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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

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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