Monitoring in a Lotka-Volterra model

Abstract

The problem of monitoring arises when in an ecosystem, in particular in a system of several populations, observing some components, we want to recover the state of the whole system in function of time. Due to the difficulty to construct exactly this state process, we look for an auxiliary system called observer, the solution of which reproduces this process with certain approximation. This means, that the solution of the observer tends to that of the original system. For this work an important concept is observability which means, that from the observation it is possible to recover the state process in a unique way, however without determining a constructive method to obtain it. If observability holds for the original system, it guarantees the existence of an auxiliary matrix which makes it possible to construct an observer of the system. The considered system of populations is described by the classical Lotka-Volterra model with one predator and two preys and the construction of its observer is illustrated with a numerical example. Finally, it is shown how the observer can be used for the estimation of the level of an abiotic effect on the population system.