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Stability analysis and bifurcations in a diffusive predator-prey system

Pages: 92 - 100, Issue Special, September 2009

 Abstract        Full Text (162.6K)              

Leonid Braverman - Athabasca University, 1 University Drive, Athabasca, AB T9S 3A3, Canada (email)
Elena Braverman - Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4, Canada (email)

Abstract: We consider a predator-prey system with logistic-type growth and linear diffusion for the prey, Holling type II functional response and the nonlinear diffusion $\nabla \left( \sigma n b \nabla b)$ for the predator, where $n$ is the prey (nutrient) and $b$ is the predator (bacteria) density, respectively. This corresponds to a collective-type behavior for predators: they spread faster when numerous enough at a front line. We present the complete linear stability analysis for this case, discuss some results of numerical simulations: the asymptotic behavior of the model (with the zero Neumann boundary conditions in a 2-D domain) was similar to the relevant Lotka-Volterra system of ordinary differential equations.

Keywords:  Diffusive predator-prey system, nonlinear diffusion term, Holling type II functional response, linear stability analysis, Hopf bifurcation
Mathematics Subject Classification:  Primary: 92D25 Secondary: 35K57, 35Q51, 35Q80

Received: July 2008;      Revised: August 2009;      Published: September 2009.