Finite time and global existence for solutions to some singular reaction-diffusion systems

S. Gaucel , Michel Langlais (1, 2)

1 ANUBIS - Tools of automatic control for scientific computing, Models and Methods in Biomathematics
2 IMB - Institut de Mathématiques de Bordeaux

This note is dedicated to the question of global existence for so- lutions 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 co- exist. 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.

Mots clés: global existence; blow–up time; oscillations; singular reaction–diffusion systems; predator–prey model in insular environment; invasion and persistence.; invasion and persistence

Domaines: Equations aux dérivées partielles [math.AP]; Ecosystèmes

S. Gaucel, Michel Langlais. Finite time and global existence for solutions to some singular reaction-diffusion systems. Discrete and Continuous Dynamical Systems - Series B, 2007, 8, pp.61-72. ⟨hal-00195450⟩