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RESEARCH PRODUCT

EMERGING PROPERTIES IN POPULATION DYNAMICS WITH DIFFERENT TIME SCALES

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The aim of this work is to show that at the population level, emerging properties may occur as a result of the coupling between the fast micro-dynamics and the slow macrodynamics. We studied a prey-predator system with different time scales in a heterogeneous environment. A fast time scale is associated to the migration process on spatial patches and a slow time scale is associated to the growth and the interactions between the species. Preys go on the spatial patches on which some resources are located and can be caught by the predators on them. The efficiency of the predators to catch preys is patch-dependent. Preys can be more easily caught on some spatial patches than others. Perturbation theory is used in order to aggregate the initial system of ordinary differential equations for the patch sub-populations into a macro-system of two differential equations governing the total populations. Firstly, we study the case of a linear process of migration for which the aggregated system is formally identical to the slow part of the full system. Then, we study an example of a nonlinear process of migration. We show that under these conditions emerging properties appear at the population level.