The Stabilizing Role of Mutation; or, Towards understanding Network Topologies under Evolutionary Dynamics

It is well-known that the (multi-population) replicator dynamics (RD) is sensitive to assumptions such as infinite populations or that the system evolves in continuous time, with qualitatively different results if these assumptions are violated.
I briefly recapitulate some previous work, where we have shown that RD is also highly sensitive to the assumption that mutation occurs on a different time scale from selection, complementing earlier results in the literature. We have proposed the concept of mutation limits as a way of relating the equilibria of RD to a class of replicator-mutator dynamics (RMD) where mutation can have arbitrary (non-uniform) directions.
Building on these results on mutation limits, I present some recent results showing that mutation stabilizes RD in two-player constant-sum games leading to asymptotic stability of the associated equilibria.
This stability allows a stochastic discrete-time system, in this case a simple "reinforcement" dynamics, to be approximated in probability by RMD such that we can infer the long-term dynamics of the stochastic system, which in general is not possible for RD.
Thus, mutation creates a bridge between the deterministic RMD and stochastic discrete-time systems, showing that the assumption of separate time-scales for selection and mutation, while being useful, induces qualitative changes possibly rendering the analysis less applicable to biological systems.
I close with some illustrations on the range of our results, sketching what they mean for networked populations and the evolution of network topologies in certain systems.

If you would like to participate, please contact Ursula Krützfeldt for link and password (kruetzfeldt@evolbio.mpg.de).