# Measures of success in a class of evolutionary models with fixed population size and structure

## Abstract

We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the Price equation and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth–death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.

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## Abbreviations

$$B$$ :

Total (population-wide) expected offspring number

$$b_i$$ :

Expected offspring number of individual $$i$$

$$d_i$$ :

Death probability of individual $$i$$

$$\mathcal{E }$$ :

Evolutionary process

$$e_{ij}$$ :

Edge weight from $$i$$ to $$j$$ in the BD process on graphs

$$f_0(x), \, f_1(x)$$ :

Reproductive rates of types 0 and 1, respectively, in the frequency-dependent Moran process

$$\mathcal M _\mathcal{E }$$ :

Evolutionary Markov chain

$$N$$ :

Population size

$$p_{\mathbf{s}\rightarrow \mathbf{s}^{\prime }}$$ :

Probability of transition from state $$\mathbf{s}$$ to state $$\mathbf{s}^{\prime }$$ in $$\mathcal M _\mathcal{E }$$

$$p^{(n)}_{\mathbf{s}\rightarrow \mathbf{s}^{\prime }}$$ :

Probability of $$n$$-step transition from state $$\mathbf{s}$$ to state $$\mathbf{s}^{\prime }$$ in $$\mathcal M _\mathcal{E }$$

$$r$$ :

Reproductive rate of type 1 in the BD process on graphs

$$R$$ :

Set of replaced positions in a replacement event

$$\mathfrak{R }$$ :

Replacement rule

$$s_i$$ :

Type of individual $$i$$

$$\mathbf{s}$$ :

Vector of types occupying each position; state of $$\mathcal M _\mathcal{E }$$

$$u$$ :

Mutation rate

$$w_i$$ :

Fitness of individual $$i$$

$$\bar{w}^1, \bar{w}^0, \bar{w}$$ :

Average fitness of types 1 and 0, and of the whole population, respectively

$$x_1, x_0$$ :

Frequencies of types 1 and 0, respectively

$$\alpha$$ :

Offspring-to-parent map in a replacement event

$$\Delta ^\mathrm{sel }x_1$$ :

Expected change due to selection in the frequency of type 1

$$\pi _\mathbf{s}$$ :

Probability of state $$\mathbf{s}$$ in the mutation-selection stationary distribution

$$\pi ^*_\mathbf{s}$$ :

Probability of state $$\mathbf{s}$$ in the rare-mutation dimorphic distribution

$$\rho _1, \rho _0$$ :

Fixation probabilities of types 1 and 0

$$\langle \; \rangle$$ :

Expectation over the mutation-selection stationary distribution

$$\langle \; \rangle ^*$$ :

Expectation over the rare-mutation dimorphic distribution

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## Acknowledgments

The authors thank Martin A. Nowak, our editor Mats Gyllenberg, and an anonymous referee for helpful conversations and suggestions. B.A. is supported by the Foundational Questions in Evolutionary Biology initiative of the John Templeton Foundation.

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Correspondence to Benjamin Allen.

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Allen, B., Tarnita, C.E. Measures of success in a class of evolutionary models with fixed population size and structure. J. Math. Biol. 68, 109–143 (2014). https://doi.org/10.1007/s00285-012-0622-x

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### Keywords

• Evolution
• Stochastic
• Axioms
• Fixation probability
• Evolutionary success
• Price equation

### Mathematics Subject Classification (2000)

• 92D15
• Problems related to evolution