Narrow Results

We consider a predator-prey model in a multi-patch environment. We assume the existence of two time scales: the migration process takes place on the behavioural level and is thus much faster than the population dynamics. Each population is subdivided into subpopulations which correspond to the spatial distribution. The model is thus a large system of ordinary differential equations. We assume that the migration rates are fastly oscillating: it is the case for some aquatic populations for example. Indeed, these populations undergo regular vertical movements in the water column every day. In order to study our model, we use a reduction method which allows us to simplify the initial model. It is then possible to bring to light that some properties emerge from the coupling between the fast migration process and the slow population dynamics. We give an explicit example of the emerging property.

Has applied mathematics disappeared behind the computer screen? High level computer languages and high capacity computers for simulation followed by data mining give powerful tools for studying complex systems. The two examples presented here are a system of difference equations from population genetics and a system of Volterra integral equations from demography, both perturbed by random noise. We analyze these systems using averaging methods and then compare the derived solution with computer simulations of the system. In the first case, mathematical analysis guides mining of the simulated solution data base. In the second, mathematical analysis reveals deterministic chaotic behavior in the averaged system that confounds the simulation/data-mining approach.