Abstract
We propose the entropy of random Markov trajectories originating and terminating at the same state as a measure of the stability of a state of a Markov process. These entropies can be computed in terms of the entropy rates and stationary distributions of Markov processes. We apply this definition of stability to local maxima and minima of the stationary distribution of the Moran process with mutation and show that variations in population size, mutation rate, and strength of selection all affect the stability of the stationary extrema.
Highlights
This work is motivated by the stationary stability theorem [1], which characterizes local maxima and minima of the stationary distribution of the Moran process with mutation in terms of evolutionary stability
The stationary distribution of a Markov process gives the probability that the process will be in each state in the long run [7]
Variations of fundamental evolutionary parameters alter the stability of equilibria, agreeing with intuitive expectations
Summary
This work is motivated by the stationary stability theorem [1], which characterizes local maxima and minima of the stationary distribution of the Moran process with mutation in terms of evolutionary stability. The theorem says that for sufficiently large populations, the local maxima and minima of the stationary distribution satisfy a selective-mutative equilibria criterion that generalizes the celebrated notion of evolutionary stability [2]. This means that the stationary distribution encodes the usual information about evolutionary stability. We propose the random trajectory entropy (RTE) of paths originating and terminating at a state as a measure of stability of the state [3,4] This is an information-theoretic quantity that is computable from the entropy rate and stationary distribution of a process, and varies continuously with the critical evolutionary parameters (as does the stationary distribution). We will see that RTE captures the behavior of the Moran process with mutation intuitively, leading to a simple method for equilibrium selection for finite populations—generally a significant problem in evolutionary game theory [5,6]
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