Stationary Stability for Evolutionary Dynamics in Finite Populations

Harper, Marc; Fryer, Dashiell E. A.

doi:10.3390/e18090316

Cited by 11 publications

(11 citation statements)

References 72 publications

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“…Our first goal is to establish the random trajectory entropy (RTE) of a state as a measure of stability of the state. We are particularly concerned with the local and global extrema of the stationary distribution, shown in [ 1 ] to have a close connection with evolutionary stability.…”

Section: Resultsmentioning

confidence: 99%

“…For the Moran process with mutation, we use a special case of the formulation [ 1 ]; see also [ 8 , 9 , 10 ]. Let a population be composed of n types

of size N with

individuals of type

so that

.…”

Section: Resultsmentioning

confidence: 99%

“…Define a matrix of mutations M where

may be a function of the population state for our most general results, but we will typically assume in examples that for some constant value

, the mutation matrix takes the form

for

and

. A typical mutation rate is

[ 1 , 14 ].…”

Section: Resultsmentioning

confidence: 99%

“…For convenience, fix a population size

and N even, and let the rate of mutation be

. Then, we have that

is the unique stationary maximum [ 1 ]. As

increases, the stationary probability at the maximum increases more quickly than the entropy rate (which is not monotonic in this case).…”

Section: Resultsmentioning

confidence: 99%

“…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. Specifically, 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 ].…”

Section: Introductionmentioning

confidence: 99%

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Entropic Equilibria Selection of Stationary Extrema in Finite Populations

Harper

Fryer

2018

Entropy

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We propose the entropy of random Markov trajectories originating and terminating at a 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.

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Section: Resultsmentioning

confidence: 99%

“…For the Moran process with mutation, we use a special case of the formulation [ 1 ]; see also [ 8 , 9 , 10 ]. Let a population be composed of n types

of size N with

individuals of type

so that

.…”

Section: Resultsmentioning

confidence: 99%

“…Define a matrix of mutations M where

may be a function of the population state for our most general results, but we will typically assume in examples that for some constant value

, the mutation matrix takes the form

for

and

. A typical mutation rate is

[ 1 , 14 ].…”

Section: Resultsmentioning

confidence: 99%

“…For convenience, fix a population size

and N even, and let the rate of mutation be

. Then, we have that

is the unique stationary maximum [ 1 ]. As

increases, the stationary probability at the maximum increases more quickly than the entropy rate (which is not monotonic in this case).…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Entropic Equilibria Selection of Stationary Extrema in Finite Populations

Harper

Fryer

2018

Entropy

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The volunteer’s dilemma in finite populations

Konrad

Morath

2021

J Evol Econ

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We study the long-run stochastic stability properties of volunteering strategies in finite populations. We allow for mixed strategies, characterized by the probability that a player may not volunteer. A pairwise comparison of evolutionary strategies shows that the strategy with a lower probability of volunteering is advantaged. However, in the long run there are also populations of volunteering types. Monomorphisms with the more volunteering types are more frequent if the populations have fewer members, and if the benefits from volunteering are larger. Such monomorphisms with volunteering cease to exist if the population becomes infinitely large. In contrast, the disadvantage of volunteering disappears if the ratio of individual benefits and costs of volunteering becomes infinitely large.

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