Qualitative stability patterns for Lotka-Volterra systems on rectangles

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

Box invariance of hybrid and switched systems

Abstract: This paper investigates the concept of box invariance for classes of hybrid and switched systems. After motivating and defining the notion, we present a concise summary of results on its characterization for single-domain dynamical systems. The notion is then extended to the case of hybrid and switched systems. We provide sufficient conditions for a hybrid or switched system to be box invariant. Models of many real systems, especially those drawn from biology, have been found to be box invariant. This paper illustrates the concept using a pharmacodynamic model of blood glucose metabolism. Copyright © 2006 IFA