# American Institute of Mathematical Sciences

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July  2021, 26(7): 3491-3504. doi: 10.3934/dcdsb.2020242

## Moran process and Wright-Fisher process favor low variability

 Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, 1015 Floyd Avenue, Richmond, VA 23284-2014, USA

* Corresponding author: Jan Rychtář

Received  October 2019 Revised  May 2020 Published  August 2020

We study evolutionary dynamics in finite populations. We assume the individuals are one of two competing genotypes, $A$ or $B$. The genotypes have the same average fitness but different variances and/or third central moments. We focus on two frequency-independent stochastic processes: (1) Wright-Fisher process and (2) Moran process. Both processes have two absorbing states corresponding to homogeneous populations of all $A$ or all $B$. Despite the fact that types $A$ and $B$ have the same average fitness, both stochastic dynamics differ from a random drift. In both processes, the selection favors $A$ replacing $B$ and opposes $B$ replacing $A$ if the fitness variance for $A$ is smaller than the fitness variance for $B$. In the case the variances are equal, the selection favors $A$ replacing $B$ and opposes $B$ replacing $A$ if the third central moment of $A$ is larger than the third central moment of $B$. We show that these results extend to structured populations and other dynamics where the selection acts at birth. We also demonstrate that the selection favors a larger variance in fitness if the selection acts at death.

Citation: Jan Rychtář, Dewey T. Taylor. Moran process and Wright-Fisher process favor low variability. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3491-3504. doi: 10.3934/dcdsb.2020242
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Fixation probabilities for the Wright-Fisher process (left) and the Moran process (right). Darker areas correspond to higher fixation probabilities; $0.1$ corresponds to a random drift. For $N = 10$, we considered genotypes $G_i; i = 1, \ldots, 11$ that have a fitness $i+1$ with probability $1/i$ and $1$ with probability $1-1/i$. The average fitness of $G_i$ is $2$, the variance is $i-1$. For every pair of $i,j$, we run $10^5$ simulations starting with a single $G_i$ individual among $G_j$ individuals. Note that the individuals' fitness is sometimes more than double the expected value
Left: A graphical representation of the genotypes $G_p$ for $p\in\{0.1, 0.2, \ldots, 0.9\}$. A fitness of $G_p$ is $\mu-x_p$ with probability $p$ and $\mu+y_p$ with probability $1-p$ where $\mu = 4$, $x_p = (1-p)y_p/p$ and $y_p = \sqrt{((1-p)^2/p + (1-p))^{-1}}$. The average fitness of $G_i$ is $\mu$ represented by the horizontal dotted line, the variance is $1$. The genotypes are color coded by $p$. The center of a disc corresponds to the fitness, the area of a disc corresponds to the probability of attaining such a fitness. The thick black curve is the third central moment $s_p$ of the genotype $G_p$ with values of the right $y$ axis. Right: Fixation probabilities for the Moran process. The darker the color, the larger the fixation probability; $0.2$ corresponds to a random drift. For $N = 5$ and every pair of $p_i,p_j$, we run $10^7$ simulations starting with a single $G_{p_i}$ individual among $G_{p_j}$ individuals
Fixation probabilities for the Db-Moran process. At every step, an individual is selected to die with the probability inversely proportional to their fitness and is then replaced by a copy of a randomly selected remaining individuals. For $N = 10$, we considered genotypes $G_i; i = 1, \ldots, 11$ as in Figure 1; a fitness of $G_i$ is $i+1$ with probability $1/i$ and $1$ with probability $1-1/i$. The average fitness of $G_i$ is $2$, the variance is $i-1$. For every pair of $i,j$, we run $10^5$ simulations starting with a single $G_i$ individual among $G_j$ individuals
Errors in approximation of $1/y$ by Taylor polynomials $P_n(y)$ at $y_0$. The approximation by a higher degree polynomial is better only on $(0,2y_0)$ and worse on $(2y_0,\infty)$
The mean square errors (MSE) of the estimate of $\mathbb{E}[\pi_A'|\pi_A]-\pi_A$ by $E_2$ (left) and by $E_3$ (right). The darker the color, the larger the error. For $N = 10$, we considered genotypes $G_i; i = 1, \ldots, 21$; a fitness of genotype $G_i$ is $i+1$ with probability $1/i$ and $1$ with probability $1-1/i$. The average fitness is $2$, the variance is $i-1$. For every $\pi_A\in\{1/N, \ldots, (N-1)/N\}$, we run $10^6$ simulations to estimate $\mathbb{E}[\pi_A'|\pi_A]-\pi_A$ numerically by the average different $\widehat{E}$. We then calculated the MSE as $\frac{1}{N-1}\sum_{\pi_A\in\left \{\frac1N, \ldots, \frac{N-1}{N}\right\}} \left(\widehat{E}-E_2\right)^2$
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