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2006, 3(1): 189-204. doi: 10.3934/mbe.2006.3.189

A nonlinear $L^2$-stability analysis for two-species population dynamics with dispersal

Salvatore Rionero 1,

1. 

University of Naples Federico II, Department of Mathematics and Applications ''R. Caccioppoli", Complesso Universitario Monte S. Angelo. Via Cinzia, 80126 Napoli, Italy

Received  December 2004 Revised  April 2005 Published  November 2005

The nonlinear $L^2$-stability (instability) of the equilibrium states of two-species population dynamics with dispersal is studied. The obtained results are based on (i) the rigorous reduction of the $L^2$-nonlinear stability to the stability of the zero solution of a linear binary system of ODEs and (ii) the introduction of a particular Liapunov functional V such that the sign of $\frac{dV}{dt}$ along the solutions is linked directly to the eigenvalues of the linear problem.
Keywords: Liapunov direct method, nonlinear stability, reaction diffusion equations., two-species population dynamics.
Mathematics Subject Classification: 70K20, 76E30, 34D20, 37B25, 70K15, 93D30, 92A15,35K5.
Citation: Salvatore Rionero. A nonlinear $L^2$-stability analysis for two-species population dynamics with dispersal. Mathematical Biosciences & Engineering, 2006, 3 (1) : 189-204. doi: 10.3934/mbe.2006.3.189
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