The \Lambda-lookdown model with selection

The -lookdown model with selection

B. Bah and E. Pardoux
July 27, 2019
Abstract

The goal of this paper is to study the lookdown model with selection in the case of a population containing two types of individuals, with a reproduction model which is dual to the -coalescent. In particular we formulate the infinite population “-lookdown model with selection”. When the measure gives no mass to , we show that the proportion of one of the two types converges, as the population size tends to infinity, towards the solution to a stochastic differential equation driven by a Poisson point process. We show that one of the two types fixates in finite time if and only if the -coalescent comes down from infinity. We give precise asymptotic results in the case of the Bolthausen–Sznitman coalescent. We also consider the general case of a combination of the Kingman and the -lookdown model.

Subject classification

60G09, 60H10, 92D25.

Keywords

Look-down with selection, Lambda coalescent, Fixation and non fixation.

1 Introduction

In this paper we consider the lookdown (which is in fact usually called the “modified lookdown”) model with selection where we replace the usual reproduction model by a population model dual to the -coalescent. We first recall the models from [20] and [9], and then we will describe the variant which will be the subject of the present paper.

Pitman [20] and Sagitov [21] have pointed at an important class of exchangeable coalescents whose laws can be characterized by an arbitrary finite measure on [0, 1]. Specifically, a -coalescent is a Markov process () on (the set of partition of ) started from the partition and such that, for each integer , its restriction to (the set of partitions of ) is a continuous time Markov chain that evolves by coalescence events, and whose evolution can be described as follows.
Consider the rates

(1.1)

Starting from a partition in with non-empty blocks, for each every possible merging of blocks (the other blocks remaining unchanged) occurs at rate , and no other transition is possible. This description of the restricted processes determines the law of the -coalescent .
Note that if , then only pairwise merging occurs, and the corresponding -coalescent is just a time rescaling (by ) of the Kingman coalescent. When which we will assume except in the very last section of this paper, a realization of the -coalescent can be constructed (as in [20]) using a Poisson point process

(1.2)

on with intensity measure where . We will assume that the measure has infinite total mass. Each atom of influences the evolution as follows :

  • for each block of run an independent Bernoulli () random variable;

  • all the blocks for which the Bernoulli outcome equals 1 merge immediately

    into one single block, while all the other blocks remain unchanged.

In order to obtain a construction for a general measure , one can superimpose onto the -coalescent independent pairwise mergers at rate .

The lookdown construction was first introduced by Donnelly and Kurtz in 1996 [9]. Their goal was to give a construction of the Fleming-Viot superprocess that provides an explicit description of the genealogy of the individuals in a population. Donnelly and Kurtz subsequently modified their construction in [10] to include more general measure-valued processes. Those authors extended their construction to the selective and recombination case [11].

We are going to present our model which we call -lookdown model with selection. An important feature of our model is that we will describe it for a population of infinite size, thus retaining the great power of the lookdown construction. As far as we know, this has not yet been done in the case of models with selection except in our previous publication [4], where we considered a model dual to Kingman’s coalescent.

We consider the case of two alleles and , where has a selective advantage over . This selective advantage is modelled by a death rate for the type individuals. We will consider the proportion of individuals. The type individuals are coded by 1, and the type individuals by 0. We assume that the individuals are placed at time on levels each one being, independently from the others, 1 with probability , 0 with probability , for some . For each and , let denote the type of the individual sitting on level at time . The evolution of is governed by the two following mechanisms.

  1. Births Each atom of the Poisson point process corresponds to a birth event. To each , we associate a sequence of i.i.d Bernoulli random variables with parameter . Let

    and

    At time , those levels with =1 and modify their label to . In other words, each level in immediately adopts the type of the smallest level participating in this birth event. For the remaining levels, we reassign the types so that their relative order immediately prior to this birth event is preserved. More precisely

    We refer to the set as a multi-arrow at time , originating from min , and with tips at all other points of This procedure is usually referred to as the modified lookdown construction of Donnelly and Kurtz. In the original construction, the types of the levels in the complement of remained unchanged at time , hence the types , for got erased from the population at time .

  2. Deaths Any type 1 individual dies at rate , his vacant level being occupied by his right neighbor, who himself is replaced by his right neighbor, etc. In other words, independently of the above arrows, crosses are placed on all levels according to mutually independent rate Poisson processes. Suppose there is a cross at level at time . If , nothing happens. If , then

We refer the reader to Figure 1 for a pictural representation of our model. Note that the type of the newborn individuals are found by “looking down”, while the type of the individual who replaces a dead individual is found by looking up. So maybe our model could be called “look-down, look-up”.

Figure 1: The graphical representation of the -lookdown model with selection of size . Solid lines represent type individuals, while dotted lines represent type individuals.

Since we have modelled selection by death events, the evolution of the first individuals depends upon the next ones, and , the proportion of type individuals among the lowest levels, is not a Markov process. We will show however that for each the collection of r.v.’s is well defined (which is not obvious in our setup) and constitutes an exchangeable sequence of –valued random variables. We can then apply de Finetti’s theorem, and prove that a.s for any fixed , where is a –valued Markov process, which is a solution to the stochastic differential equation (which we call the –Wright–Fisher SDE with selection)

(1.3)

where , and is a Poisson point measure on with intensity . The process represents the proportion of type individuals at time in the infinite size population. Note that uniqueness of a solution to (1.3) is proved in [8].

The paper is organized as follows. We both construct our process, and establish the crucial exchangeability property satisfied by the -lookdown model with selection in section 2. In section 3 we establish the convergence of to the solution to (1.3). In section 4 we show that one of the two types fixates in finite time if and only if the -coalescent comes down from infinity. Moreover, in the case of no fixation, we show that as , and discuss when and when . In the case of the Bolthausen–Sznitman coalescent (which does not come down from infinity), we precise the law of , and study the speed at which either of the two types invades the whole population. Finally, we extend our results to the case in the last section 5.

In this paper, we use to denote the set of positive integers , and to denote the set . We suppose that the measure fulfills the condition

(1.4)

and in all the paper except in section 5, we assume that .

2 The lookdown process, exchangeability

2.1 Some results for general

Throughout the paper, the notation

is used for the th moment of the finite measure on [0, 1] for arbitrary real . Note that is a decreasing function of with for , while may be either finite or infinite for . For observe from (1.1) that is the rate at which jumps to its absorbing state {[n]} from any state with blocks. Let denote a random variable with distribution , defined on some background probability space with expectation operator , so . Recall the formula (1.1) for the transition rates of the -coalescent, which we rewrite as

For any partition with a finite number of blocks, the total rate of transitions of all kinds in a -coalescent, which can be rewritten as

By monotone convergence,

2.2 Construction of our process

In this section, we will construct the process corresponding to a given initial condition defined in the Introduction.

Recall the Poisson point process defined in (1.2). For each and , let

We have

Lemma 2.1.

For each and ,

Proof : Each atom of affects at least 2 of the first individuals with probability

Consequently

The result follows.

2.2.1 -lookdown model without selection

In this subsection, we essentially follow [10]. For each , one can define the vector with values in , by

  1. .

  2. At any birth event and such that , for each , evolves as follows

Using the above lemma, we see that the process has finitely many jumps on for all , hence its evolution is well defined. From this definition, one can easily deduce that the evolution of the type at levels 1 up to depends only upon the types at levels up to . Consequently, if , the restriction of to the first levels yields , in other words :

Hence, the process is easily defined by a projective limit argument as a -valued process.

2.2.2 -lookdown model with selection

This section is devoted to the construction of the infinite population lookdown model with selection.

For each , we consider the process obtained by applying all the arrows between , and only the crosses on levels 1 to . Using the fact that we have a finite number of crosses on any finite time interval, it is not hard to see that the process is well defined by applying the model without selection between two consecutive crosses, and applying the recipe described in the Introduction at a death time. More generally, our model is well defined if we suppress all the crosses above a curve which is bounded on any time interval . Note also that, if we remove or modify the arrows and or the crosses above the evolution curve of a type individual, this does not affect her evolution as well as that of those sitting below her.

At any time , let denote the lowest level occupied by a individual. Of course, if , then , for all . If for any , a.s, then the process is well defined by taking into account only those crosses below the curve , and evolves as follows. When in state , jumps to

  1. at rate ;

  2. at rate , ,

where we have used the notation defined by (1.1). In other words, the infinitesimal generator of the Markov process is given by:

(2.1)

Now, we are going to show that the process is well defined. For this, we study two cases.

Case 1: as

For each , we define

and

We have . For each , we define

Consider first the event

Recall the Poisson point measure defined in (1.2). Now, for each , we define the process, with values in , by

  1. .

  2. At any birth event evolves as follows

  3. Suppose there is a cross on level at time . If or and , nothing happens. If and , then

In other words, the process is obtained by applying all the arrows between , and only the crosses on levels 1 to . On the event , we have a finite number of such crosses on any finite time interval, and is constructed as explained above. Now, let

By a projective limit argument, we can easily deduce that the process is well defined on the set . Our model is defined on the event .

Now we consider the event . We first work on the event . This means that the allele fixates in finite time. It implies that for each is finite as well. Consider first the process defined on , i.e we take into account all the arrows between , and only the crosses on levels to . This process is well defined on the time interval . However, on the interval hence the process is well defined in . We next consider the process defined on . This process is well defined on the time interval . But on the interval , there is at most one , whose position is completely specified from the previous step. Iterating that procedure, and using again a projective limit argument, we define the full -lookdown model with selection.

If , but for some , the construction is easily adapted to that case. In fact some arguments in section 4 below show that this cannot happen with positive probability.

Case 2 :

Let

We now show that a.s. on the set . Indeed, for any stopping time and , define to be the event that there is at least one cross on each of the levels on the the interval , and to be the event that no birth arrow points to a level less than or equal to on the time interval . It is plain that the quantity

is deterministic, independent of , and that . Now clearly

Hence

or equivalently

Let now

We deduce from the last inequality and the strong Markov property that for any ,

consequently . This being true for all , the claim follows.

If , the idea is to show that there exists an increasing mapping such that a.s. for large enough, any individual sitting on level at any time never visits a level below , with the convention that if that individual dies, we replace him by his neighbor below. Once this is true, the evolution of the individuals sitting on levels is not affected by deleting the crosses above level . Hence it is well defined. If this holds for all large enough, the whole model is well defined.

Let

For each , we will show that an individual sitting on a high enough level at any time never visits a level below . In order to prove this, we couple our model with the following one.

On the interval , we erase all the arrows pointing to levels above , and pretend that all individuals above level , , are of type , i.e coded by 1, and we apply all the crosses above level . This model is clearly well defined since until there is only one , all other sites being occupied by 1’s. We next extend this model for as follows :

For each , let denote the lowest level occupied by a individual. At time , for all . At any time , we shall have for , and for . Again all crosses are kept, and we keep only those arrows whose tip hits a level .

This model is well defined. For each , we define as the first time where all the first individuals of this model are of type . We have

Lemma 2.2.

If , then for each ,

Proof : The result follows from and the fact that the process of arrows from 1 to 2 is a Poisson process with rate .

Now, let and denote the process which describes the position at time of the individual sitting on level at time in the present model.

We will prove below that the individual who sits on level at time 0 will remain below the level on the time interval . If she does not visit any level below before time , she will never visit any level below at any time, and moreover any individual who visits level before time will remain above the individual who was sitting at level at time 0 until , hence will never visit any level below .

Since the “true” model has more arrows and less “active crosses” than the present model, if we show that in the present model a.s. there exists such that the individual who starts from level at time 0 never visits a level below , we will have that in the true model a.s. for large enough the evolution within the box is not altered by removing all the crosses above . A projective limiting argument allows us then to conclude that the full model is well defined.

The result will follow from the Borel-Cantelli lemma and the following lemma.

Lemma 2.3.

If , then for each ,

where

Proof : It is clear from the definition of that there exists a death process , which is independent of conditionally upon , and such that

where

On the other hand, we have

All we need to prove is that

The process is a jump Markov death process which takes values in the space . When in state , jumps to at rate (recall that all crosses are kept in the present model). In other words the infinitesimal generator of is given by

Let . The process given by

(2.2)

is a martingale. Applying (2.2) with the particular choice , there exists a martingale such that and

(2.3)

We note that is a martingale under . This is due to the fact that the Poisson process of crosses above is independent of . We first deduce from (2.3) that

Using the fact that is a pure death process, we obtain the identity

which, together with (2.3), implies

From (2.3), it is easy to deduce that (recall that )

which implies that

The result is proved .


From now on, we equip the probability space with the filtration defined by , where , and stands for the class of –null sets of . Any stopping time will be defined with respect to that filtration.

2.3 Exchangeability

In this subsection, we will show that the -lookdown model with selection preserves the exchangeability property, by an argument similar to that which we developed in [4].

Let denote the group of permutations of the set . For all and , we define the vectors

We should point out that is a permutation of and it is clear from the definitions that

(2.4)

The main result of this subsection is

Theorem 2.4.

If are exchangeable random variables, then for all , are exchangeable.

We first establish two lemmas, which treat repectively the case of resampling and of death events (we refer the reader to (1.2) for the definition of the collection ).

Lemma 2.5.

For any finite stopping time , any –valued –measurable random variable , if the random vector is exchangeable, and is the first time after of an arrow pointing to a level or a death at a level , then conditionally upon the fact that , for some and , where , the random vector is exchangeable.

Note that is the list of the types of the individuals sitting on levels just after a birth event during which one of the individuals sitting on a level between and has put children on levels up to .

Proof : For the sake of simplifying the notations, we condition upon , and . We start with some notation.

the levels selected by the point between levels and are

We define

Thanks to (2.4), we deduce that, for , ,

(2.5)

On the event , we have :

This implies that

For , define the mapping by :

where

In other words, is the vector from which the coordinates with indices have been suppressed. The right hand side of (2.5) is equal to