Differential Systems with Positive Variables

Abstract

The variables of biological, ecological, or chemical systems are often positive, because they measure concentrations, numbers, ...We study polynomial n-dimensional differential systems defined in the positive orthant. We use the tools of the usual linear algebra to exploit the specific structure of such systems, and obtain some indications on their behavior. In some cases, we are able to exhibit functions that decrease along the trajectories and therefore to give sufficient conditions for a regular global behavior: that is, all the trajectories either converge towards the equilibria or are unbounded. Our main example will be the n-dimensional Lotka-Volterra system (arising in biological modeling of species interactions). We apply the above results to stabilize the Lotka-Volterra system by controlling either the total growth rates of some species, or, alternatively, the individual growth rates of some species.