American Institute of Mathematical Sciences

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2009, 2009(Special): 92-100. doi: 10.3934/proc.2009.2009.92

Stability analysis and bifurcations in a diffusive predator-prey system

Leonid Braverman 1, and  Elena Braverman 2,

1. 

Athabasca University, 1 University Drive, Athabasca, AB T9S 3A3, Canada
2. 

Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4, Canada

Received  July 2008 Revised  August 2009 Published  September 2009

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: linear stability analysis, nonlinear diffusion term, Diffusive predator-prey system, Holling type II functional response, Hopf bifurcation.
Mathematics Subject Classification: Primary: 92D25 Secondary: 35K57, 35Q51, 35Q80.
Citation: Leonid Braverman, Elena Braverman. Stability analysis and bifurcations in a diffusive predator-prey system. Conference Publications, 2009, 2009 (Special) : 92-100. doi: 10.3934/proc.2009.2009.92
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