Moran-type bounds for the fixation probability in a frequency-dependent Wright–Fisher model

Abstract

We study stochastic evolutionary game dynamics in a population of finite size. Individuals in the population are divided into two dynamically evolving groups. The structure of the population is formally described by a Wright–Fisher type Markov chain with a frequency dependent fitness. In a strong selection regime that favors one of the two groups, we obtain qualitatively matching lower and upper bounds for the fixation probability of the advantageous population. In the infinite population limit we obtain an exact result showing that a single advantageous mutant can invade an infinite population with a positive probability. We also give asymptotically sharp bounds for the fixation time distribution.

Appendices

Appendix 1: Moran process

The goal of this section is to obtain an analogue of Theorem 3 (i. e., of the results stated in full detail in Theorems 4 and 6) for the frequency-dependent Moran process introduced in Nowak et al. (2004), Taylor et al. (2004). The main result of this section is stated in Theorem 8.

For a given integer $N\ge 2,$ the Moran process which we denote by $Y^{(N)}_t,$ $t\in {\mathbb Z}_+,$ is a discrete-time birth and death Markov chain on ${\varOmega }_{_N}$ with transition kernel $P^{(N)}_{i,j}:=P\bigl (Y^{(N)}_{t+1}=j\bigl |Y^{(N)}_t=i\bigr )$ defined as follows. The chain has two absorbtion states, 0 and N,  and for any $i\in {\varOmega }_{_N}^o,$

$$\begin{aligned} P^{(N)}_{i,j}= \left\{ \begin{array}{ll} \frac{N-i}{N}\xi _{_N}(i)&{}\quad \text{ if }~j=i+1\\ \frac{i}{N}\bigl (1-\xi _{_N}(i)\bigr )&{}\quad \text{ if }~j=i-1\\ 1-P^{(N)}_{i,i-1}-P^{(N)}_{i,i+1}&{}\quad \text{ if }~j=i\\ 0&{}\quad \text{ otherwise }. \end{array} \right. \end{aligned}$$

The process in this form, with a general selection parameter $w\in (0,1],$ was introduced in Nowak et al. (2004). We remark that even though (Taylor et al. 2004) formally considered only the basic variant with $w=1,$ their main theorems hold for an arbitrary $w\in (0,1].$

Similarly to (3), we define

$$\begin{aligned} \widehat{p}_{_N}(i)=P\left( Y^{(N)}_\tau =N|Y^{(N)}_0=i\right) , \end{aligned}$$

where $\tau =\inf \bigl \{t>0:Y^{(N)}_t=0~\text{ or }~Y^{(N)}_t=N\bigr \}.$ Since the Moran model is a birth-death process, the fixation probabilities are known explicitly (Nowak et al. 2004; Taylor et al. 2004) [see, for instance, Example 6.4.4 in Durrett (2010a) for a general birth and death chain result]:

$$\begin{aligned} \widehat{p}_{_N}(i)=\frac{1+\sum _{j=1}^{i-1} \prod _{k=1}^j \frac{g_{_N}(k)}{f_{_N}(k)}}{1+\sum _{j=1}^{N-1} \prod _{k=1}^j\frac{g_{_N}(k)}{f_{_N}(k)}}. \end{aligned}$$

(43)

In what follows we however bypass a direct use of this formula.

The result following is an analogue of Lemma 1 for the Moran process.

Lemma 6

Let Assumption 2 hold. Then for any $N\ge N_0$ and $i\in {\varOmega }_{_N}^o,$

$$\begin{aligned} E\bigl (\gamma ^{Y^{(N)}_{t+1}}\bigl |Y^{(N)}_t=i\bigr )\le \gamma ^i \quad \text{ and } \quad E\bigl (\alpha ^{Y^{(N)}_{t+1}}\bigr |Y^{(N)}_t=i\bigr )\ge \alpha ^i. \end{aligned}$$

(44)

Proof

We will only prove the first inequality in (44). The proof of the second one can be carried out in a similar manner. We have:

$$\begin{aligned} E\left( \gamma ^{Y^{(N)}_{t+1}}|Y^{(N)}_t=i\right)&= \gamma ^{i+1}\frac{N-i}{N}\xi _{_N}(i)+\gamma ^{i-1}\frac{i}{N}\bigl (1-\xi _{_N}(i)\bigr ) \\&\qquad + \gamma ^i\Bigl (1-\frac{N-i}{N}\xi _{_N}(i)-\frac{i}{N}\bigl (1-\xi _{_N}(i)\bigr )\Bigr ) \\&= \gamma ^i + \gamma ^{i-1}(1-\gamma )\frac{i}{N} - \gamma ^{i-1}(1-\gamma )\xi _{_N}(i)\Bigl (\gamma +(1-\gamma )\frac{i}{N}\Bigr ). \end{aligned}$$

Since by virtue of (2) and Assumption 2,

$$\begin{aligned} \frac{i}{N}-\xi _{_N}(i)\Bigl (\gamma +(1-\gamma )\frac{i}{N}\Bigr )\le 0, \end{aligned}$$

we conclude that $E\bigl (\gamma ^{Y^{(N)}_{t+1}}\bigl |Y^{(N)}_t=i\bigr )\le \gamma ^i$. $\square$

In the same way as Lemma 1 implies Theorem 4, the above result yields the following bounds for the fixation probabilities in the Moran process:

$$\begin{aligned} \frac{1-\gamma ^i}{1-\gamma ^N}\le \widehat{p}_{_N}(i) \le \frac{1-\alpha ^i}{1-\alpha ^N}. \end{aligned}$$

(45)

More precisely, we have:

Theorem 8

Let Assumption 1 hold. Then:

• (i) For any $N\ge N_0$ and $i\in {\varOmega }_{_N}^o,$ the inequalities in (45) hold true.

• (ii) Furthermore, for any $i\in {\mathbb N},$

  $$\begin{aligned} \lim _{N \rightarrow \infty } \widehat{p}_{_N}(i)=1-\lambda ^{-i}, \end{aligned}$$

  (46)

  where $\lambda$ is defined in (19).

A proof of the limit in (46) is outlined at the end of the appendix, after the following Remark.

Remark 4

The identities for the optimal values of $\alpha$ and $\gamma$ given in (7) suggest that in some cases one of the bounds in (45) might be asymptotically tight for large populations. The purpose of this remark is to explore conditions for the equalities $\lambda ^{-1}=\alpha$ or $\lambda ^{-1}=\gamma$ to hold true. By the definition given in (19), $\lambda =\lim _{N\rightarrow \infty } \frac{f_{_N}(i)}{g_{_N}(i)}$ for any $i\in {\mathbb N}.$ Moreover, for any $N\ge N_0,$

$$\begin{aligned} \frac{f_{_N}(1)}{g_{_N}(1)}=\frac{1-w+wb}{1-w+wd+(c-d)(N-1)^{-1}} \end{aligned}$$

(47)

and

$$\begin{aligned} \frac{f_{_N}(N-1)}{g_{_N}(N-1)}=\frac{1-w+wa+(b-a)(N-1)^{-1}}{1-w+wc}. \end{aligned}$$

(48)

It follows from (47) that

$$\begin{aligned} \lambda =\frac{1-w+wb}{1-w+wd}= \left\{ \begin{array}{cl} \sup \limits _{N\ge N_0} \frac{f_{_N}(1)}{g_{_N}(1)}&{}\quad \text{ if }~c\ge d\\ \inf \limits _{N\ge N_0} \frac{f_{_N}(1)}{g_{_N}(1)}&{}\quad \text{ if }~c\le d. \end{array} \right. \end{aligned}$$

Furthermore, (48) implies that

$$\begin{aligned} \frac{1-w+wa}{1-w+wc}= \left\{ \begin{array}{cl} \inf \limits _{N\ge N_0} \frac{f_{_N}(N-1)}{g_{_N}(N-1)}&{}\quad \text{ if }~b\ge a\\ \sup \limits _{N\ge N_0} \frac{f_{_N}(N-1)}{g_{_N}(N-1)}&{}\quad \text{ if }~b\le a \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \frac{1-w+wa+(b-a)(N_0-1)^{-1}}{1-w+wc}= \left\{ \begin{array}{cl} \max \limits _{N\ge N_0} \frac{f_{_N}(N-1)}{g_{_N}(N-1)}&{}\quad \text{ if }~b\ge a\\ \min \limits _{N\ge N_0} \frac{f_{_N}(N-1)}{g_{_N}(N-1)}&{}\quad \text{ if }~b\le a. \end{array} \right. \end{aligned}$$

In principle, the last three identities contain all the information which is needed to identify necessary and sufficient conditions for the occurrence of either $\alpha =\lambda ^{-1}$ or $\gamma =\lambda ^{-1},$ where it is assumed, as in (7), that the optimal bounds $\alpha =\inf _{N\ge N_0}\min _{i\in {\varOmega }^o_{_N}} \frac{g_{_N}(i)}{f_{_N}(i)}$ and $\gamma =\sup _{N\ge N_0}\max _{i\in {\varOmega }^o_{_N}} \frac{g_{_N}(i)}{f_{_N}(i)}$ are employed. For instance, in the generic prisoner’s dilemma case $b>d>a>c$ one can set

$$\begin{aligned} \gamma ^{-1}=\lambda =\inf _{N\ge N_0}\min _{i\in {\varOmega }^o_{_N}} \frac{f_{_N}(i)}{g_{_N}(i)} \end{aligned}$$

(49)

provided that $\frac{\displaystyle 1-w+wb}{\displaystyle 1-w+wd } \le \frac{\displaystyle 1-w+wa}{\displaystyle 1-w+wc },$ which is equivalent to

$$\begin{aligned} (1-w)(c+b-a-d)\le w(ad-bc). \end{aligned}$$

This leads us to consider the following possible scenarios for a prisoner’s-dilemma-type underlying game, that is assuming that $b>d>a>c:$

• 1. $ad\ge bc$ and $c+b-a-d\le 0$ (for instance, $b=4,d=3,a=2,c=1$). In this case (49) holds for any $w\in (0,1].$

• 2. $ad>bc$ and $c+b-a-d> 0$ (for instance, $b=5,d=3,a=2,c=1$). In this case (49) holds if and only if

  $$\begin{aligned} \frac{1-w}{w}\le \frac{ad-bc}{c+b-a-d} \quad \Leftrightarrow \quad w\ge \Bigl (1+\frac{ad-bc}{c+b-a-d}\Bigr )^{-1}. \end{aligned}$$

• 3. $ad=bc$ and $c+b-a-d> 0$ (for instance, $b=6,d=3,a=2,c=1$). In this case (49) holds only if $w=1.$

• 4. $ad<bc$ and $c+b-a-d\ge 0$ (for instance, $b=7,d=3,a=2,c=1$). In this case (49) holds for no $w\in (0,1].$

• 5. It remains to consider the case when $ad<bc$ and $c+b-a-d< 0.$ We will now verify that this actually cannot happen. To get a contradiction, assume that this scenario is feasible and let $\varepsilon =\min \{b-d,a-c\},$ $d'=d+\varepsilon ,$ $c'=c+\varepsilon .$ Then

  $$\begin{aligned} c'+b-a-d'=c+b-a-d< 0 \end{aligned}$$

  (50)

  and, since $ad<bc$ and $a<b,$

  $$\begin{aligned} ad'=ad+a\varepsilon <bc'=bc+b\varepsilon . \end{aligned}$$

  (51)

  But, due to the choice of $\varepsilon$ we made, we should have either $b=d'$ or $a=c'.$ In the former case (50) and (51) imply the combination of inequalities $c'-a< 0$ and $a<c',$ while in the latter they yield $b-d'< 0$ and $d'<b,$ neither of which is possible.

The limit in (46) has been computed in Antal and Scheuring (2006) (technically, in the specific case $w=1$), see in particular formula (39) there. The proof in Antal and Scheuring (2006) relies on (43) and involves some semi-formal approximation arguments. We conclude this appendix with the outline of a formal proof of this result which is based on a different approach, similar to the one employed in the proof of Theorem 6.

Toward this end observe first that, provided that both processes have the same initial state, the fixation probabilities of the Markov chain $Y^{(N)}$ coincide with those of a Markov chain ${\widetilde{Y}}^{(N)}=\bigl ({\widetilde{Y}}^{(N)}_t\bigr )_{t\in {\mathbb Z}_+}$ on the state space ${\varOmega }_{_N}$ with transition kernel ${\widetilde{P}}^{(N)}_{i,j}:=P\bigl ({\widetilde{Y}}^{(N)}_{t+1}=j\bigl |{\widetilde{Y}}^{(N)}_t=i\bigr )$ which is defined as follows. Similarly to $Y^{(N)},$ the chain ${\widetilde{Y}}^{(N)}$ has two absorbtion states, 0 and N. Furthermore, for any $i\in {\varOmega }_{_N}^o,$

$$\begin{aligned} {\widetilde{P}}^{(N)}_{i,j}= \left\{ \begin{array}{cl} \frac{P^{(N)}_{i,i+1}}{P^{(N)}_{i,i-1}+P^{(N)}_{i,i+1}}&{}\quad \text{ if }~j=i+1\\ \frac{P^{(N)}_{i,i-1}}{P^{(N)}_{i,i-1}+P^{(N)}_{i,i+1}}&{}\quad \text{ if }~j=i-1\\ 0&{}\text{ otherwise }. \end{array} \right. \end{aligned}$$

The advantage of using the chain ${\widetilde{Y}}^{(N)}$ over $Y^{(N)}$ rests on the fact that while both $P^{(N)}_{i,i-1}$ and $P^{(N)}_{i,i+1}$ converge to zero as $N\rightarrow \infty ,$

$$\begin{aligned} \lim _{N\rightarrow \infty } {\widetilde{P}}^{(N)}_{i,i+1}= & {} \lim _{N\rightarrow \infty } \frac{\frac{N-i}{N}\xi _{_N}(i)}{\frac{N-i}{N}\xi _{_N}(i)+ \frac{i}{N}\bigl (1-\xi _{_N}(i)\bigr )}=\lim _{N\rightarrow \infty } \frac{\xi _{_N}(i)}{\xi _{_N}(i)+ \frac{i}{N}} \\= & {} \frac{\lambda }{1+\lambda } \end{aligned}$$

and, consequently, $\lim _{N\rightarrow \infty } {\widetilde{P}}^{(N)}_{i,i-1}=\frac{1}{1+\lambda }.$ In other words, as $N\rightarrow \infty ,$ the sequence of Markov chains ${\widetilde{Y}}^{(N)}$ converges weakly to the nearest-neighbor random walk (birth and death chain) $\widetilde{Z}=\bigl (\widetilde{Z}_t\bigr )_{t\in {\mathbb Z}_+}$ on ${\mathbb Z}_+$ with absorbtion state at zero and transition kernel defined at $i\in {\mathbb N}$ as follows:

$$\begin{aligned} P\bigl (\widetilde{Z}_{t+1}=j\bigl |\widetilde{Z}_t=i)= \left\{ \begin{array}{cl} \frac{\lambda }{1+\lambda }&{}\quad \text{ if }~j=i+1\\ \frac{1}{1+\lambda }&{}\quad \text{ if }~j=i-1\\ 0&{}\quad \text{ otherwise }. \end{array} \right. \end{aligned}$$

If $\lambda >1,$ then, similarly to (28), we have $P\bigl (\lim _{t\rightarrow \infty } \widetilde{Z}_t=+\infty \bigr )>0.$ The rest of the proof is similar to the argument following (28) in the proof of Theorem 6, with the processes ${\widetilde{Y}}^{(N)}$ and $\widetilde{Z}$ considered instead of, respectively, $X^{(N)}$ and Z. The only two exceptions are:

• 1. $P\bigl (\lim _{t\rightarrow \infty } \widetilde{Z}_t=0\bigl |\widetilde{Z}_t=i\bigr )=\lambda ^i$ for any $i\in {\mathbb N}.$ This follows from the solution to the “infinite-horizon” variation of the standard gambler’s ruin problem (Durrett 2010a) (take the limit as $N\rightarrow \infty$ in (43) assuming that $\frac{g_{_N}(k)}{f_{_N}(k)}=\lambda ^{-1}$ for all $N,k\in {\mathbb N}$).

• 2. The last four lines in (33) should be suitably replaced. For instance, one can use the following bound:

  $$\begin{aligned}&P\Bigl ({\widetilde{Y}}^{(N)}_1\ne 0,\ldots , {\widetilde{Y}}^{(N)}_{K-1}\ne 0,\,\max _{0\le t \le K-1}{\widetilde{Y}}^{(N)}_t<m\Bigr ) \\&\qquad +\sum _{j=m}^{N-1} P\Bigl (\max _{0\le t \le K-1}{\widetilde{Y}}^{(N)}_t=j\Bigr )\cdot \bigl (1-\widehat{p}_{_N}(j)\bigr ) \\&\quad \le \Bigl [1-P\bigl ({\widetilde{Y}}^{(N)}_1=m-2,{\widetilde{Y}}^{(N)}_2=m-3,\ldots , {\widetilde{Y}}^{(N)}_{m-1}=0\bigl |{\widetilde{Y}}^{(N)}_0=m-1\bigr )\Bigr ]^{\frac{K}{m-1}} \\&\qquad + \bigl (1-\widehat{p}_{_N}(m)\bigr ) \\&\quad \le \Bigl [1-\prod _{i=1}^{m-1}\frac{P^{(N)}_{i,i-1}}{P^{(N)}_{i,i-1}+P^{(N)}_{i,i+1}}\Bigr ]^{\frac{K}{m-1}}+\frac{\gamma ^m}{1-\gamma ^N} \\&\quad =\Bigl [1-\prod _{i=1}^{m-1}\frac{g_{_N}(i)}{g_{_N}(i)+f_{_N}(i)}\Bigr ]^{\frac{K}{m-1}}+\frac{\gamma ^m}{1-\gamma ^N} \\&\quad \le \Bigl [1-\Bigl (\frac{\alpha }{1+\alpha }\Bigr )^{m-1}\Bigr ]^{\frac{K}{m-1}}+\frac{\gamma ^m}{1-\gamma ^N}. \end{aligned}$$

We leave the details to the reader.

Appendix 2

In this short appendix we discuss an interpretation of the drift $\mu _{_N}(i)$ which is not used anywhere the paper, but it is worthwhile to state it explicitly as it is of independent interest. For $i\in {\varOmega }_{_N}$ and $j=1,\ldots ,N,$ let

$$\begin{aligned} S_{N,i}(j)= \left\{ \begin{array}{lll} 1&{}\text{ if }&{}j\le i\\ 0&{}\text{ if }&{}j> i \end{array} \right. \quad \text{ and } \quad F_{N,i}(j)= \left\{ \begin{array}{lll} f_{_N}(i)&{}\text{ if }&{}j\le i\\ g_{_N}(i)&{}\text{ if }&{}j>i. \end{array} \right. \end{aligned}$$

Thus, if one enumerates and labels the individuals at the state $X^{(N)}_t=i$ in such a way that the first i individuals are of type A and get labeled 1,  and the remaining $N-i$ individuals are of type B and get labeled 0,  then $S_{N,i}(j)$ and $F_{N,i}(j)$ represent, respectively, the label and the fitness of the j-th individual. Furthermore, using the above enumeration, let (u, v),  $u<v,$ be a pair of individuals chosen at random. That is,

$$\begin{aligned} P(u=j,v=k)=\frac{2}{N(N-1)}\quad \text{ for } \text{ any }\quad j,k\in {\varOmega }_{_N}\backslash \{0\},\, j<k. \end{aligned}$$

Let

$$\begin{aligned} H_{N,i}:=\frac{2i(N-i)}{N(N-1)}=E\bigl (S_{N,i}(v)-S_{N,i}(u)\bigr ) \end{aligned}$$

be the heterozygosity (Durrett 2010b, Section 1.2) of the Wright–Fisher process $X^{(N)}$ at state $i\in {\varOmega }_{_N}^o,$ that is the probability that two individuals randomly chosen from the population when $X^{(N)}_t=i$ have different types. In this notation,

$$\begin{aligned} \mu _{_N}(i)= & {} \frac{N(N-1)}{2}H_{_N}(i)\frac{f_{_N}(i)-g_{_N}(i)}{if_{_N}(i)+(N-i)g_{_N}(i)} \\= & {} \frac{N(N-1)}{2}\cdot \frac{E\bigl (F_{N,i}(v)-F_{N,i}(u)\bigr )}{\sum _{j=1}^N F_{N,i}(j)} \\= & {} \frac{1}{2}\frac{\sum _{j,k=1}^N \bigl |F_{N,i}(k)-F_{N,i}(j)\bigr |}{\sum _{j=1}^N F_{N,i}(j)}, \end{aligned}$$

suggesting that the drift $\mu _{_N}(i)$ can serve as a measure of heterozygosity that is suitable for our game-theoretic framework.