Stability of the solutions of the mesoscopic equation that corresponds to the replicator equation.

The replicator equation is a deterministic nonlinear equation arising in evolutionary game theory describing the evolving lifeforms in terms of frequencies of strategies. It is related to a mean field approach and therefore it has a macroscopic character: the description is referred to the frequencies (densities) of agents playing the corresponding strategies.  However, the macroscopic approach is not sufficient to describe the dynamics of complex living systems by reducing  the complexity of the overall systems. In some applications to consider the agents as discrete interacting units is important in order to capture the complexity of (biological) phenomena.

We propose a class of kinetic type equations that describe the replicator  dynamics at the mesoscopic level. Under suitable assumptions we show the asymptotic (exponential) stability of the solutions to such kinetic equations in the case when the corresponding macroscopic equation is asymptotically stable. To obtain the mesoscopic model corresponding to the replicator equation we follow the techniques developed by N. Bellomo with coautors applying tools of the kinetic theory of active particles for complex living systems.

In perspective, the obtained results could be used for analysing the asymptotic behaviour in time of mathematical model which describes tumour-immune system competition.