A hybrid model of population dynamics with refuge-regime: regularization and self-organization

A mathematical model of the dynamics of the predator and prey populations in the form of a hybrid dynamical system consisting of two two-dimensional systems switching between each other is proposed. Switching of the systems allows us to reproduce a special refuge-regime when the prey number is very small and predators have complications to find preys. The sliding modes are studied using Filippov approach. Regularization of the system by using two switching lines to avoid chattering is provided. For the regularized model the limit sets are established. A scenario of the system self-organization preventing the unbounded populations' growth is proposed. A sensitivity study is carried out with respect to a parameter defining the switching lines. An important result of the research is that sufficiently small changing of the switching lines does not change the qualitative behavior of the system.