1. Cross-Diffusion Driven Instability in a Predator-Prey System with Cross-Diffusion
- Author
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Maria Carmela Lombardo, Marco Sammartino, Eleonora Tulumello, Tulumello, E, Lombardo, MC, and Sammartino, M
- Subjects
Wavefront ,Work (thermodynamics) ,Partial differential equation ,Ginzburg-Landau equation ,Applied Mathematics ,Nonlinear diffusion ,Turing instability ,Mathematical analysis ,FOS: Physical sciences ,Pattern formation ,Pattern Formation and Solitons (nlin.PS) ,Mechanics ,Nonlinear Sciences - Pattern Formation and Solitons ,Instability ,Nonlinear system ,Amplitude ,Quintic Stuart-Landau equation ,Quantitative Biology::Populations and Evolution ,Amplitude equation ,Settore MAT/07 - Fisica Matematica ,Marginal stability ,Mathematics - Abstract
In this work we investigate the process of pattern formation induced by nonlinear diffusion in a reaction-diffusion system with Lotka-Volterra predator-prey kinetics. We show that the cross-diffusion term is responsible of the destabilizing mechanism that leads to the emergence of spatial patterns. Near marginal stability we perform a weakly nonlinear analysis to predict the amplitude and the form of the pattern, deriving the Stuart-Landau amplitude equations. Moreover, in a large portion of the subcritical zone, numerical simulations show the emergence of oscillating patterns, which cannot be predicted by the weakly nonlinear analysis. Finally when the pattern invades the domain as a travelling wavefront, we derive the Ginzburg-Landau amplitude equation which is able to describe the shape and the speed of the wave., Comment: 15 pages, 5 figures
- Published
- 2014