Haldane's formula in Cannings models: the case of moderately strong selection.

For a class of Cannings models we prove Haldane's formula, [Formula: see text], for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for [Formula: see text] and [Formula: see text]. Here, [Formula: see text] is the selective advantage of an individual carrying the beneficial type, and [Formula: see text] is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele's frequency process with slightly supercritical Galton-Watson processes in the early phase of fixation.

Introduction

Analysing the probability of fixation of a beneficial allele that arises from a single mutant is one of the classical problems in population genetics, see Patwa and Wahl (2008) for a historical overview. A rule of thumb known as Haldane’s formula states that the probability of fixation of a single mutant of beneficial type with small selective advantage and offspring variance  in a large population of individuals, whose total number N is constant over the generations, is approximately equal to . Originally, this was formulated for the (prototypical) model of Wright and Fisher, in which the next generation arises by a multinomial sampling from the previous one (which leads to in the neutral case), with the “reproductive weight” of an individual of beneficial type being increased by the (small) factor . A natural generalization of the Wright-Fisher model are the Cannings models; here one assumes exchangeable offspring numbers in the neutral case (Cannings 1974; Ewens 2004), and separately within the sets of all individuals of the beneficial and the non-beneficial type in the selective case (Lessard and Ladret 2007).

The reasoning in the pioneering papers by Fisher (1922), Haldane (1927) and Wright (1931) was based on the insight that, as long as the beneficial type is rare, the number of individuals carrying the beneficial type is a slightly supercritical branching process for which the survival probability iswhere is the offspring expectation and is the offspring variance (see Athreya 1992, Theorem 3). The heuristics then is that the branching process approximation should be valid until the beneficial allele has either died out or has reached a fraction of the population that is substantial enough so that the law of large numbers dictates that this fraction should rise to 1.

Notably, Lessard and Ladret (2007) obtained (for fixed population size N) the resultas a special case of their explicit analytic representation of within a quite general class of Cannings models and selection mechanisms.

An interesting parameter regime as is that of moderate selection,which is between the classical regimes of weak and strong selection. Is the Haldane asymptoticsvalid in the regime (2)?

If one could bound in this regime the o(s)-term in (1) by , then (1) would turn into (3). Such an estimate seems, however, hard to achieve in the analytic framework of Lessard and Ladret (2007).

The main result of the present paper is a proof of the Haldane asymptotics using an approximation by Galton–Watson processes in the regime of moderately strong selection, which corresponds to (2) for . Hereby, we assume that the Cannings dynamics admits a paintbox representation, whose random weights are exchangeable and of Dirichlet-type, and fulfil a certain moment condition, see Sect. 3. Here, the effect of selection is achieved by a decrease of the reproductive weights of the non-beneficial individuals by the factor .

An approximation by Galton–Watson processes was used in González Casanova et al. (2017) to prove the asymptotics (3) in the regime of moderately strong selection for a specific Cannings model that arises in the context of experimental evolution, with the next generation being formed by sampling without replacement from a pool of offspring generated by the parents.

In the case the method developed in the present paper would fail, because then the Galton–Watson approximation would be controllable only up to a time at which the fluctuations of the beneficial allele (that are caused by the resampling) still dominate the trend that is induced by the selective advantage. However, in Boenkost et al. (2021) we proved the Haldane asymptotics (3) for the case of moderately weak selection, i.e. under Assumption (2) with . There a backward point of view turned out to be helpful, which uses a representation of the fixation probability in terms of sampling duality via the Cannings ancestral selection graph developed in Boenkost et al. (2021) (see also González Casanova and Spanó 2018).

The results of the present paper together with those of Boenkost et al. (2021) do not cover the boundary case between moderately strong and moderately weak selection. We conjecture that the Haldane asymptotics (3) is valid also in this case.

A class of Cannings models with selection

This section is a short recap of Boenkost et al. (2021) Section 2; we include it here for self-containedness.

Paintbox representation in the neutral case

Neutral Cannings models are characterized by the exchangeable distribution of the vector of offspring sizes; here the are non-negative integer-valued random variables which sum to N. An important subclass are the mixed multinomial Cannings models. Their offspring size vector arises in a two-step manner: first, a vector of random weights is sampled, which is exchangeable and satisfies and , .

In the second step, a die with N possible outcomes and outcome probabilities is thrown N times, and counts how often the outcome i occurs. Hence, given the random weights the offspring numbers are Multinomial-distributed. Following Kingman’s terminology, we speak of a paintbox representation for , and call the underlying (random) paintbox.

This construction is iterated over the generations : Let be independent copies of , and denote the individuals in generation g by (i, g), . Assume that each individual , in generation , chooses its parent in generation g, with conditional distributionwhere given the choices of the parents for individuals are independent and identically distributed. This results in exchangeable offspring vectors which are independent and identically distributed over the generations g.

For notational simplicity we do not always display dependence of on the generation g, and write instead. From time to time however we want to emphasise the dependence of on N and therefore write instead of .

Some exchangeable offspring vectors do not have a paintbox representation, for example a random permutation of the vector . Prototypical paintboxes are , which leads to the Wright-Fisher model, and the class of Dirichlet-distributed random weights. In particular, the offspring distribution with Dirichlet-distributed paintbox can be seen as a limiting case of the offspring distribution for the model of experimental evolution considered in Baake et al. (2019) and González Casanova et al. (2017).

A paintbox representation with selection

Let , , be as in the previous section, and let . Assume each individual carries one of two types, either the beneficial type or the wildtype. Depending on the type of individual (i, g) we setif (i, g) is of wildtype and if (i, g) is of beneficial type. The probability that an individual is chosen as parent is now given byfor all . Parents are chosen independently for all and the distribution does not change over the generations. If (i, g) is the parent of the child inherits the type of its parent. In particular, this reproduction mechanism leads to offspring numbers that are exchangeable among the beneficial as well as among wildtype individuals.

The Cannings frequency process

In the previous section we gave a definition for a Cannings model which incorporates selection, by decreasing the random weight of each wildtype individual by the factor . This allows to define the Cannings frequency process with state space [N] which counts the number of beneficial individuals in each generation g.

Assume there are beneficial individuals at time g; due to the exchangeability of we may assume that the individuals are the beneficial ones. Given , the probability that individual is of beneficial type is then due to (4) equal toand is the same for all . Hence, given and given there are k beneficial individuals in generation g, the number of beneficial individuals in generation has distributionthis defines the transition probabilities of the Markov chain .

Main result

Before we state our main result we specify the assumptions on the paintbox and the strength of selection.

Definition 1

(Dirichlet-type weights) We say that a random vector with exchangeable components is of Dirichlet-type ifwhere are independent copies of a random variable Y with .

We assume thatfor some , which implies the finiteness of all moments of Y. The relevance (and possible relaxations) of Condition (8) are discussed further in Remark 2(a), see also the comment in Remark 1(a).

Remark 1

The biological motivation for considering Dirichlet-type weights comes from seasonal reproductive schemes. At the beginning of a season a set (of size N) of individuals is alive. These individuals and their offspring reproduce and generate a pool of descendants within that season. Only a few individuals from this pool survive till the next season. The number N in the model is assumed to be the total number of individuals that make it to the next season. Dirichlet-type weights arise in the asymptotics of an infinitely large pool of offspring; then sampling with and without replacement coincide. Condition (8), which we will require for the proof of Theorem 1 (see also Remark 2), guarantees that the pool of descendants of a single individual is not too large in comparison to the pool of descendants generated by the other individuals. The simplifying assumption implies that the weight of a parent cannot be equal to zero. Observe, however, that weights of single parents can be arbitrarily small if (e.g.) Y has a density which is continuous and strictly positive in zero.

The case of a deterministic Y corresponds to , i.e. the classical Wright-Fisher model. If Y has a Gamma()-distribution, then is Dirichlet -distributed.

Theorem 1 in Huillet and Möhle (2021), gives a classification of a large class of Cannings models with a paintbox of the form (7) with regard to the convergence of their rescaled genealogies.

Let be a sequence of Cannings offspring numbers that are represented by the paintboxes . It is well known (and easily checked) that If is of the form (1) with , which is clearly implied by (8), then (see Huillet and Möhle (2021) Theorem 1 (i)) the right hand side of (9) converges to as .

In view of Remark 1(c) we have for the asymptotic neutral offspring varianceReplacing Y by does not affect (7), hence we can and will assume in the proofs, which simplifies (10) to (and makes (10) consistent with the notation of Huillet and Möhle (2021)). Under the assumption , which is implied by (8) as well, the following asymptotics is valid We will discuss the relevance of this asymptotics in Remark 2(b), and prove it in Lemma 2(b).

Turning to the selective advantage, we assume that for a fixed the sequence obeyswhich we call the regime of moderately strong selection, thus generalizing the corresponding notion introduced in Sect. 1. (Note that (12) has an analogue in the regime of moderately weak selection as discussed in Boenkost et al. (2021)). In order to connect to (2) we definewhich is equivalent to , with (12) translating toWe now state our main result on the asymptotics (as ) of the fixation probability of the Cannings frequency process defined in Sect. 2.3. Note that the Markov chain has the two absorbing states 0 and N, with the hitting time of being a.s. finite for all N.

Theorem 1

(Haldane’s formula) Assume that Conditions (7), (8) and (12) are fulfilled. Let be the number of beneficial individuals in generation g, with . Let , then

We give the proof of Theorem 1 in Sect. 5, after preparing some auxiliary results in Sect. 4. Next we give a strategy of the proof and its main ideas, with an emphasis on the role of Condition (12). In Remark 2 we discuss possible relaxations of Condition (8) and the boundary case

The proof of Theorem 1 is divided into three parts, corresponding to three growth phases of . Concerning the first phase we show that the probability to reach the level is , for some small and ; this is the content of Proposition 1. The proof is based on stochastic domination from above and below by slightly supercritical Galton–Watson processes and with respective offspring distributions (29) and (30).

To construct a Galton–Watson stochastic upper bound of in its initial phase, we recall that the transition probabilities of are mixed Binomial specified by (6). Using (7) we approximate (5) from above byAs we will show in Lemma 4, this is possible with probability for some and , and for with . We will then be able to dominate the mixed Binomial distribution (6) by the mixed Poisson distribution with random parameter (15), again up to an error term of order . Noting that (15) is a sum of independent random variables, we arrive at the upper Galton–Watson approximation for a single generation. For any small this can be repeated for generations, which (as an application of Lemma 1 will show) is enough to reach either the level 0 or the level with probability .

To obtain a Galton–Watson stochastic lower bound of in its initial phase, we adapt an approach that was used in González Casanova et al. (2017) in a related situation. As in Sect. 2.1, number the individuals in generation g by (i, g), now with , being the beneficial individuals, and denote by the number of children of the individual (i, g), . As will be explained in the proof of Lemma 5, as long as has not reached the level , the distribution of can be bounded from below by a mixed binomial distributionwith probability for some sufficiently small , again for . A suitable stopping and truncation at the level will give the Galton–Watson process approximation from below for the first phase.

We will verify in Sect. 4.1 that both slightly supercritical branching processes and reach the level with probability .

As to the second phase, we will argue in Sect. 5.2 that, after reaching the level the Cannings frequency process will grow to a macroscopic fraction with high probability. If the frequency of beneficial individuals is at least (but still below ), then in a single generation the frequency of beneficial individuals grows in expectation at least by . Hence, generations after has reached the level , the expected value of the process reaches the level . Similarly one bounds the variance produced in a single generation and derives from this an estimate for the variance accumulated over generations. This bound being sufficiently small, an application of Chebyshev’s inequality yields that (after generations) crosses the level with probability tending to 1 after reaching the level .

In Sect. 5.3) we deal with the last phase, and will show that the fixation probability tends to 1 as if we start with at least individuals of beneficial type. Here we use the representation for the fixation probability that is based on a sampling duality between the Cannings frequency process and the Cannings ancestral selection process (CASP) which was provided in Boenkost et al. (2021). For a subregime of moderately weak selection the claim will follow quickly from the representation formula combined with a concentration result for the equilibrium distribution of the CASP that was proved in Boenkost et al. (2021). To complete the proof we will then argue that both the CASP and the representation of the fixation probability depend on the selection parameter in a monotone way.

Remark 2

With some additional work the assumption (8) of the existence of some exponential moment of Y can be relaxed to some weaker moment condition. In order not to overload the present paper, we restrict here to a sketch.

In Lemma 5 we couple the frequency process of the beneficial individuals with Galton–Watson processes for generations. By means of the estimates in Lemma 3 and Lemma 4 we show that these couplings hold for a single generation with probability for some appropriate . Since we need the couplings to hold for generations, it suffices that the couplings hold in a single generation with probability for some (since in this case the probability of the coupling to fail is and therefore can be neglected with regard to (14)). Such probability bounds can also be obtained under weaker assumptions on the distribution of the random variable Y. Assume e.g. that Y has a regularly varying tail, i.e. for some and L is a slowly varying function. For the proof of Lemma 5 we need to estimate the probability of the event figuring in Lemma 3 with and the probability of the event figuring in Lemma 4 with . To show that these probabilities are of order we only need that (since the remaining probability in Lemma 3 can be estimated with Hoeffding’s inequality, see Hoeffding (1963)) with in Lemma 4 and in Lemma 3. The asymptotics (3.2) in Mikosch and Nagaev (1998) states that . Consequently, we need to choose such that as well as . This works for all choices of , provided that .

It would be nice to have a proof of the asymptotics (14) under the assumption that the 4th moment of Y is finite, even without the assumption of a regularly varying tail.

The investigation of the analogue to (14) in the absence of finite second moments, i.e. for Cannings models with heavy-tailed offspring distributions, is the subject of ongoing research, and will be treated in a forthcoming paper.

Relation (11) will be used in the proof of Lemma 6. Moreover, this relation is also instrumental in the companion paper Boenkost et al. (2021) (on the regime of moderately weak selection). The special case in Lemma 2 a) shows that the assumption implies This gives a rate of decay for the triple coalescence probability (and is the moment condition (3.6) in Boenkost et al. (2021)).

Condition (8) (on the existence of an exponential moment of Y) guarantees the Haldane asymptotics (14) for Cannings models with weights of Dirichlet type also in the whole regime of moderately weak selection without any further assumption. In particular the assumption on the finiteness of a negative moment of Y in Boenkost et al. (2021), Lemma 3.7 b), is unnecessary. Indeed, in the proof of Lemma 2(a) we show that . As shown in the proof of Lemma 3.7(b) in Boenkost et al. (2021) Condition (8) guarantees that for a sequence with and for all we can estimate from above by for appropriate constants . Consequently, for N sufficiently large we have for some appropriate constant , that is Condition (3.8) in Boenkost et al. (2021) is fulfilled.

It seems a mathematically intriguing question whether in the regime of moderate selection all Cannings models which admit a paintbox representation with Dirichlet-type weights and are in Kingman domain of attraction, also follow the Haldane asymptotics (14).

An example of a sequence of Cannings models (with weights not of Dirichlet-type) which fulfil Möhle’s condition but do not follow the Haldane asymptotics, is the following. In each generation a randomly chosen individual gets weight , and all the other individuals have a weight of . Then we have and , therefore by Möhle’s criterion the genealogy lies in the attraction of Kingman’s coalescent. However, the Haldane asymptotics would predict that the survival probability is of order , which for is . Since the fixation probability of a beneficial allele cannot be smaller than the fixation probability under neutrality (which is ), (14) must be violated in this example.

The present work together with the approach in Boenkost et al. (2021) does not cover the boundary case . A quick argument why our arguments cannot be extended simply to the boundary case is the following. We show that once the beneficial type exceeds (in the order of magnitude) the frequency it goes to fixation with high probability. In the regime we use couplings with Galton–Watson processes to show that this threshold is reached with probability . However, these couplings are not guaranteed as soon as collisions occur, i.e. when beneficial individuals are replacing beneficial individuals. By the well known ”birthday problem“ collisions are common as soon as individuals are of the beneficial type. Therefore we require , i.e. .

In the light of the results of the present paper and of Boenkost et al. (2021), there is little reason to doubt that the assertion of Theorem 1 should fail in the boundary case . However, the question remains open (and intriguing) whether then the backward or the forward approach (or a combination of both) is the appropriate tool for the proof.

Auxiliary results

Slightly supercritical Galton–Watson processes

Throughout this subsection, is a sequence of positive numbers converging to 0, is a fixed positive number, and , are Galton–Watson processes with offspring expectationoffspring variance and uniformly bounded third moments . Unless stated otherwise we assume that . We writefor the survival probability of and observeThe derivation and discussion of the asymptotics (18) has a venerable history, a few key references being Haldane (1927), Kolmogorov (1938), Eshel (1981), Hoppe (1992), Athreya (1992) Theorem 3, Haccou et al. (2005) Theorem 5.5.

Lemma B.3 in González Casanova et al. (2017) gives a statement on the asymptotic probability that either quickly dies out or reaches a certain (moderately) large threshold. The following lemma improves on this in a twofold way. It dispenses with the assumption for a fixed and more substantially, it gives a quantitative estimate for the probability that, given non-extinction, the (moderately) large threshold is reached quickly.

Lemma 1

Fix , and let . Then, for all with .

Proof

Observe thatIn Part 1 of the proof we will estimate the first probability on the r.h.s. of (20); this will give the above-mentioned improvement of Lemma B.3 in González Casanova et al. (2017). Part 2 of the proof deals with the second probability on the r.h.s. of (20).

Part 1. Like in the proof of Lemma B.3 in González Casanova et al. (2017) we obtain an upper bound on the time at which the process reaches the level given survival, by considering the process consisting of the immortal lines of conditioned to non-extinction. (For simplicity of notation we drop a superscript N in .) Let denote the survival probability of as in (17). The offspring distribution of arises from that of as(see Lyons and Peres 2017 Proposition 5.28). In particular one hasDenote, as usual, for a random variable X and an event A by . Furthermore,because of the assumed uniform boundedness of the third moments of . These two relations together with (16), (18) and the fact that immediately give a lower bound for , implying that for any Hence the process , when conditioned on survival, is bounded from below by the counting process of immortal lines, which in turn is bounded from below by the process with offspring distribution So far we closely followed the proof in González Casanova et al. (2017), but now we deviate from that proof to obtain the rate of convergence claimed in (19).

An upper bound for the time also gives an upper bound for the time . The idea is now to divide an initial piece of generations into parts, each of generations. Because of the immortality of and the independence between these parts we obtain immediately thatWe then bound from below by an application of the Paley-Zygmund inequality in its formwhere X is a non-negative random variable (with finite second moment). For a supercritical Galton–Watson process with offspring expectation m and offspring variance the n-th generation offspring expectation and n-th generation offspring variance are given by and (see Athreya and Ney (1972), p.4). Hence, we obtainWe choose the smallest such thatObserve that which ensures that for N large enough. We now estimate using (24) as followsApplying (23) with yieldswhich because of (25) implies . If after time the process is still smaller than our desired bound , we can iterate this argument times and arrive atwith . This gives the desired bound for the first term in (20).

Part 2. We now turn to the second term on the r.h.s. of (20). DefineObviously , and so it suffices to proveThis proof follows closely that of the second part of Lemma B.3 in González Casanova et al. (2017); we include it here for completeness.

We observewhere the first and the last equality follow from the branching property and from (21), respectively. We have shown in (22) that and hence we can concludeFinally, an application of Markov’s inequality yields (26)

Estimates on the paintbox

The following lemma provides the asymptotics (11) as well as the moment bounds for the Dirichlet-type weights that were addressed in Remark 2(b).

Lemma 2

(Moments of the weights) Let be iid positive random variables with and We abbreviate .

(a) Assume for some . Then(b) Assume . Then

a) Consider the event . First we note thatLet K be so large that . Hoeffding’s inequality applied to the sample mean of i.i.d. copies of the bounded random variable implies that decays exponentially fast. Since is bounded this yields the claim.

(b) We observe thatLet and Markov’s inequality applied to together with the assumption implies . Likewise, . Furthermore, let K be so large that . Again, Hoeffding’s inequality applied to the sample mean of i.i.d.copies of the bounded random variable together with monotonicity imply that decays exponentially; a fortiori we have .

Let . We have since on we have and on and we have . Hence, it remains to show that Define and observeAbbreviate . On we have and henceThusBy Cauchy-Schwarz we have . Similarly, , since vanishes due to the independence of Y and . The remaining terms in (27) are as well, which completes the proof of Lemma 2.

We now prove a bound on the deviations for the total weight of k individuals.

Lemma 3

(Large deviations bound for a moderate number of random weights) Let and satisfy (7), (8), and let for some . Then for all there exists a positive constant depending only on and the distribution of Y such that

This follows by a combination of two Cramér bounds. Indeed, the l.h.s. of (28) is by assumption bounded from above byAbbreviating with such that we estimate the latter probability from above bydenoting by I(y) the rate function of Y. Due to (8) I(y) exists around and is strictly positive for (see Dembo and Zeitouni (1994) Theorem 2.2.3). This yields an upper bound of with .

The next lemma gives stochastic upper and lower bounds for the sums of the random weights in terms of sums of the independent random variables .

Lemma 4

(Bounds for the random weights) Assume that Conditions (7) and (8) are fulfilled and . Let , then for for some .

It suffices to showFor we havewhere I(y) is the rate function of Y. Condition (8) ensures that for , see Dembo and Zeitouni (1994) Theorem 2.2.3.

For any one has . This yieldsUsing Cramér (1938) Theorem 1, the probability on the r.h.s. can, with a suitable , be estimated from above bywhich gives the desired result.

Proof of the main result

Recall from (13) that we denote the order of the selection strength by . To simplify notation we will drop the subscript and simply write . As mentioned already in the sketch of the proof of Theorem 1 we assume without loss of generality that

The proof of the Theorem is divided into three parts, which correspond to three phases of growth for the Cannings frequency process . The initial phase is decisive: due to Proposition 1, the probability that reaches the level for some sufficiently small is given by the r.h.s. of (14). Lemmas 6 and 7 then guarantee that, once having reached the level , the process reaches N with high probability. The proof of the Theorem is then a simple combination of these three results and the strong Markov property. Indeed, with as in Proposition 1, Lemmas 6 and 7, and with fulfilling the requirements specified there, the fixation probability in the l.h.s. of (14) can be rewritten as

First phase: from 1 to

In this section we show that as long as the process can be upper and lower bounded (with sufficiently high probability) by two slightly supercritical branching processes and . To construct the upper bound we take the highest per capita selective advantage, which occurs when only a single individual is beneficial. Using Lemmas 3 and 4, we will approximate the thus arising mixed binomial distribution by a mixed Poisson distribution, which leads for to the offspring distributionwhere is the random variable figuring in (7). To arrive at the lower bounding Galton–Watson process we note that the per capita selective advantage is bounded from below by the one when beneficial individuals are present in the parent generation, as long as the process has not reached the level . Again using Lemmas 3 and 4 we will show that the offspring distribution of can be chosen as the mixed binomial distribution

Lemma 5

(Coupling with Galton–Watson processes) Let and be such that and , and putThen can be defined on one and the same probability space together with two branching process and with offspring distributions (30) and (29), respectively, such that for with as in Lemma 4.

Applying the latter estimate g times consecutively yields immediately the following corollary:

Corollary 1

Let and be as in Lemma 5. If , then for all

of Lemma 5. We proceed inductively, assuming that for we have constructed , and up to generation such that (31) holds for . Together with we will construct and , and check the asserted probability bound for the coupling.

Given and the weights  in generation , the number of beneficial individuals in generation g has the binomial distribution (6). Aiming first at the construction of the upper bound , we relate (6) to (29) in terms of stochastic order. For , a Bin(N, p)-distributed random variable B is stochastically dominated by a Pois-distributed random variable P ifsee (1.21) in Klenke and Mattner (2010). Indeed, in this case the probability of the outcome zero is not larger for a Pois()-distributed random variable than for a Bernoulli(p)-distributed random variable , which yields , where denotes the usual stochastic ordering of the random variables. Consequentlywith and being independent copies of and , respectively. In particular, for and we haveHence Condition (33) holds ifGiven , the success probability of the binomial distribution (6) is bounded from above viaThus by Lemma 3, (34) is fulfilled with probability with as in Lemma 3. In this sense the number of beneficial offspring is dominated by a Pois-distributed random variable with high probability. Applying Lemma 4 yields that with probability the following chain of inequalities is valid:In this way can be coupled with a branching process with a mixed Poisson offspring distribution of the form (29).

The lower bound also uses a comparison with a Galton–Watson process, now with a mixed binomially distributed offspring distribution:

Number the individuals in generation by , with , being the beneficial individuals. Given , we use a sequence of coin tossings to determine which of the individuals from generation g are the children of . The first N tosses determine which individuals are the children of . Denoting the number of these children by , the next tosses (with an updated success probability) determine which individuals are the children of , etc. Observe that as long as , and given and , then for has distributionNote that the success probability in (35) can be estimated from below byAs long as , Lemma 3 ensures that for with probability . Lemma 4, in turn, yields that the r.h.s. of (36) is bounded from below bywith probability at least .

Thus, if , then the distribution of specified in (35) is bounded from below bywith probability .

If , then we have . Consequently can be coupled with a Galton–Watson process with offspring distribution of the form (30) such that also the lower estimate in (32) is fulfilled. This completes the proof of Lemma 5.

We are now ready to prove that reaches the level with probability .

Proposition 1

(Probability to reach the critical level) Assume Conditions (7), (8) and (12) are fulfilled and define with , then

We use the couplings of with the slightly supercritical branching processes and from Corollary 1 and show that both processes reach the level with probability . Let and E be the event that the stochastic ordering between and holds until generation , that isWe show below that the stopping time fulfilsFor some g that is polynomially bounded in N, the r.h.s. of (32) is bounded from above by . Thus, combining Corollary 1 and (38) we deduceWe are now going to bound (37) from above by estimating the corresponding probability for and the stopping time . More precisely,To obtain an upper bound for the probability of to reach the level it suffices to estimate the survival probability of . For notational simplicity let us write for the event and similarly for the event . We haveThe survival probability of will now be estimated by means of (18). To this purpose we calculate the expectation and variance of the offspring distribution (29).

The expectation is and the variance is given byEquation (18) yields that the survival probability of the process is given by . The lower bound in (37) follows by similar arguments by considering the process instead.

It remains to show (38). Define and as the stopping times that the process reaches 0 and the process reaches the upper bound , respectively, i.e.with the convention that the infimum over an empty set is infinity. Thenby an application of Lemma 1 and Corollary 1 and as defined there. To keep the notation simple we denote by terms of the order for some Proceeding with (39) we obtainagain by an application of Lemma 1. Note thatIn order to show that (40) is it suffices to prove thatConsidering again the event and applying (18) one obtainssince the events E and imply that for some and the probability for to die out after reaching is . One more application of (18) yieldswhich finishes the proof.

Second phase: from to

In this section we show that , once having reached the level , will reach the level with probability tending to 1 as .

Lemma 6

(From to with high probability) Assume with , let and define the stopping timeThen there exists some such that

By monotonicity it is enough to prove the claim for . By definition we haveNext we lower-bound by the process , , with conditional distributionas long as . If we assume that is distributed as a slightly supercritical branching process with Pois distributed offspring, whereWe will see that by this definition in each generation the expectation of increases by the factor , see (45) and (46). The generation-wise increase of the variance conditioned on the current state can be estimated from above by a factor , see (45) and (47), leading to an iterative estimate on the variance of the form (48). As long as , the success probability in the mixed Binomial distribution on the r.h.s. of (41) dominates the corresponding one on the r.h.s. of (42). Consequently, starting and X both in we can couple them, such that as long as did not cross the level . In particular, we have for the stopping time To show we will estimate the first and second moment of for a suitably chosen and then use Chebyshev’s inequality to show that is above with sufficiently high probability. For this purpose we consider m(x) and v(x), the one-step conditional expectation and variance of at , that isFrom the definition of as a branching process above we have for Next we show that m(x) and v(x) fulfil relations similar to (45) also for , which will allow to estimate the expectation and the variance of .

For we have due to (42)In the penultimate equality we used which results from the factand (11). Consequently, we have for all , recalling (43),Next we analyse v(x), again for . In view of (42), a decomposition of the variance givesBecause of the negative correlation of the , the sum of the first and the second term is not larger than , which because of (11) is . Since the third term is , we have for all Combining (46) and (47) allows us to estimate the variance for , again by decomposing the variance:Iterating this argument yieldsChoose the minimal such that , which yields recalling the initial condition Applying Chebyshev’s inequality with , we obtainfor some small due to the assumptions on .

Since , this impliesand due to (44) this finishes the proof.

Third phase: from to N

Lemma 7 below concerns the last step of the proof, showing that once the process has reached the level , it goes to fixation with high probability. Our proof relies on a representation of the fixation probability of in terms of (a functional of) the equilibrium state of the counting process of the potential ancestors in the time discrete Cannings ancestral selection graph as provided by Boenkost et al. (2021). The process is called Cannings ancestral selection process (CASP) in Boenkost et al. (2021); for fixed N, it is a recurrent, [N]-valued Markov chain whose transition probabilities are specified in Boenkost et al. (2021) Section 2.3.

Theorem 3.1 and Formula (3.2) (see also Corollary 3.3) in Boenkost et al. (2021) provide the following sampling duality representation of the fixation probability of when started with k individuals:Intuitively, this says that goes extinct if and only if a random sample of (random) size , drawn without replacement from the population of size N, avoids the k beneficial individuals.

Formula (49) impliesThe representation of the transition probabilities of in Boenkost et al. (2021) Section 2.3 in terms of two half steps yields that for fixed N CASPs with different selection parameters can be coupled in such a way that is increasing in . Take a sequence satisfying and Condition (1.2) in Boenkost et al. (2021), i.e.Let be the equilibrium state belonging to (and to the same Dirichlet-type paintbox as that of ). The central limit result Boenkost et al. (2021), Corollary 6.10, implies that in probability as . Because of the just mentioned monotonicity in the selection coefficient, the same convergence holds true for the sequence . The following lemma is thus immediate from (50) and dominated convergence:

Lemma 7

(From to N with high probability) Let be a Cannings frequency process with for some . Assume that Conditions (7), (8) and (12) are fulfilled. Define . Then

Discussion

The analysis of fixation probabilities of slightly beneficial mutants is at the heart of population genetics; some seminal and more modern references are given in the Introduction. Our main result concerns Haldane’s asymptotics (3) for the fixation probability in a regime of moderate selection.

Our framework is that of Cannings models with selection (as reviewed in Sect. 2), where the corresponding neutral genealogies are assumed to belong to the domain of attraction of Kingman’s coalescent. This class of models is motivated by seasonal reproduction cycles in which within each season a large number of offspring is generated but only a comparatively small number (concentrated around a carrying capacity N) of randomly sampled offspring survive to the next season. In this setting it is reasonable to approximate sampling without replacement by sampling with replacement. Thus, under the assumption of neutrality, the probability that the j-th offspring that survives till the next generation is a child of parent i is approximately given by the random weightwhere are the sizes of (potential one-generation) offspring of parents . These sizes are assumed to be independent and identically distributed in the present paper, leading to the concept of weights of Dirichlet type. The subsequent generation then arises by a multinomial sampling with random weights, and to add selection the weights of wildtype parents are decreased by the factor . For a closely related model with a specific distribution of (and sampling without replacement) in the context of Lenski’s long-term evolution experiment see González Casanova et al. (2017) and Baake et al. (2019).

We prove Haldane’s asymptotics in the case of moderately strong selection, see Theorem 1, in which the selection strength obeysfor some and a large population size N. In the companion paper Boenkost et al. (2021) the range of moderately weak selection was considered, i.e. in the casefor some Since in the regime of moderate selection, selection acts in this case on a faster timescale than genetic drift.

In Boenkost et al. (2021) an ancestral selection graph for the just described class of Cannings models with selection was defined, and it was shown that the fixation probability is equal to the expeced value , where is the number of lines of the ancestral selection graph in its equilibrium. While we could analyse directly the asymptotics of that quantity in the regime of moderately weak selection, we were facing too large fluctuations of in the regime of moderately strong selection in order to be successful with this approch. Conversely, it turned out that the classical idea of branching process approximation is suitable precisely in that latter regime.

For highly skewed offspring distributions an asymptotics for the fixation probability arises which is different from (3). In cases where the neutral genealogy is attracted by a Beta-coalescent, Okada and Hallatschek (2021) argue that the fixation probability is proportionally to , if . Thus the probability of fixation is substantially smaller than in Haldane’s asymptotics, which is reasonable since the offspring variance is diverging as . Notably, since the evolutionary timescale of Cannings models in the domain of attraction of Beta-coalescents is of the order , the case corresponds to the regime of moderate selection; note also that the case of coalescents being in the domain of attraction of a Kingman coalescent corresponds formally to .