Aggregation, emergence and immergence in hierarchically organized systems

The aim of this work is to show that couplings between fast micro-dynamics and slow macro-dynamics can make emerge global properties. Emergence corresponds to a bottom-up coupling, that is to the effect exerted by a micro-level at a macro-level. Immergence corresponds to the inverse process, that is to an up-down coupling. As an example, we present a prey-predator model 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 growth and interactions between the populations. Preys go on spatial patches where resources are located and where predators can attack them. The efficiency of predators is patch-dependent. Perturbation theory is used in order to aggregate the initial system of ordinary differential equations for patch sub-populations into a macro-system of two differential equations governing total populations. First, we study a case of density-independent migrations, for which no emergence occurs. Then, we study some examples of density-dependent migrations. In this case, emerging properties appear at the population level.