Qualitative stability patterns for Lotka-Volterra systems on rectangles

Laurent Tournier (1) , Jean-Luc Gouzé (1)

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We present a qualitative analysis of the Lotka-Volterra differential equation within rectangles that are transverse with respect to the flow. In similar way to existing works on affine systems (and positively invariant rectangles), we consider here nonlinear Lotka-Volterra n-dimensional equation, in rectangles with any kind of tranverse patterns. We give necessary and sufficient conditions for the existence of symmetrically transverse rectangles (containing the positive equilibrium), giving notably the method to build such rectangles. We also analyse the stability of the equilibrium thanks to this transverse pattern. We finally propose an analysis of the dynamical behavior inside a rectangle containing the positive equilibrium, based on Lyapunov stability theory. More particularly, we make use of Lyapunov-like functions, built upon vector norms. This work is a first step towards a qualitative abstraction and simulation of Lotka-Volterra systems.

Mots clés: Dynamical systems; Lotka-Volterra equation; Transverse rectangle; Qualitative analysis; Vector norms Lyapunov functions

Domaines: Systèmes dynamiques [math.DS]

Laurent Tournier, Jean-Luc Gouzé. Qualitative stability patterns for Lotka-Volterra systems on rectangles. [Research Report] RR-6346, INRIA. 2007, pp.16. ⟨inria-00186247v2⟩