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

Sebastién Gaucel; Michel Langlais

doi:10.3934/dcdsb.2007.8.61

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2007, Volume 8, Issue 1: 61-72. Doi: 10.3934/dcdsb.2007.8.61

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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
2.
Université Victor Segalen Bordeaux 2, case 26, UMR CNRS 5251 IMB & INRIA Futurs Anubis, 146, rue Léo Saignat, 33076 Bordeaux Cedex

Received: August 31, 2006

Revised: October 31, 2006

Published: December 2006

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Abstract

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

Mathematics Subject Classification: Primary: 35K57, 35B05; Secondary: 34D23 92D25.

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