Self-organized populations interacting under pursuit-evasion dynamics

• We introduce a hierarchy of models for interacting preys–predators populations.
• We discuss on numerical grounds the formation of remarkable patterns.
• We analyze a model of the hierarchy and prove global existence of solutions.

We discuss the modeling of interacting populations through pursuit-evasion–or attraction–repulsion–principles : preys try to escape chasers, chasers are attracted by the presence of preys. We construct a hierarchy of models, ranging from ODEs systems with finite numbers of individuals of each population, to hydrodynamic systems. First-order macroscopic models look like generalized “two-species Keller–Segel equations”. But, due to cross-interactions, we can show that the system does not exhibit any blow up phenomena in finite time. We also obtain second-order models, that have the form of systems of balance laws, derived from kinetic models. We bring out a few remarkable features of the models based either on mathematical analysis or numerical simulations.