Some remarks on a singular reaction-diffusion system arising in predator-prey modeling

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July 2007, 8(1): 61-72. doi: 10.3934/dcdsb.2007.8.61

Some remarks on a singular reaction-diffusion system arising in predator-prey modeling

Sebastién Gaucel ^1, and Michel Langlais ^2,

1.	Unité MIA MathRisq, INRA, Domaine de Vilvert, F-78352 Jouy-en-Josas cedex, France
2.	Université Victor Segalen Bordeaux 2, case 26, UMR CNRS 5251 IMB & INRIA Futurs Anubis, 146, rue Léo Saignat, 33076 Bordeaux Cedex, France

Received September 2006 Revised November 2006 Published April 2007

Abstract
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This note is dedicated to the question of global existence for solutions to a two component singular system of reaction-diffusion equations modeling predator-prey interactions in insular environments. Depending on a 2D parameter space, positive orbits of the underlying ODE system undergo interesting dynamics, e.g., finite time existence and global existence may coexist. These results are partially extended to the reaction-diffusion system in the case of identical diffusivities. Our analysis relies on an auxiliary non singular reaction-diffusion system whose solutions may or may not blow up in finite time. Numerical simulations illustrate our analysis, including a numerical evidence of spatio-temporal oscillations.

Keywords: Global existence, oscillations, predator-prey model in insular environment, invasion and persistence., blow-up time, singular reaction-diffusion systems.

Mathematics Subject Classification: Primary: 35K57, 35B05; Secondary: 34D23 92D2.

Citation: Sebastién Gaucel, Michel Langlais. Some remarks on a singular reaction-diffusion system arising in predator-prey modeling. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 61-72. doi: 10.3934/dcdsb.2007.8.61

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