I spent several months in 2006 working on this project at IPFW. Results have been published in the August 2008 issue of the International Journal of Bifurcation and Chaos:
Bifurcations, and Temporal and Spatial Patterns of a Modified Lotka-Volterra Model
Edward A. McGehee, Noel Schutt, Desiderio A. Vasquez, Enrique Peacock-López
My 2006 Indiana Academy of Science abstract:
Pattern Formation in a Reaction-diffusion System.
Noel R. Schutt and Desiderio A. Vasquez, Department of Physics, IPFW, Fort Wayne IN
Patterns arise in chemical, physical, ecological, and biological systems due to the interaction of reaction and diffusion. The reaction presents a stable steady state, which loose stability due to different diffusivities of chemical or biological species. In this work, we analyze numerically the two-dimensional patterns arising in a modified Lotka-Volterra model, which applies to ecological systems. We carry out a linear stability analysis for steady solutions of the reaction-diffusion system. We also solve numerically the nonlinear equations using a five-point finite difference approximation, together with an implicit Euler method for time evolution. The reaction mechanism alone (without diffusion) allows for two stable steady states. Introducing diffusion, one of the states looses stability allowing for pattern formation, while the other provides a stable homogeneous state. The interaction between the two states allows for the formation of complex patterns. This leads to the development of patterns with different wavelengths depending on the initial conditions. We also report localized patterns of very few wavelengths, confined inside a non-patterned state.
- BSU (PowerPoint)
This animation gallery shows the development Turing patterns for a representative selection parameters used in this research. Each run has two animations: X is the prey or activator; Y is the predator or inhibitor. Colors represent population density. The frames are snapshots of the system taken at intervals of 2000 time steps. Note the different ranges used in the plots.
All files are in QuickTime format.