A Mixed Lyapunov-Max-Plus Algebra Approach to the Stability Problem for a two Species Ecosystem Modeled with Timed Petri Nets

Consider the interaction of populations, in which there are exactly two species, one of which the predators eats the other the prey thereby affecting each other , s growth rates. In the study of this growth rate dynamics Lotka-Volterra models have been used. This paper proposes a new modeling and analysis methodology which consists in considering the growth rate dynamics as a discrete event system. A discrete event system, is a dynamical system whose state evolves in time by the occurrence of events at possibly irregular time intervals. Place-transitions Petri nets (commonly called Petri nets) are a graphical and mathematical modeling tool applicable to discrete event systems in order to represent its states evolution. Timed Petri nets are an extension of Petri nets that model discrete event systems where now the timing at which the state changes is taken into consideration. One of the most important performance issues to be considered in a predator-pray system is its stability. Lyapunov stability theory provides the required tools needed to aboard the stability problem for the predator-prey system treated as a discrete event system modeled with timed petri nets. Employing Lyapunov methods, a sufficient condition for the stabilization problem is obtained. It is shown that it is possible to restrict the discrete event systems , state space in such a way that boundedness is guaranteed. By proving boundedness one confirms a dominant oscillating behavior of both populations dynamics performance. However, the oscillating frequency results to be unknown. This inconvenience is overcome by considering a specific recurrence equation, in the max-plus algebra, which is assigned to the timed Petri net graph.