Abstract

Aggregation methods allow one to replace a large scale dynamical system (micro-system) by a reduced dynamical system (macro-system) governing a small number of global variables. This aggregation of variables can be performed when two time scales exist, a fast time scale and a slow time scale. Perturbation theory allows to obtain an approximated aggregated dynamical system which describes the behaviour of a few number of slow time varying variables which are constants of motion of the fast part of the micro-system. Aggregation methods are applied to the case of the devastation of the great barrier reef by the starfishes. We recall the Antonelli/Kazarinoff model which implies a stable limit cycle for the corals and starfish populations. This prey-predator model describes the interactions between two species of corals and the starfish. Then, we generalize the Antonelli/Kazarinoff model to the case of two spatial patches with a fast part describing the starfish migration on the patches and the human manipulation of the communities by divers and, a slow part describing the growth and the interactions between the populations. We obtain an aggregated model governing the total coral densities on the patches and the total starfish population. This model can exhibit stable limit cycle oscillations and a Hopf bifurcation. The critical value of the bifurcation parameter is expressed in terms of the proportions of coral species and starfish on the two patches. This implies for example that rather than random killing of starfish by the Australian military, it may be better to send teams of divers to outbreaking reefs when they first occur who will then manipulate the community structure to increase protection