey populations. We conclude that growing the domain reasonably fast does not affect the spatial and temporal behavior of the populations, and in the case of chaos adds additional parameters to be considered.