Authors: Claude Lobry ^1,2; Tewfik Sari ^3,4,5

• 1 Université Nice Sophia Antipolis (1965 - 2019)
• 2 Modelling and Optimisation of the Dynamics of Ecosystems with MICro-organisme
• 3 Laboratoire de Mathématiques Informatique et Applications
• 4 Information – Technologies – Analyse Environnementale – Procédés Agricoles
• 5 Laboratoire de Mathématiques Informatique et Applications [UHA]

The Rosenzweig-MacArthur model is a system of two ODEs used in population dynamics to modelize the predator-prey relationship. For certain values of the parameters the differential system exhibits a unique stable limit cycle. When the dynamics of the prey is faster than the dynamics of the predator, during oscillations along the limit cycle, the density of preys take so small values that it cannot modelize any actual population. This phenomenon is known as the "atto-fox" problem. In this paper we assume that the populations are living in two patches and are able to migrate from one patch to another. We give conditions for which the migration can prevent the density of prey being too small.