The concept of repulsivity in dynamical systems as motivated by persistence problems in population biology

Abstract

An alternative to stability analysis in population biology is proposed for cases in which persistence problems such as coexistence of species and protectedness of genetic polymorphisms are of primary interest. In order to arrive at a general mathematical description, the concept of repulsivity of certain sets with respect to dynamical systems (continuous-time as well as discrete-time) defined on metric spaces is introduced. A first basic result linking this concept to the existence of Lyapunov functions is derived in analogy to the respective results from stability theory.