Event Detail

Event Type: 
Thursday, June 5, 2014 - 02:00
Willamette West, Valley Library

Speaker Info

OSU Mathematics

In this work we consider a model for pattern-producing vegetation in semi-arid regions of the world proposed by Klausmeier. It is a coupled nonlinear diffusion-advection evolutionary PDE system describing the vegetation density and water amount. The model was studied extensively by J.Sherratt who took into consideration the field data for the parameter values and considered the complexity of the regions of stability. In this thesis we study numerical approximations to the Klausmeier system.

First we consider a simplified zero-dimensional nonlinear ODE system similar to the Klausmeier model. The parameter-dependent equilibria of the system change qualitative behavior as the rainfall varies with respect to the plant loss and plant dispersion coefficients. Three parameter sets are chosen to be descriptive of the system's different states. We consider several Finite Difference schemes to approximate the solutions to this ODE system. These include explicit and various implicit combinations of discretization for the nonlinear term. Initial conditions are chosen to be small random perturbations around the equilibria states for each of the parameter sets. Suitable restrictions on the choice of the time step are considered and catalogued.

Next we work with the original PDE system to discern asymptotic pattern formation. Using the same range of methods as in the ODE case, we discretize the system in space and time and apply periodic boundary conditions along with the realistic parameter values. Conclusive evidence of banded vegetation patterns is obtained.