the recovery of a recessive allele in a mendelian diploid model

THE RECOVERY OF A RECESSIVE ALLELE IN A
MENDELIAN DIPLOID MODEL
Anton Bovier, Loren Coquille, Rebecca Neukirch
To cite this version:
Anton Bovier, Loren Coquille, Rebecca Neukirch. THE RECOVERY OF A RECESSIVE
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THE RECOVERY OF A RECESSIVE ALLELE
IN A MENDELIAN DIPLOID MODEL
ANTON BOVIER, LOREN COQUILLE, AND REBECCA NEUKIRCH
A BSTRACT. We study the large population limit of a stochastic individual-based model
which describes the time evolution of a diploid hermaphroditic population reproducing
according to Mendelian rules. In [25] it is proved that sexual reproduction allows unfit
alleles to survive in individuals with mixed genotype much longer than they would in
populations reproducing asexually. In the present paper we prove that this indeed opens
the possibility that individuals with a pure genotype can reinvade in the population after
the appearance of further mutations. We thus expose a formal description of a mechanism
by which a recessive allele can re-emerge in a population. This can be seen as a statement
of genetic robustness exhibited by diploid populations performing sexual reproduction.
1. I NTRODUCTION
In population genetics, the study of Mendelian diploid models of fixed population size
began more than a century ago (see e.g. [2,11,13,14,16,17,24,27,28]), while their counterparts of variable population size models were studied in the context of adaptive dynamics
from 1999 onwards [20]. The approach of adaptive dynamics is to introduce competition
kernels to regulate the population size instead of maintaining it constant, see [19, 21, 22].
Stochastic individual-based versions of these models appeared in the 1990s, see [3–6,
12, 15]. They assume single events of reproduction, mutation, natural death, and death
by competition happen at random times to each individual in the population. An important and interesting feature of these models is that different limiting processes on different
time-scales appear as the carrying capacity tends to infinity while mutation rates and mutation step-size tend to zero (see [1, 3,6, 12, 23]). One of the major results in this context is
the convergence of a properly rescaled process to the so called Trait Substitution Sequence
(TSS) process, which describes the evolution of a monomorphic population as a jump
process between monomorphic equilibria. More generally, Champagnat and Méléard [6]
obtained the convergence to a Polymorphic Evolution Sequence (PES), where jumps occur
between equilibria that may include populations that have multiple co-existing phenotypes. The appearance of co-existing phenotypes is, however, exceptional and happens
only at so-called evolutionary singularities. From a biological point of view, this is somewhat unsatisfactory, as it apparently fails to explain the biodiversity seen in real biological
systems.
1991 Mathematics Subject Classification. 60K35,92D25,60J85.
We acknowledge financial support from the German Research Foundation (DFG) through the Hausdorff
Center for Mathematics, the Cluster of Excellence ImmunoSensation, and the Priority Programme SPP1590
Probabilistic Structures in Evolution. L.C. has been partially supported by the LabEx PERSYVAL-Lab
(ANR-11-LABX-0025-01) through the Exploratory Project CanDyPop and by the Swiss National Science
Foundation through the grant No. P300P2 161031.
We would like to thank Pierre Collet and Vincent Beffara for their help on the theory of dynamical systems
and fruitful discussions.
1
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
2
Most of the models considered in this context assume haploid populations with a-sexual
reproduction. One exception is the paper [7] by Collet, Méléard and Metz in 2013, and
then a series of papers by Coron and co-authors [8–10]. In [7], the Trait Substitution
Sequence is derived in a Mendelian diploid model under the assumption that the fitter
mutant allele and the resident allele are co-dominant.
The main reason why both in haploid models and in the model considered in [7] the
evolution along monomorphic populations is typical is that the time scales for the fixation
of a new trait and the extinction of the resident trait are the same (both of order ln K)
(unless some very special fine-tuning of parameters occurs that allows for co-existence).
This precludes (at least in the rare mutation scenarios considered) that an initially less fit
trait survives long enough until after possibly several new mutations occurred that might
create a situation where this trait may become fit again and recover.
In a follow-up paper to [7], two of the present authors [25], it was shown that, if instead
one assumes that the resident allele is recessive, the time to extinction of this allele is
dramatically increased. This will be discussed in detail in Section 1.2 and paves the way
for the appearance of a richer limiting process.
The general framework in [7] and [25] is the following. Each individual is characterised
by a reproduction and death rate which depend on a phenotypic trait determined by its
genotype, which here is determined by two alleles (e.g. A and a) on one single locus. The
evolution of the trait distribution of the three genotypes aa, aA and AA is studied under
the action of (1) heredity, which transmits traits to new offsprings according to Mendelian
rules, (2) mutation, which produces variations in the trait values in the population onto
which selection is acting, and (3) of competition for resources between individuals.
The paper [25] proves that sexual reproduction allows unfit alleles to survive in individuals with mixed genotype much longer than they would in populations reproducing
asexually. This opens the possibility that while this allele is still alive in the population,
the appearance of new mutants alters the fitness landscape in such a way that is favourable
for this allele and allow it to reinvade in the population, leading to a new equilibrium with
co-existing phenotypes. The goal of this paper is to rigorously prove that such a scenario
indeed occurs under fairly natural assumptions.
1.1. The stochastic model. The individual-based microscopic Mendelian diploid model
is a non-linear birth-and-death process. We consider a model for a population of a finite number of hermaphroditic individuals which reproduce sexually. Each individual i is
characterised by two alleles, ui1 ui2 , taken from some allele space U ⊂ R. These two alleles
define the genotype of the individual i. We suppress parental effects, which means that
we identify individuals with genotype u1 u2 and u2 u1 . Each individual has a Mendelian
reproduction rate with possible mutations and a natural death rate. Moreover, there is
an additional death rate due to ecological competition with the other individuals in the
population. Let
fu1 u2 ∈ R+
Du1 u2 ∈ R+
K∈N
cu1 u2 ,v1 v2
∈ R+
K
the per capita birth rate (fertility) of an individual with genotype u1 u2 .
the per capita natural death rate of an individual with genotype u1 u2 .
the carrying capacity, a parameter which scales the population size.
the competition effect felt by an individual with genotype u1 u2 from
an individual of genotype v1 v2 .
Ru1 u2 (v1 v2 ) ∈ {0, 1} the reproductive compatibility of the genotype v1 v2 with u1 u2
µK ∈ R+
the mutation probability per birth event. Here it is independent of the
genotype.
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
m(u, dh)
3
mutation law of a mutant allelic trait u + h ∈ U, born from an
individual with allelic trait u.
Scaling the competition function c down by a factor 1/K amounts to scaling the population
size to order K. We are interested in asymptotic results when K is large. We assume rare
mutation, i.e. µK 1. If a mutation occurs at a birth event, only one allele changes from
u to u + h where h is a random variable with law m(u, dh).
At any time t, there is a finite number, Nt , of individuals, each with genotype in U 2 .
We denote by u11 (t)u12 (t), ..., u1Nt (t)u2Nt (t) the genotypes of the population at time t. The
population, νt , at time t is represented by the rescaled sum of Dirac measures on U 2 ,
Nt
1X
δi i .
(1.1)
νt =
K i=1 u1 (t)u2 (t)
Formally, νt takes values in the set of re-scaled point measures


n


X


1

1 1
n n
2
i
i
n
≥
0,
u
u
,
...,
u
MK = 
δ
u
∈
U
,

u1 u2 1
2
1
2


K

i=1
(1.2)
on U 2 , equipped with the vague topology. Define hν, gi as the integral of the measurable
function g : U 2 → R with respect to the measure ν ∈ MK . Then hνt , 1i = NKt and for any
u1 u2 ∈ U 2 , the positive number hνt , 1u1 u2 i is called the density at time t of the genotype
u1 u2 . The generator of the process is defined as in [7]: first we define, for the genotypes
u1 u2 , v1 v2 and a point measure ν, the Mendelian reproduction operator:
(Au1 u2 ,v1 v2 F)(ν)
"
!
!
!
!#
δu1 v1
δu1 v2
δu2 v1
δu2 v2
1
F ν+
+F ν+
+F ν+
+F ν+
− F(ν),
=
4
K
K
K
K
and the Mendelian reproduction-cum-mutation operator:
!
!!
Z "
δu1 +h,v1
δu1 +h,v2
1
(Mu1 u2 ,v1 v2 F)(ν) =
F ν+
+F ν+
m(u1 , h)
8 R
K
K
!
!!
δu2 +h,v1
δu2 +h,v2
+ F ν+
+F ν+
m(u2 , h)
K
K
!
!!
δu2 ,v1 +h
δu1 ,v1 +h
+F ν+
m(v1 , h)
+ F ν+
K
K
!
!!
#
δu1 ,v2 +h
δu2 ,v2 +h
+ F ν+
+F ν+
m(v2 , h) dh − F(ν).
K
K
(1.3)
(1.4)
The process (νt )t≥0 is then a MK -valued Markov process with generator LK , given for
any bounded measurable function F : MK → R by:
(LK F)(ν)
!
!
!
Z
Z
δu1 u2
=
Du1 u2 +
cu1 u2 ,v1 v2 ν(d(v1 v2 )) F ν −
− F(ν) Kν(d(u1 u2 ))
K
U2
U2
!
Z
Z
fv1 v2 Ru1 u2 (v1 v2 )
+
(1 − µK ) fu1 u2
(Au1 u2 ,v1 v2 F)(ν)ν(d(v1 v2 )) Kν(d(u1 u2 ))
hνRu1 u2 , f i
U2
U2
!
Z
Z
fv1 v2 Ru1 u2 (v1 v2 )
+
µK fu1 u2
(Mu1 u2 ,v1 v2 F)(ν)ν(d(v1 v2 )) Kν(d(u1 u2 )). (1.5)
hνRu1 u2 , f i
U2
U2
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
4
The first non-linear term describes the competition between individuals. The second
f v2 Ru1 u2 (v1 v2 )
and last linear terms describe the birth with and without mutation. There, fu1 u2 v1KhνR
u1 u2 , f i
is the reproduction rate of an individual with genotype u1 u2 with an individual with genotype v1 v2 . Note that νRu1 u2 is the population restricted to the pool of potential partners of
an individual of genotype u1 u2 .
For all u1 u2 , v1 v2 ∈ U 2 , we make the following Assumptions (A):
(A1) The functions f, D and c are measurable and bounded, which means that there
exists f¯, D̄, c̄ < ∞ such that
0 ≤ fu1 u2 ≤ f¯, 0 ≤ Du1 u2 ≤ D̄ and 0 ≤ cu1 u2 ,v1 v2 ≤ c̄.
(1.6)
(A2) fu1 u2 − Du1 u2 > 0 and there exists c > 0 such thatR c ≤ cu1 u2 ,v1 v2 .
(A3) There exists a function, m̄ : R → R+ , such that m̄(h)dh < ∞ and m(u, h) ≤ m̄(h)
for any u ∈ U and h ∈ R.
For fixed K, under the Assumptions (A1)+(A3) and assuming that E(hν0 , 1i) < ∞,
Fournier and Méléard [15] have shown existence and uniqueness in law of a process with
infinitesimal generator LK . For K → ∞, under mild restrictive assumptions, they prove
the convergence of the process νK in the space D(R+ , MK ) of càdlàg functions from R+ to
MK , to a deterministic process, which is the solution to a non-linear integro-differential
equation. Assumption (A2) ensures that the population does not tend to infinity in finite
time or becomes extinct too fast.
1.2. Previous works. Consider the process starting with a monomorphic aa-population,
with one additional mutant individual of genotype aA. Assume that the phenotype difference between the mutant and the resident population is small. The phenotype difference is
assumed to be a slightly smaller death rate compared to the resident population, namely:
Daa = D,
DaA = D − ∆.
(1.7)
for some small enough ∆ > 0. The mutation probability for an individual with genotype
u1 u2 is given by µK . Hence, the time until the next mutation in the whole population is of
order Kµ1 K . Now assume that the demographic parameters introduced in Section 1.1 depend
continuously on the phenotype. In particular, they are the same for individuals bearing the
same phenotype.
In [7] it is proved that if the two alleles a and A are co-dominant and if the allele A is
slightly fitter than the allele a, namely
Daa = D,
DaA = D − ∆,
DAA = D − 2∆,
(1.8)
then in the limit of large population and rare mutations (ln K eV K for some V >
0), the suitably time-rescaled process converges to the TSS model of adaptive dynamics,
essentially as shown in [3] in the haploid case. In particular, the genotypes containing the
unfit allele a decay exponentially fast after the invasion of AA (see Figure 1).
If in place of co-dominance we assume, as in [25], that the fittest phenotype A is dominant, namely
Daa = D, DaA = D − ∆, DAA = D − ∆,
(1.9)
then this has a dramatic effect on the evolution of the population and, in particular, leads
to a much prolonged survival of the unfit phenotype aa. Indeed, it was know for some
time (see e.g. [24]) that in this case the unique stable fixpoint (0, 0, n̄AA ) corresponding to
a monomorphic AA population is degenerate, i.e. its Jacobian matrix has zero-eigenvalue.
1
µK K
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
5
This implies that in the deterministic system, the aa and aA populations decay in time
only polynomially fast to zero, namely like 1/t2 and 1/t, respectively. This is in contrast
to the exponential decay in the co-dominant scenario (see Figure 1). In [25] it was shown
that the deterministic system remains a good approximation of the stochastic system as
long as the size of the aA population remains much larger than K 1/2 and therefore that the
a-allele survives for a time of order at least K 1/2−α , for any α > 01 Note that this statement
is a non trivial fact, since it is not a consequence of the law of large numbers, because the
time window diverges as K grows. In summary, the unfit recessive a-allele survives in the
population much longer due to the slow decay of the aA-population.
F IGURE 1. Evolution of the model from a resident aa population at equilibrium with a small amount of mutant aA, and when the alleles a and A
are co-dominant (left) or when the mutant phenotype A is dominant (right).
It is argued in [25] that if we choose the mutation time scale in such a way that there
remain enough a-alleles in the population when a new mutation occurs, i.e.
1
ln K K 1/2−α as K → ∞, for some α > 0,
(1.10)
µK K
and if the new mutant can coexist with the unfit aa-individuals, then the aa-population can
potentially recover. This is the starting point of our paper.
1.3. Goal of the paper. The goal of this paper is to show that under reasonable hypothesis, the prolonged survival of the a-allele after the invasion of the A-allele can indeed lead
to a recovery of the aa-type. To do this, we assume that there will occur a new mutant
allele, B, that on the one hand has a higher fitness than the AA-phenotype but that (for simplicity) has no competition with the aa-type. The possible genotypes after this mutation
are aa, aA, AA, aB, AB, and BB, so that even for the deterministic system we have now to
deal with a 6-dimensional dynamical system whose analysis if far from simple.
Under the assumption of dominance of the fittest phenotype, and mutation rate satisfying (1.10), we consider the model described in Section 1.1 starting at the time of the
second mutation, that is (with probability converging to 1 as K → ∞) the AA population
being close to its equilibrium and the aA population having decreased to a size of order
KµK , while the aa-population is of the order of the square of the aA-population. We assume that there just occurred a mutation to a fitter (and most dominant) allele B: we thus
start with a quantity K1 of genotype AB. We will start with a population where AA is close
1
In [25] only state that survival occurs up to time K 1/4−α . However, taking into account that it is really
only the survival of the aA-population that needs to be ensured, one can easily improve this to K 1/2−α .
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
6
to its equilibrium, the populations of aa and aA are already small (of order ε2 and ε), and
by mutation a single individual of genotype AB appears.
By using well known techniques [3, 6], we know that the AB-population behaves as a
super-critical branching process and reaches the level ε with positive probability in a time
of order ln K, without perturbing the 3-system (aa, aA, AA).
We see in numerical solutions to the deterministic system that a reduced fertility together with a reduced competition between a and B phenotypes constitutes a sufficient
condition for the recovery of the aa-population. For simplicity and in order to prove
rigorous results, we suppose that there can be no reproduction between individuals of phenotypes a and B, nor competition between them, and we reduce the number of remaining
parameters as much as possible (see Section 2). We study the deterministic system which
corresponds to the large population limit of the stochastic counterpart, and we show that
(for an initial quantity ε of aA, ε2 of aa and ε3 of AB) the system converges to a fixed
point denoted by paB consisting of the two coexisting populations aa and BB. If no further
assumptions are made, we will show that the number of individuals bearing an a allele
decreases to level ε1+∆/(1−∆) (where ∆ is defined in (1.7)) before aa grows and stabilises at
order 1.
α
If ∆ < 1−2α
, this control on the a allele is in principle sufficient in order for the stochastic system to exhibit the recovery of aa with positive probability in the large population
1
limit. Indeed, if the mutation time is of order K 2 −α , then the initial amount of aa and aA
genotypes is close to the typical fluctuations of those populations. Following the heuristics
of [25] (although our six-dimensional stochastic process is surely much more tedious to
study), the deterministic system should constitute a good approximation of the process if
the typical fluctuations of populations containing an a allele do not bring them to extincα
tion. If ∆ < 1−2α
this ensures that the population containing an a allele is not falling below
−1/2
order K
at any time.
In order to go deeper and control the speed of recovery of the aa-population, we look for
a parameter regime which ensures that the aa-population always grows after the invasion
of B. Ensuring this lower bound on aa is not trivial at all, and the solution we found is
to introduce an additional parameter η, which lowers the competition between the aA and
BB populations, compared to the one between AA and BB. Note that the competition does
not depend only on the phenotype, and can be interpreted as a refinement of a phenotypic
competition for resources: the strength (or ability to get resources) of an individual not
only depends on its phenotype but also on the dominance of its genotype. We show that
for η larger than some positive value (of order ∆), the aa population always grows after
the invasion of B. The time of convergence to the coexistence fixed point is thus lowered,
see Figure 5. Moreover, we point out the existence of a bifurcation: for η larger than some
threshold, the co-existence fixed point paB becomes unstable and the system converges to
another fixed point where all populations coexist.
Our contribution is a formal description of a mechanism by which a recessive allele can
re-emerge in a population. This can be seen as a statement of genetic robustness exhibited
by diploid populations performing sexual reproduction.
The structure of the paper is the following. In Section 2 we describe our assumptions
on the parameters of the model, and compute the large population limit; in Section 3 we
present our results on the evolution of the deterministic system towards the co-existence
fixed point paB , and we give a heuristic of the proof. Section 4 is dedicated to the proof of
these results.
Notation. We write x = Θ(y) whenever x = O(y) and y = O(x) as ε → 0.
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
7
F IGURE 2. Simulation of the stochastic system for f = 6, D = 0.7, ∆ =
0.1, c = 1, η = 0.02, ε = 0.014 and K = 7000.
2. M ODEL SETUP
Let G = {aa, aA, AA, aB, AB, BB} be the genotype space. Let ni (t) be the number of
individuals with genotype i ∈ G in the population at time t and set niK (t) ≡ K1 ni (t).
Definition 2.1. The equilibrium size of a monomorphic uu-population, u ∈ {a, A, B}, is the
fixed point of a 1-dimensional Lotka-Volterra equation and is given by
fuu − Duu
.
cuu,uu
n̄u =
(2.1)
Definition 2.2. For u, v ∈ {a, A, B}, we call
S uv,uu = fuv − Duv − cuv,uu n̄u
(2.2)
the invasion fitness of a mutant uv in a resident uu-population.
We take the phenotypic viewpoint and assume that the B-allele is the most dominant
one. That means the ascending order of dominance (in the Mendelian sense) is given by
a < A < B, i.e.
(1) phenotype a consists of the genotype aa,
(2) phenotype A consists of the genotypes aA, AA,
(3) phenotype B consists of the genotypes aB, AB, BB.
For simplicity, we assume that the fertilities are the same for all genotypes, and that
natural death rates are the same within the three different phenotypes. Moreover, we
assume that there can be no reproduction between a and B phenotypes.
To sumarize, we make the following Assumptions (B) on the rates:
(B1) Fertilities. For all i ∈ G, and some f > 0
fi ≡ f
(2.3)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
8
(B2) Natural death rates. The difference in fitness of the three phenotypes is realised
by choosing a slightly higher natural death-rate of the a-phenotype and a slightly
lower death-rate for the B-phenotype. For some 0 < ∆ < D,
DaB
Daa = D + ∆,
DAA ≡ DaA = D
≡ DAB ≡ DBB = D − ∆
(2.4)
(2.5)
(2.6)
(B3) Competition rates. We require that phenotypes a and B do not compete with each
other. Moreover, we introduce a parameter η ≥ 0 which lowers the competition
between BB and aA. For some 0 ≤ η < c,
ci, j
{i, j}∈G×G
aa
aA
=: AA
aB
AB
BB
aa aA AA aB AB BB
c
c c
0 0 0
c
c c
c c c−η
c
c c
c c c
0
c c
c c c
0
c c
c c c
0 c−η c
c c c
A biological interpretation for this kind of competition could be that it is coded
in the alleles which food an individual with a given genotype prefers. Since an
AB-individual shares one B-allele with a BB-individual, they compete stronger for
the same food than AA with BB since those have completely different alleles.
(B4) Reproductive compatibility. We require that phenotypes a and B do not reproduce
with each other.
(Ri ( j)){i, j}∈G×G
aa
aA
≡ AA
aB
AB
BB
aa aA AA aB AB BB
1
1 1
0 0 0
1
1 1
1 1 1
1
1 1
1 1 1
0
1 1
1 1 1
0
1 1
1 1 1
0
1 1
1 1 1
Observe that, under Assumptions (B),
S AB,AA = f − (D − ∆) − cn̄AA = f − D + ∆ − c
S aa,BB = f − D − ∆.
f −D
= ∆,
c
(2.7)
(2.8)
Therefore, the mutant AB has a positive invasion fitness in the population AA, as well as
aa in the BB population (due to the absence of competition between them).
2.1. Birth rates. We assume that there is no recombination between phenotypes a and B.
Thus,
(1) the pool of possible partners for the phenotype a consists of phenotypes a and A;
the total population of this pool is denoted by
Σ3 := naa + naA + nAA ,
(2.9)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
9
(2) the pool of possible partners for the phenotype A consists of the three phenotypes
a, A, and B; the total population of this pool is denoted by
Σ6 := naa + naA + nAA + naB + nAB + nBB ,
(2.10)
(3) the pool of possible partners for the phenotype B consists of phenotypes A and B;
the total population of this pool is denoted by
Σ5 := naA + nAA + naB + nAB + nBB .
(2.11)
Computing the reproduction rates with the Mendelian rules as described in (1.5) leads to
the following (time-dependant) birth-rates bi = bi (n(t)):
baa = f
naa naa + 21 naA
1
n 1n
2 aB 2 aA
+ 12 naB
+f
,
naa + naA + nAA
naA + nAA + naB + nAB + nBB
1
1
1
n
n
+
n
+
n
aA
aa
aA
aB
2
2
2
+f
,
naa + naA + nAA + naB + nAB + nBB
naa
1
n
2 aA
+ nAA
1
n 1n
2 aA 2 aB
(2.12)
+ 12 nAB + 21 naB (nAA + nAB )
+f
naa + naA + nAA
naA + nAA + naB + nAB + nBB
1
n + nAA naa + naA + 12 naB + 41 naA nAB
2 aA
+f
,
naa + naA + nAA + naB + nAB + nBB
baA = f
bAA = f
1
1
n
n
2 AB 2 aA
baB = f
1
n
2 aA
+ naB
1
n
2 aB
+f
+ 21 nAB + nBB
1
n
2 aA
+ nAA + nAB
1
n
2 aB
1
n
2 aA
+ nAA
1
n
2 aA
+ nAA + 21 nAB
naa + naA + nAA + naB + nAB + nBB
naA + nAA + naB + nAB + nBB
bBB = f
naA + nAA + naB + nAB + nBB
bAB = f
+ nAA + 12 nAB
(2.13)
+f
1
n 1n
2 aA 2 aB
+f
1
n
2 aA
+ nAA
1
n
2 aB
(2.14)
naa + naA + nAA + naB + nAB + nBB
+ 21 nAB + nBB
naA + nAA + naB + nAB + nBB
+ 12 nAB + nBB
,
, (2.15)
+ 12 nAB + nBB
,
naa + naA + nAA + naB + nAB + nBB
(2.16)
(naB + nAB + 2nBB )2
.
naA + nAA + naB + nAB + nBB
1
4
(2.17)
2.2. Death rates. The death rates are the sum of the natural death and the competition:
daa
daA
dAA
daB
dAB
dBB
= naa (D + ∆ + c(naa + naA + nAA )),
= naA (D + c(naa + naA + nAA + naB + nAB ) + (c − η)nBB ),
= nAA (D + c(naa + naA + nAA + naB + nAB + nBB )),
= naB (D − ∆ + c(naA + nAA + naB + nAB + nBB )),
= nAB (D − ∆ + c(naA + nAA + naB + nAB + nBB )),
= nBB (D − ∆ + (c − η)naA + c(nAA + naB + nAB + nBB )).
(2.18)
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
10
2.3. Large population limit. By [15] or [7], for large populations, the behaviour of the
stochastic process is close to the solution of a deterministic equation.
Proposition 2.3 (Generalisation of Proposition 3.2 in [7]).
Let T > 0 and C ⊂ R6+ be a compact set. Assume that the initial condition nK (0) =
1
(n (0), naA (0), nAA (0), naB (0), nAB (0), nBB (0)) converges almost surely to a deterministic
K aa
vector x0 = (x10 , x20 , x30 , x40 , x50 , x60 ) ∈ C, as K → ∞.
Let ñ(t, x0 ) denote the solution to
ṅ(t) = b(n(t)) − d(n(t)) ≡ F(n(t)),


X


rm i.e. ṅi (t) = bi (n(t)) − Di +
ci, j n j (t) ni (t),
(2.24)
for all i ∈ G
(2.25)
j∈G
with initial condition x0 , where (bi )i∈G and (di )i∈G are given in (2.12)-(2.17) and (2.18)(2.23). Then, for all T > 0,
lim sup |niK (t) − ñi (t, x0 ))| = 0,
K→∞ t∈[0,T ]
a.s.,
(2.26)
for all i ∈ G.
2.4. Initial condition. Fix ε > 0 sufficiently small. For the results below, we will consider the dynamical system (2.24) starting with the initial condition:
n̄A ≥ nAA (0) ≥ n̄A − Θ(ε),
naA (0) = ε,
(2.27)
(2.28)
naa (0) = Θ(ε2 ),
(2.29)
nAB (0) = ε3 ,
nBB (0) = 0,
naB (0) = 0.
(2.30)
(2.31)
(2.32)
Remark. In all the figures below, the choice of parameters is the following:
f = 6, D = 0.7, ∆ = 0.1, c = 1, ε = 0.01,
and the parameter η is specified on each picture.
3. R ESULTS
We are working with a 6-dimensional dynamical system, and computing all the fixed
points analytically is impossible for a general choice of the parameters. We can however
compute those which are relevant for our study. We will call pA (resp. pB ) the fixed points
corresponding to the monomorphic AA (resp. BB) population at equilibrium, and paB the
fixed point corresponding to the coexisting aa and BB populations. Setting the relevant
populations to 0 and solving ṅ(t) = 0, we get:
pA = (0, 0, n̄A , 0, 0, 0)
pB = (0, 0, 0, 0, 0, n̄B )
paB = (n̄a , 0, 0, 0, 0, n̄B )
(3.1)
(3.2)
(3.3)
, n̄A = f −D
, and n̄B = f −D+∆
. Note that the BB equilibrium population is
where n̄a = f −D−∆
c
c
c
the same in pB and paB . This is due to the non-interaction between phenotypes a and B.
Our general result is that starting with initial conditions (2.27)-(2.32), that is close to
pA (with small coordinates in directions aa, aA and AB), and under minimal assumptions
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
11
��
F IGURE 3. General qualitative behaviour of {ni (t), i ∈ G} and projection of
the dynamical system on the coordinates aa, AA and BB. The re-invasion
of the aa population happens sooner and sooner as η grows (η = 0.02 for
both pictures).
on the parameters, the system gets very close to pB before finally converging to paB , see
Figure 3.
Theorem 3.1. Consider the dynamical system (2.24) started with initial conditions (2.27)(2.32). Suppose the following Assumptions (C) on the parameters hold:
(C1) ∆ sufficiently small,
(C2) f sufficiently large,
(C3) 0 ≤ η < c/2.
Then the system converges to the fixed point paB . More precisely, for any fixed δ > 0, as
ε → 0, it reaches a δ-neighbourhood of paB in a time of order Θ(ε−1/(1+ηn̄B −∆) ).
Moreover, it holds:
(1) for η = 0, the amount of allele a in the population decays to Θ(ε1+∆/(1+∆) ) before
reaching Θ(1),
(2) for η > 4∆
, the amount of a allele in the population is bounded below by Θ(ε) for
n̄B
all t > 0.
Remark. For η large, we prove that the fixed point paB is unstable. We observe numerically
that the system is attracted to a fixed point where all the 6 populations coexist, but we do
not prove this.
Let us now briefly discuss the linear stability of the relevant fixed points and give an
heuristics of the proof of Theorem 3.1.
3.1. Linear stability analysis. The Jacobian matrix JF := (∂Fi /∂n j )i j of the map F defined in (2.24) can be explicitly computed at pA and paB and the situation is as follows:
• The eigenvalues of JF (pA ) are 0, ∆ > 0 and −( f − D), −( f + ∆), −( f − ∆) (double)
which are all strictly negative under Assumptions (C). The fixed point pA is thus
unstable.
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
12
• The eigenvalues of JF (paB ) are 0 (double), and −(2 f − D), −( f − D + ∆), −( f − D −
∆), −(( f − D)(5 f − 4D) + f ∆)/(4( f − D) + ηn̄B ) which are strictly negative under
Assumptions (C). The linear analysis thus does not imply the stability of paB but
the Phase 4 of our proof will (see Section 4.5) .
It turns out that JF (pB ) is singular but as the invasion fitness of aa is positive, i.e.
S aa,BB > 0 (see (2.7)), this implies that a small perturbation in the first coordinate will
be amplified, and thus implies the instability of the fixed point pB .
3.2. Heuristics of the proof. Recall we start the dynamical system (2.24) with initial
conditions (2.27)-(2.32). A numerical solution of the system is provided on Figure 4.
Remark. Assumption C1 of Theorem 3.1 is needed throughout the proof in order to be
able to use the results of [25] which rely on the Center Manifold Theorem (a line of fixed
points becomes an invariant line under small enough perturbation).
Phase 1. Time period: until nAB = ε0 .
The mutant population, consisting of all individuals of phenotype B, first grows
up to ε0 exponentially fast with rate ∆ without perturbing the behaviour of the
3-system (aa, aA, AA). The rate of growth corresponds to the invasion fitness of
AB in the resident population AA, see (2.7). Following [25], AA stays close to n̄A ,
while aA and aa continue to decay like 1/t and 1/t2 respectively. The duration T 1
of this phase is such that Θ(ε3 )et∆ = Θ(1) ⇔ T 1 = Θ(| log ε|).
Phase 2. Time period: until naA = Θ(nAA ).
The evolution is a perturbation of an effective 3-system (AA, AB, BB) which behaves exactly the same as in [25], since the parameters satisfy the same hypotheses
(slightly lower death rate for phenotype B than for phenotype A, and constant competition parameters). A comparison result (following Theorem 4.5 below) shows
that this 3-system is almost unperturbed until naA = Θ(nAA ). If that happens in a
time T 2 diverging with ε (which we ensure throughout the calculation), we thus
know that BB approaches n̄B , while nAB ∝ 1/t and nAA ∝ 1/t2 .
The important fact in this phase is that the amount of allele a in the population
decays for η small while it increases for large enough η. Indeed, let us derive some
bounds on ΣaA,aB = naA + naB . The population ΣaA,aB reproduces by taking the dominant allele in a population of order Θ(1) and the allele a in itself. Thus its birth
rate satisfy bΣaA,aB ≈ f ΣaA,aB . We can compute its death rate exactly and use that
nBB ≈ Σ5 ≈ n̄B :
dΣaA,aB = ΣaA,aB (D − ∆ + cΣ5 ) − ηnaA nBB + ∆naA
≈ f ΣaA,aB − naA (ηn̄B − ∆),
(3.4)
(3.5)
Σ̇aA,aB ≈ naA (ηn̄B − ∆)
= Θ(ΣaA,aB · nAB )(ηn̄B − ∆)
(3.6)
(3.7)
The last equality comes from the fact that aA newborns have mainly their a allele
coming from ΣaA,aB and their A allele coming from AB. Using the 1/t decay of AB
we get:
Σ̇aA,aB ≈
Θ(ΣaA,aB )
(ηn̄B − ∆)
Θ(1) + Θ(1)t
(3.8)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
13
As ΣaA,aB (T 1 ) = Θ(ε) we deduce that ΣaA,aB (t) = Θ(ε)(Θ(1) + Θ(1)t)Θ(ηn̄B −∆) , and
thus naA = Θ(nAB ·ΣaA,aB ) = Θ(ε)(Θ(1) + Θ(1)t)Θ(ηn̄B −∆) /(Θ(1) + Θ(1)t). By solving
naA = Θ(nAA ) = Θ(n2AB ) we get the order of magnitude of T 2 = Θ(ε−1/(1+ηn̄B −∆) ).
Note that for η = 0, ΣaA,aB (T 2 ) = Θ(ε1+∆/(1−∆) ). Moreover, (3.7) implies that for
η > ∆/n̄B , we have Σ̇aA,aB > 0, which proves points 1 and 2 of Theorem 3.1.
Phase 3. Time period: until aa reaches equilibrium.
The fact that naA = Θ(nAA ) has a crucial effect on the birth rate of aa (see (2.12))
since the term (naa + 21 naA )/(naa + naA + nAA ) becomes of order Θ(1). As long as AA
stays smaller than Θ(ε), we get a lower bound on naa which grows exponentially
fast since f is chosen large enough (Assumption C2):
baa ≥ f naa Θ(1),
daa ≤ naa (D + ∆ + Θ(ε)),
ṅaa ≥ naa ( f Θ(1) − D − ∆ − Θ(ε)).
(3.9)
(3.10)
(3.11)
As aa grows, it makes ΣaA,aB grow, and thus AA and AB as well. We have to show
that this could not prevent aa from reaching equilibrium. We do not give a detailed
argument here, but essentially, the presence of the macroscopic BB population prevents all the non-aa populations to grow too much. Note that if η is too large, then
aA could get a positive fitness and grow to a macroscopic level. That is why we
have to impose Assumption C3, which will become clearer heuristically in the next
phase. We recall that aa does not compete with BB and thus it grows exponentially
fast with rate f − (D + ∆) until an ε0 -neighbourhood of the fixed point where aa
and BB coexist. The rate of growth corresponds to the invasion fitness of aa in
the resident population BB, see (2.7). Note that, due to Assumption C2, this rate
is much larger than the invasion rate of BB into AA. That is why the fourth phase
looks very steep on Figure 4, see the stretched version on Figure 6. This phase
lasts a time T 3 = Θ(| log ε|).
Phase 4. The Jacobian matrix of the field (2.24) at the fixed point paB has two zero, and 4
negative eigenvalues. paB is thus a non-hyperbolic equilibrium point of the system
and linearisation fails to determine its stability properties. Instead, we use the
result of center manifold theory ( [18,26]) that asserts that the qualitative behaviour
of the dynamical system in a neighbourhood of the non-hyperbolic critical point
paB is determined by its behaviour on the center manifold near paB . Using the
Center Manifold Theorem, we show that asymptotically as f → ∞, the field is
attractive for η < c · rmax where rmax ' 0.593644 is the maximum of the rational
function (4.334). Thus paB is a stable fixed point which is approached with speed
1
as long as η < c · rmax . For higher values of η, numerical solutions show that the
t
system converges to a fixed point where the 6 populations co-exist, but we do not
prove this.
4. P ROOF
Definition 4.1. Let x, y, z ∈ {aa, aA, AA, aB, AB, BB} and h ∈ R. We define
T x=y = inf{t > 0 : n x (t) = ny (t)},
(4.1)
T x=δy = inf{t > 0 : n x (t) = δny (t)},
(4.2)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
14
F IGURE 4. Numerical solution of the deterministic system for η = 0.02, logplot.
T hx = inf{t > 0 : n x (t) > h}
T hx+y
T hx+y+z
(4.3)
= inf{t > 0 : n x (t) + ny (t) > h}
(4.4)
= inf{t > 0 : n x (t) + ny (t) + nz (t) > h}
(4.5)
Moreover, let
∆ > ε0 > ε > 0.
(4.6)
The value ε0 is the small order 1 level in the Phase 1, see the proof heuristics (Section
3.2). We consider ∆ fixed and sufficiently small, and will first send ε → 0 and then ε0 → 0.
4.1. Preliminaries. We first prove general facts which will be useful through the proof.
Lemma 4.2. Let c > 0 and n(t) be such that
• ṅ(t) ≤ g(t) − c · n(t) for all t ∈ T ⊂ R+ ,
• c · n(0) ≤ g(0),
if c · n(t) = g(t) ⇒ c · ṅ(t) ≤ ġ(t) for all t ∈ T then c · n(t) ≤ g(t) for all t ∈ T ,
Proof. This is an easy analysis exercise.
Proposition 4.3. If naB (0) < nAB (0) then naB (t) ≤ nAB (t).
Proof. Intuitively this inequality comes from the fact that phenotype a individuals cannot
reproduce with phenotype B. Indeed, if we consider the couples that could give rise to an
AB (resp. aB) individual, they are of the form (Ag1 , Bg2 ) (resp. (ag1 , Bg2 )), with g1 , g2 ∈
{a, A, B} and the combination (AA, Bg2 ) is possible whereas (aa, Bg2 ) is impossible. Here
is the rigorous derivation of the result: We compare the birth- and the death-rates of nAB
and naB
dAB
daB
= D − ∆ + c(naA + nAA + naB + nAB + nBB ) =
,
(4.7)
naB
nAB
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
baB
bAB
15
+ 21 nAB + nBB
= f naB
+ IaB ,
naA + nAA + naB + nAB + nBB
1
n + 12 nAB + nBB
2 aB
+ IAB .
= f nAB
naA + nAA + naB + nAB + nBB
1
n
2 aB
(4.8)
(4.9)
We see that the death-rates of the two populations are the same, whereas the birth-rates
differ only in a factor which comes from the reproduction of the other populations. If we
take a closer look to these factors IaB , IAB under the assumption that naB = nAB we see that


1
1

n + 21 nAB + nBB
n + 12 nAB + nBB

2 aB
2 aB
1


IAB = f 2 naA + nAA 
+
naA + nAA + naB + nAB + nBB naa + naA + nAA + naB + nAB + nBB
(4.10)


1
1
n + 12 nAB + nBB
n + 12 nAB + nBB


2 aB
2 aB
 .
+
= IaB + f nAA 
naA + nAA + naB + nAB + nBB naa + naA + nAA + naB + nAB + nBB
(4.11)
Thus IAB > IaB . Hence, ṅAB > ṅaB and nAB (t) stays above naB (t) for all t > 0.
4.2. Phase 1: Perturbation of the 3-system (aa, aA, AA) until AB reaches Θ(1).
We start with initial conditions given by (2.27)-(2.32). We will show that the mutant
population, consisting of all individuals of phenotype B, grows up to some ε0 > ε without
perturbing the behaviour of the 3-system (aa, aA, AA) in this time. Let
T 1 := T εaB+AB+BB
.
0
(4.12)
Proposition 4.4. With the initial conditions (2.27)-(2.32), for all t ∈ [0, T 1 ], it holds,
(1) naB (t) ≤ Θ(εε0 ), naA (t) ≤ Θ(ε), naa (t) ≤ Θ(ε2 ) and n̄A − Θ(ε0 ) ≤ nAA (t) ≤ n̄A .
(2) nBB (t) = Θ(n2AB (t)).
(3) nAB (t) grows
with rate ∆. It reaches the level ε0 in a time at most of
exponentially
1
3 ∆−Θ(ε0 )
.
order Θ log (ε0 /ε )
Proof. Until T 1 the perturbation of the dynamics of the 3-system (aa, aA, AA) is at most
of order ε0 . Thus we have n̄A − Θ(ε0 ) ≤ nAA (t) ≤ n̄A + Θ(ε0 ), as well as naa , naA ≤ Θ(ε0 ).
With this rough bounds we will find finer bounds.
(1) The ∆ reduced death rate of the mutant AB gives it a positive fitness, and the
growth is exponential until it reaches a macroscopic level. For an upper bound on
the time T εaB+AB+BB
, we have to construct a minorising process for nAB . Indeed, let
0
us compare the birth and death rates:
2 f nAA
= nAB ( f − Θ(ε0 )),
nAA + Θ(ε0 )
≤ nAB (D − ∆ + cn̄A + Θ(ε0 )) = nAB ( f − ∆ + Θ(ε0 )).
bAB ≥ 21 nAB
(4.13)
dAB
(4.14)
Hence, we get for the minorising process
ṅAB ≥ nAB (∆ − Θ(ε0 )),
nAB (t) ≥ ε e
3 (∆−Θ(ε0 ))t
,
(4.15)
(4.16)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
��
��
F IGURE 5. Log-plots of {ni (t), i ∈ G} for η = 0 (top), η = 0.003 (center)
and η = 0.014 (bottom).
16
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
17
. For an lower bound on
and the time T 1 is at most of order Θ log((ε0 /ε )
the time T 1 , we have to construct a majorising process for nAB . We compare the
birth and death rates:
3
1
∆−Θ(ε0 )
bAB ≤ nAB ( f + Θ(ε0 )),
dAB ≥ nAB (D − ∆ + cn̄A − Θ(ε0 )) = nAB ( f − ∆ − Θ(ε0 )).
(4.17)
(4.18)
Hence, we get for the majorising process
ṅAB ≤ nAB (∆ + Θ(ε0 )),
(4.19)
nAB (t) ≤ ε e
,
(4.20)
1
and the time T 1 is at least of order Θ log((ε0 /ε3 ) ∆−Θ(ε0 ) .
(2) Heuristically, the newborns of genotype aA are still in majority produced by recombination of AA and aA, because the mutant population is not large enough to
contribute. The newborns of genotype aB are in majority produced by reproduction of the aA-population with the B-population. Finally, the newborns of genotype aa are in majority produced by recombination of aA and aA, because the only
mutant which could perturb it is aB which is of smaller order.
(a) We show that naa ≤ n2aA or according to Lemma 4.2 ṅaa − 2ṅaA naA ≤ 0 when
naa = n2aA .
Observe that ṅaa − 2ṅaA naA = baa − 2naA baA − daa + 2naA daA . The biggest
contributing terms of baa − 2naA baA and daa − 2naA daA at naa = n2aA are
3 (∆+Θ(ε0 ))t
baa − 2naA baA =
daa − 2naA daA =
f
n2 − 2Σ6f nAA n2aA ,
4Σ5 aB
−n2aA ( f − ∆ + Θ(ε0 )).
(4.21)
(4.22)
Thus we get as long as naB < naA :
ṅaa − 2ṅaA naA = baa − 2naA baA − daa + 2naA daA
≤ n2aA f − 2Σ6f nAA − ∆ + Θ(ε0 ) +
(4.23)
f
n2
4Σ5 aB
< 0.
(4.24)
(b) We show that naB really stays smaller than naA , precisely we show that naB ≤
naA nAB or equivalently according to Lemma 4.2 ṅaB − ṅaA nAB − ṅAB naA ≤ 0 at
naB = naA nAB .
The biggest contributing terms are
baB − nAB baA − naA bAB =naA nAB 4Σf 5 + 4Σf 6 − Σf6 nAA − 2Σf 5 nAA − 2Σf 6 nAA
+ naA nBB 2Σf 5 + 2Σf 6 − Σf5 nAA − Σf6 nAA ,
(4.25)
daB − nAB daA − naA dAB = − naA nAB (D + cΣ6 − ηnBB ).
(4.26)
Thus we get
ṅaB − ṅaA nAB − ṅAB naA ≤ naA nAB − f +
f
2Σ5
+ Θ(ε0 ) − naA nBB 2 f −
f
Σ5
− Θ(ε0 ) < 0.
(4.27)
(c) We show that naA ≤ Θ(ε1−ε0 ).
We construct a majorising process on aA. The biggest contributing terms are
baA ≤
f
n n
Σ6 AA aA
+ naA Θ(ε0 ),
daA ≥ naA ( f − Θ(ε0 )),
(4.28)
(4.29)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
18
and we get that
ṅaA ≤ Θ(ε0 )naA ,
Θ(ε0 )t
naA (t) ≤ εe
(4.30)
,
(4.31)
what shows that until time T 1 , naA ≤ Θ(ε1−ε0 ).
(d) We show n̄A − ε ≤ Σ5 ≤ n̄A + 2∆ε0 .
We construct a minorising and a majorising processes on Σ5 :
bΣ5 ≤ f Σ5 + Θ(naa ),
(4.32)
bΣ5 ≥ f Σ5 − Θ(n2aA ),
dΣ5 ≥ Σ5 (D + cΣ5 ) − (∆ + 2ηnaA )(naB + nAB + nBB ),
dΣ5 ≤ Σ5 (D + cΣ5 + cnaa ),
(4.33)
(4.34)
(4.35)
Σ̇5 ≤ Σ5 ( f − D − cΣ5 ) + (∆ + 2ηnaA )(naB + nAB + nBB ),
(4.36)
Σ̇5 ≥ Σ5 ( f − D − cΣ5 − cnaa ).
(4.37)
At the upper bound we have Σ̇5 ≤ 0 and at the lower bound Σ̇5 ≥ 0, which
ensure the claimed bounds by Lemma 4.2.
(3) The newborns of genotype BB are in majority produced by recombination of AB
with itself. Indeed, by comparison of the birth- and death-rates,
naB + nAB + nBB
f
+
n2
naA + nAA + naB + nAB + nBB naA + nAA + naB + nAB + nBB AB
≤ f nBB Θ(ε0 ) + n̄fA n2AB + Θ(ε30 ),
(4.38)
bBB ≤ f nBB
bBB ≥ f nBB Θ(ε0 ) +
f
n2 ,
4n̄B AB
(4.39)
dBB ≥ nBB (D − ∆ + cn̄A − Θ(ε0 )) = nBB ( f − ∆ − Θ(ε0 )),
dBB ≤ nBB (D − ∆ + cn̄B + Θ(ε0 )) = nBB ( f + Θ(ε0 )).
(4.40)
(4.41)
we get the upper bound for the process
ṅBB ≤ −nBB ( f (1 − Θ(ε0 )) − ∆ − Θ(ε0 )) +
f 2
n ,
n̄A AB
(4.42)
and the lower bound
ṅBB ≥ −nBB ( f + Θ(ε0 )) +
f
n2 .
4n̄B AB
(4.43)
By applying Lemma 4.2 to n = nBB and g = n2AB (with constants in front), as
nBB (0) = 0 < nAB (0) = ε3 and by Proposition 4.4 (2) ṅAB ≥ 0 for all t ∈ [0, T 1 ] , we
deduce that nBB (t) ≤ Θ(n2AB (t)) for all t ∈ [0, T 1 ].
1
∆−Θ(ε
0)
.
Note that Proposition 4.4 implies that T 1 = T εaB+AB+BB
= T εAB
≤ Θ log ε0 /ε3
0
0
4.3. Phase 2: Perturbation of the 3-system (AA, AB, BB) until naA = Θ(nAA ).
Let δ > 0 (to be chosen sufficiently small in the sequel). Let
T 2 := T aA=δAA ∧ T aB=δAB ∧ T aa=aA∧aB
(4.44)
We will show that for t ∈ [T 1 , T 2 ] the system behaves as a main 3-system (AA, AB, BB)
plus perturbations of order δ. The 3-system (AA, AB, BB) behaves exactly the same as
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
19
in [25] since the parameters satisfy the same hypotheses (slightly lower death rate for
phenotype B than for phenotype A individuals, and constant competition parameters).
Moreover, the crucial role of the parameter η is that the population containing an allele
a only continues to grow in this phase when η is large enough. This is due to the smaller
competition that aA feels from BB, the aA population is thus higher and induces the growth
of aB.
We start by considering how the growth of aA- and aB-populations can perturb the
3-system (AA, AB, BB).
Lemma 4.5. Let nup
. (t) be the population of the unperturbed 3- system (AA, AB, BB). The
3-system (AA, AB, BB) satisfies
 1

 f nAB +nBB 2

 2
 ,
ṅBB ≥ ṅup
−
(n
+
n
)
+
cn
(4.45)
aA
aB
BB
BB
 (nAA +nAB +nBB )2


 1

 f 4 naB + 12 nAB + nBB

+ cnBB ,
ṅBB ≤ ṅup
(4.46)
BB + (naA + naB ) 
Σ5



 f (nAB + nAA ) 21 nAB + nBB
 ,


+
cn
(4.47)
ṅAB ≥ ṅup
−
(n
+
n
+
n
)

AB 
aa
aA
aB 
AB

(nAA + nAB + nBB )2
f
f
1
1
1
n
n
+
n
+
n
+
n
n
+
n
,
(4.48)
ṅAB ≤ ṅup
+
aA
aB
AB
BB
aB
AB
AA
AB
Σ5
2
2
Σ5
2

 1
2

 f naA + 1 nAB +nAA
up
2
2
 ,
+
cn
ṅAA ≥ ṅAA − (naa + naA + naB )  (n
(4.49)
AA
AA +nAB +n BB )2
f
1
(4.50)
ṅAA ≤ ṅup
AA + 2Σ5 naA 2 naA + nAB + nAA .
Proof. We consider the rates of AA, AB and BB under the perturbation of aa, aA and aB:
1
1
f
n
n
+
n
+
n
aB
aB
AB
BB
2
f 1
4
2
bBB =
n + nBB +
(4.51)
2 AB
Σ5
Σ5
2
1
1
f 12 nAB + nBB (naA + naB )
n
+
n
+
n
f
n
aB
aB
AB
BB
4
2
=bup
+
,
(4.52)
BB −
Σ5 (nAA + nAB + nBB )
Σ5
up
dBB =dBB
+ cnBB (naB + naA ) − ηnaA nBB .
(4.53)
Thus,
+ 12 nAB + nBB + ηnaA nBB ,
(4.54)
2
f (naA + naB ) 12 nAB + nBB
up
ṅBB ≥ṅBB −
− cnBB (naA + naB ).
(4.55)
Σ5 (nAA + nAB + nBB )
For the AB-population we get:
2 f 12 nAB + nBB 12 nAB + nAA
f naa nAA 12 naB + 21 nAB + nBB
f (nAA + nAB )
bAB =
−
+
naB
Σ5
Σ5 Σ6
2Σ5
1
1
f naA 21 naB + 12 nAB + nBB
f naB nAA f naA 2 naB + 2 nAB + nBB
+
+
+
(4.56)
2Σ6
2Σ5
2Σ6
f
f
f naB nAB
f
f
1
1
=bup
+
n
n
+
n
+
n
+
+
n
n
+
+
aA
aB
AB
BB
aB
AA
AB
2
2
2Σ5
2Σ6
2Σ5
2Σ6
2Σ5
ṅBB ≤ṅup
BB +
f
n 1n
Σ5 aB 4 aB
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
−
f naa nAA
up
dAB =dAB
ṅAB ≤ṅup
AB
1
n
2 aB
+ 12 nAB + nBB
Σ5 Σ6
2f
−
+ cnAB (naB + naA ),
+ Σf5 naA 21 naB + 21 nAB + nBB +
1
1
f naB + nAB +nBB
2
2
Σ5 Σ6
ṅAB ≥ṅup
AB −
naa nAA −
1
n
2 AB
+ nAA
1
n
2 AB
+ nBB (naA + naB )
Σ5 (nAA + nAB + nBB )
f
n n + 2Σf 5 naB nAB ,
Σ5 aB AA
1
1
2 f nAB +nAA
nAB +nBB
2
2
(naA + naB )
Σ5 (nAA +nAB +nBB )
20
,
(4.57)
(4.58)
(4.59)
− cnAB (naB + naA ).
(4.60)
And finally for the AA-population:
2
1
1
1
1
f 21 nAB + nAA
f
n
n
n
+n
+
n
f
n
n
+n
+
n
aa
AA
aA
AA
AB
aA
aA
AA
AB
f naA nAB
2
2
2
2
bAA =
+
−
+
Σ5
4Σ5
Σ5 Σ6
2Σ6
(4.61)
2
f 12 nAB + nAA (naA + naB ) f naA nAB f naa nAA 12 naA + nAA + 21 nAB
up
=bAA −
+
−
Σ5 (nAA + nAB + nBB )
4Σ5
Σ5 Σ6
f naA 21 naA + nAA + 12 nAB
,
(4.62)
+
2Σ6
up
dAA =dAA
+ cnAA (naa + naA + naB ),
(4.63)
1
1
f naA nAB f naA 2 naA + nAA + 2 nAB
up
ṅAA ≤ṅAA +
+
,
(4.64)
4Σ5
2Σ6
2
1
1
f
n
+
n
+
n
(naa + naA + naB )
aA
AB
AA
2
2
− cnAA (naa + naA + naB ).
(4.65)
ṅAA ≥ṅup
AA −
Σ5 (nAA + nAB + nBB )
As solutions of a dynamical system are continuous with respect to its parameters (in particular with respect to δ), the latter theorem shows that until T 2 , the 3-system (AA, AB, BB)
is at most perturbed by Θ(δ). We will show that T 2 diverges with ε. Thus, for small enough
√
δ, AB will have time to reach the small fixed value ε0 > 0 in this phase, and we can use
the asymptotic decay of the AB and AA populations which is proved in [25]. We now start
to analyse the growth of the small aa-, aA- and aB-populations. The sum-process Σ5 plays
a crucial role for the behaviour of the system in this phase and we need finer bounds on it:
Proposition 4.6. The sum-process Σ5 = naA + nAA + naB + nAB + nBB satisfies for all
t ∈ [T 1 , T 2 ]:
∆
∆2
∆
∆2
nAA −
nAA ≤ Σ5 ≤ n̄B −
nAA +
nAA .
(4.66)
n̄B −
cn̄B
cn̄B
cn̄B
cn̄B
Proof. We estimate a minorising process and a majorising process on Σ5 :
(nAA + nAB + nBB )(naA + nAA + naB + nAB + nBB )
bΣ5 ≤ f
naA + nAA + naB + nAB + nBB
(naA + naB )( 43 naA + nAA + 34 naB + nAB + nBB )
+f
+ Θ(δ) ≤ f Σ5 + Θ(δ),
(4.67)
naA + nAA + naB + nAB + nBB
(nAA + nAB + nBB )(naA + nAA + naB + nAB + nBB )
bΣ5 ≥ f
naA + nAA + naB + nAB + nBB
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
(naA + naB )( 34 naA + nAA + 34 naB + nAB + nBB )
+f
− Θ(δ) ≥ f Σ5 − Θ(δ),
naA + nAA + naB + nAB + nBB
dΣ5 ≤Σ5 (D − ∆ + cΣ5 ) + ∆(nAA + naA ) − 2ηnaA nBB + Θ(δ),
dΣ5 ≥Σ5 (D − ∆ + cΣ5 ) + ∆(nAA + naA ) − 2ηnaA nBB .
21
(4.68)
(4.69)
(4.70)
We get
Σ̇5 ≤ −cΣ25 + Σ5 ( f − D + ∆) − ∆nAA + Θ(δ),
(4.71)
Σ̇5 ≥
(4.72)
−cΣ25
+ Σ5 ( f − D + ∆) − ∆nAA − Θ(δ).
We start with the proof of the upper bound. We use Lemma 4.2 and show that when Σ5
reaches the upper-bound, it decays faster than the latter. Using (4.71) we compute Σ̇5 at
2
2
2
n + ∆ n2 + 2∆c nAA +
the bound. Note that if Σ5 ≤ n̄B − cn̄∆B nAA + cn̄∆ B nAA , then Σ25 ≤ n̄2B − 2∆
c AA c2 n̄2 AA
B
Θ(∆4 )n2AA , thus
Σ̇5 ≤ −∆2 nAA −
∆2 2
n
cn̄2B AA
+ Θ(δ) < 0.
(4.73)
It is left to show that Σ̇5 ≤ − cn̄∆B ṅAA + cn̄∆ B ṅAA . Since we already know (cf. Lemma 4.5)
that (AA, AB, BB) behaves like a 3-system with Θ(δ) perturbations, then AA is decreasing,
ṅAA ≤ 0, this finishes the proof of the upper bound.
2
n −
Now we check the lower bound. If Σ5 ≥ n̄B − cn̄∆B nAA − cn̄∆ B nAA then Σ25 ≥ n̄2B − 2∆
c AA
2∆2
∆2
n − c nAA . Using (4.72), the derivative of Σ5 at the lower bound is thus lower
c2 n̄2B AA
bounded by
2
Σ̇5 ≥ ∆2 nAA − cn̄∆ 2 nAA − Θ(δ) ≥ ∆2 nAA 1 − cn̄1B − Θ(δ) > 0.
(4.74)
2
B
By Lemma 4.2, it is enough to show that at the lower bound Σ̇5 ≥ − cn̄∆B ṅAA . For this we
calculate a majorising process on AA:
bAA ≤
f
n (n
Σ5 AA AA
+ nAB ) +
f
n2
4Σ5 AB
+ Θ(δ),
dAA ≥ f nAA ,
(4.75)
(4.76)
ṅAA ≤ − Σf5 nAA nBB + 4Σf 5 n2AB + Θ(δ).
(4.77)
Hence we have to show that ∆2 nAA 1 − cn̄1B − Θ(δ) ≥ cn̄∆Bfn̄A nAA nBB − 14 n2AB − Θ(δ∆),
in the case nAA nBB > 14 n2AB . This is equivalent to show that χ := nAA nBB − 41 n2AB ≤
∆n̄A
(cn̄B − 1) nAA . For this we use once again Lemma 4.2 and estimate the derivative of
f
χ from above with the help of minorising processes on AA and BB and a majorising process on AB:
bAA ≥
f
n (n
Σ5 AA AA
+ nAB ) +
f
n2
4Σ5 AB
− Θ(δ),
(4.78)
dAA ≤ ( f + ∆)nAA + Θ(δ),
ṅAA ≥
bBB ≥
(4.79)
− Σf5 nAA nBB − ∆nAA + 4Σf 5 n2AB − Θ(δ).
f
n (n + nBB ) + 4Σf 5 n2AB − Θ(δ),
Σ5 BB AB
(4.80)
(4.81)
dBB ≤ f nBB ,
ṅBB ≥
bAB ≤
− Σf5 nAA nBB + 4Σf 5 n2AB + Θ(δ).
2f
f
1
n
n
+
n
+
n
AA
BB + Σ5 nAA n BB
Σ5 AB
2 AB
dAB ≥ ( f − ∆)nAB ,
(4.82)
(4.83)
+ Θ(δ),
(4.84)
(4.85)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
ṅAB ≤
2f
n n
Σ5 AA BB
−
f
n2
2Σ5 AB
22
+ ∆nAB + Θ(δ).
(4.86)
The derivative is given by:
χ̇ = ṅAA nBB + nAA ṅBB − 12 ṅAB nAB
≤ − f χ + Θ(δ).
(4.87)
(4.88)
χ̇ ≤ −∆n̄A (cn̄B − 1)nAA + Θ(δ) < 0.
(4.89)
At the upper bound we get:
It is left to show that χ̇ ≤
f
χ − Θ(δ) we show that
n̄A
∆n̄A
(cn̄B
f
0 ≤ ( f − 2∆)χ −
− 1)ṅAA . Using the minorising process ṅAA ≥ −∆nAA −
∆n̄A
(cn̄B
n̄B
− 1)χ −
∆2 n̄A
(cn̄B
f
− 1)nAA − Θ(δ).
(4.90)
An easy calculation proves this fact and finishes the proof of the lower bound.
Lemma 4.7. For t ∈ [T 1 , T 2 ] and for ∆ sufficiently small it holds,
Σ̇aA,aB ≥ −Θ(∆)ΣaA,aB .
(4.91)
Proof. Using Propositions 4.6, we have the following bound on the process:
naA ( 12 naA + nAA + naB + nAB + nBB ) + naB (nAA + 12 naB + nAB + nBB )
− Θ(δnaA )
naA + nAA + naB + nAB + nBB
≥ f ΣaA,aB − Θ(δnaA ),
(4.92)
= ΣaA,aB (D − ∆ + cΣ5 ) − ηnaA nBB + ∆naA + cnaA naa
bΣaA,aB ≥ f
dΣaA,aB
≤ f ΣaA,aB − naA (ηnBB − ∆) + Θ(∆2 nAA )Σ2aA,aB ,
(4.93)
Σ̇aA,aB ≥ naA (ηnBB − ∆ − Θ(δ))Θ(∆2 nAA )Σ2aA,aB ≥ naA (−∆ − Θ(δ)) − Θ(δ∆2 nAA )ΣaA,aB
≥ ΣaA,aB (−∆ − Θ(δ)).
(4.94)
Lemma 4.8. For all t ∈ [T 1 , T 2 ] the aa-population is bounded by
f
f
Σ2aA,aB ≤ naa ≤
Σ2 .
4n̄B ( f + ∆)
n̄A (D + ∆) aA,aB
(4.95)
Observe that this implies T 2 = T aA=δAA ∧ T aB=δAB .
Proof. First observe that the inequality is satisfied at t = T 1 . We start with the upper bound
and show that naa would decrease at this bound. For this we estimate a majorising process
on aa:
baa ≤ naa +naAf +nAA naa 21 naA + naa + 4Σf 5 Σ2aA,aB + 2Σf 5 naA naa ,
(4.96)
daa ≥ naa (D + ∆),
ṅaa ≤
(4.97)
f
n2
naa +naA +nAA aa
+
− naa (D + ∆).
(4.98)
−Θ(δ) 2
+ Θ(Σ2aA,aB naA ) ≤ − 3 f 4n̄
ΣaA,aB < 0.
A
(4.99)
f
n n
naa +naA +nAA aa aA
+
f
Σ2
4Σ5 aA,aB
We calculate the slope of this process at the upper bound:
ṅaa ≤
f
Σ2
4Σ5 aA,aB
−
f 2
Σ
n̄A aA,aB
By Lemma 4.2, to ensure that (4.95) stays an upper bound it is enough to show that
−Θ(δ) 2
− 3 f 4n̄
ΣaA,aB ≤
A
2f
Σ̇
Σ
.
n̄A (D+∆) aA,aB aA,aB
(4.100)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
23
This is a consequence of Lemma 4.7.
For the lower bound we proceed similarly. This time, with the knowledge of the upper
bound, we estimate a minorising process on aa:
baa ≥
f 2
Σ
Σ5 aA,aB
− Θ(Σ3aA,aB ),
(4.101)
daa ≤ naa ( f + ∆),
ṅaa ≥
f 2
Σ
n̄B aA,aB
(4.102)
− naa ( f + ∆) − Θ(Σ3aA,aB ).
At the lower bound the process increases:
ṅaa ≥ n̄fB − 4n̄f B Σ2aA,aB − Θ(Σ3aA,aB ) =
3f 2
Σ
4n̄B aA,aB
− Θ(Σ3aA,aB ) > 0.
f
Σ̇
Σ
.
2n̄B ( f +∆) aA,aB aA,aB
By Lemma 4.2, it is left to show that ṅaa ≥
a majorising process on ΣaA,aB :
(4.103)
(4.104)
Thus we have to calculate
bΣaA,aB ≤ f ΣaA,aB + Θ(Σ2aA,aB ),
dΣaA,aB ≥ ( f − ∆)ΣaA,aB + naA (∆ − ηnBB )
≥ ( f − ∆)ΣaA,aB − ( f − D)ΣaA,aB
= (D − ∆)ΣaA,aB ,
(4.105)
(4.106)
(4.107)
(4.108)
Σ̇aA,aB ≤ ( f − D + ∆)ΣaA,aB + Θ(Σ2aA,aB ).
(4.109)
Thus we get
f ( f −D+∆) 2
Σ
2n̄B ( f +∆) aA,aB
−
3f 2
Σ
4n̄B aA,aB
+ Θ(Σ3aA,aB ) = − 2n̄f B Σ2aA,aB
=
3
− f −D+∆
+ Θ(Σ3aA,aB )
2
f +∆
− 2n̄f B Σ2aA,aB f2(+2D+∆
+ Θ(Σ3aA,aB ) < 0
f +∆)
(4.110)
(4.111)
This finishes the proof of the lower bound.
Let
T ≡ = inf{t > T 1 : naA (t) = naB (t)}.
(4.112)
Proposition 4.9. For all t ∈ [T 1 , T = ] it holds
naB ≤ naA = Θ(ε).
(4.113)
Proof. In this time interval the newborns of genotype aA are in majority produced by reproductions of a population of order one, namely AB or AA, with the population aA. Since
naA feels competition from a macroscopic population (AA, AB or BB) the aA-population
stays of order Θ(ε). We make this more rigorous. To show this we consider a majorising
process on aA and use Proposition 4.6, and Lemma 4.8:
f
n (n + 12 nAB ) + 2Σf 5 naB (2nAA + nAB ) + Θ(Σ2aA,aB ),
Σ5 aA BB
naA ( f + ∆ − n̄∆B nAA − ηnBB − Θ(∆2 nAA )),
f
f
nAA
2
5
−naA nBB f −ηΣ
+
n
+
∆
1
−
−
Θ(∆
n
)
+ Σ5 naB ( 21 nAB
AB
AA
Σ5
2Σ5
n̄B
baA ≤ f naA −
(4.114)
daA ≥
(4.115)
ṅaA ≤
≤ −naA nBB D+∆
+
Σ5
≤ −naA
f
Σ5
f
n
2Σ5 AB
D+∆
nBB
f
+∆ 1−
nAA
n̄B
+ 12 nAB + ∆ 1 −
− Θ(∆2 nAA ) +
nAA
n̄B
f
n (1n
Σ5 aB 2 AB
− Θ(∆2 nAA ) +
+ nAA + Θ(δ))
(4.116)
+ nAA + Θ(δ))
(4.117)
f
n (1n
Σ5 aB 2 AB
+ nAA + Θ(δ)).
(4.118)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
24
By Proposition 4.5 and [25] there exists a time t0 = Θ(1) such that the expression in the
first bracket becomes bigger than the expression in the second bracket. Thus naA decreases
after t0 and since aA does not exceed Θ(ε) until t0 it will stay smaller or equal to Θ(ε) until
T=.
We show that as soon as aB crosses aA the BB-population is already bigger than or
equal to the AA-population. First we estimate a upper bound for aB:
Lemma 4.10. For all t ∈ [T 1 , T 2 ] the aB-population is upper bounded by
naB
nAB + 2nBB + 2∆
c
naA ≡ C(t)naA .
≤
nAB + 2nAA
(4.119)
Proof. First observe that the bound is fulfilled at t = T 1 . Similarly to the proof of Lemma
4.8 we estimate a majorising process on aB given by:
ṅaB ≤ −naB 2Σf 5 (nAB + 2nAA ) − n̄∆B nAA − Θ(∆2 nAA ) + naA 2Σf 5 (nAB + 2nBB + Θ(δ)). (4.120)
By Lemma 4.2, we have to show that as soon as aB reaches the upper bound it decreases
faster than the bound, thus we calculate the slope of the majorising process at this value:
+2nBB +2∆/c)
ṅaB ≤ − 2Σf 5 nAB + 2nBB + 2∆
− Θ(∆2 nAA ) naA + ∆(nn̄AB
nAA naA + 2Σf 5 (nAB + 2nBB )naA
c
B (nAB +2nAA )
(4.121)
2
nAA )
naA + Σ∆5 12 nAB + nBB + ∆c naA
(4.122)
≤ − ∆ f −Θ(∆
cΣ5
2
nAA )
≤ ∆+Θ(∆
naA n̄B + ∆c − cf
(4.123)
Σ5
nAA )
(D − 2∆) naA ≤ 0.
≤ − ∆+Θ(∆
cΣ5
2
(4.124)
We have to show that ṅaB ≤ C(t)ṅaA + Ċ(t)naA . Since the 3-system converges towards
(0, 0, n̄B ), C(t) is a monotone increasing function and hence Ċ(t) ≥ 0. Thus if we can show
that ṅaB ≤ C(t)ṅaA we are done. For this we have to calculate the slope of the minorising
process on aA when aB would reach the upper bound. This process is given by:
ṅaA ≥ −naA 2f + ∆ − ηnBB + 2Σf 5 (nBB − nAA ) + Θ(δ) + naB 2Σf 5 (nAB + 2nAA ).
(4.125)
The slope at the upper bound is:
ṅaA ≥ −naA 2f + ∆ − ηnBB + 2Σf 5 (nBB − nAA ) −
∆f
≥ −naA ∆ − ηnBB − cΣ
+
Θ(δ)
5
D−∆
≥ naA ∆ cΣ5 + ηnBB − Θ(δ) ≥ 0.
f
2Σ5
nAB + 2nBB +
2∆
c
+ Θ(δ)
(4.126)
(4.127)
(4.128)
Since C(t) > 0 this finishes the proof.
Lemma 4.11. We have T = ≤ T 2 . Moreover it holds,
nAA (T = ) ≤ nBB (T = ) + Θ(∆).
(4.129)
Proof. We first show that T = < T 2 . Using Proposition 4.6 we construct two processes that
provide an upper bound and a lower bound on naB , respectively:
baB ≥ f naB −
baB ≤ f naB −
f
n (1n
Σ5 aB 2 AB
f
n (1n
Σ5 aB 2 AB
+ nAA ) +
+ nAA ) +
daB ≤ naB f,
daB ≥ naB ( f −
f
n (1n
Σ5 aA 2 AB
f
n (1n
Σ5 aA 2 AB
+ nBB − Θ(δ2 )),
(4.130)
+ nBB + Θ(δ)),
(4.131)
(4.132)
∆
n
n̄B AA
− Θ(∆2 nAA )),
(4.133)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
ṅaB
 1
 f ( nAB + nAA )
≤ −naB  2
−
Σ5
∆
n
n̄B AA

f ( 12 nAB + nBB + Θ(δ))

− Θ(∆ nAA ) + naA
,
Σ5
2
25
(4.134)
f ( 12 nAB + nAA )
f ( 12 nAB + nBB − Θ(δ2 ))
ṅaB ≥ −naB
+ naA
.
(4.135)
Σ5
Σ5
We first show that T = < ∞. We know that the 3-system (AA, AB, BB) converges to (0, 0, n̄B )
and that naB ≤ naA = Θ(ε) (Proposition 4.9), for t ≤ T = . We consider the worst case and
assume that naB < naA then we get from (4.135) that at some time t0 , where nAB + 2nBB is
already macroscopic,
ṅaB ≥ Θ(ε),
naB ≥ Θ(ε)t.
(4.136)
Thus the time aB needs to reach naA = Θ(ε) is of order Θ(1). This time is shorter than
T aA=δAA . Indeed, suppose the contrary, then by
4.9 naA does not exceed Θ(ε)
Proposition
aA=δAA
AA
2
before T 2 , and thus T
≥ T Θ(ε/δ) = Θ (δ/ε) which diverges with ε. A similar
aB=δAB
reasoning shows that T = < T
. Hence T = < T 2 .
It is left to show that nAA (T = ) ≤ nBB (T = ) + Θ(∆). From Lemma 4.10 we deduce that at T =
it holds
1
n
2 AB
+ nAA ≤ 12 nAB + nBB +
nAA ≤ nBB + Θ(∆).
∆
c
(4.137)
(4.138)
Lemma 4.12. For all t ∈ [T 1 , T 2 ] the AB-population is bounded by
√
(1) nAB ≥ 2 n̄B nAA − 2nAA 1 + cn̄∆B ,
q
(2) nAB ≤ 2 n̄B nAA 1 + ∆f − 2nAA .
Proof.
(1) The proof works like the one of Lemma 4.8. First observe that the bound holds at
t = T 1 . Then we calculate a minorising process on AB:
bAB ≥ f (2nAA + nAB ) −
dAB ≤ f nAB ,
ṅAB ≥ −nAB Σf5
f
(2nAA
Σ5
+ nAB )(nAA + 21 nAB + Θ(δ2 )),
(4.139)
(4.140)
2
1
n
+
n
+
Θ(δ
)
+ 2 f nAA −
AA
2 AB
2f
n
Σ5 AA
2
1
n
+
n
+
Θ(δ
)
.
AA
2 AB
(4.141)
We use Proposition 4.6 and show that this minorising process would increase quicker than
the lower-bound if AB reaches it:
√
√
ṅAB ≥ − 2Σ5f
n̄B nAA − nAA 1 + cn̄∆B
n̄B nAA − cn̄∆B nAA + Θ(δ2 )
√
+ 2 f nAA − 2Σ5f nAA n̄B nAA − cn̄∆B nAA
(4.142)
√
≥ 2Σ5f cn̄∆B nAA (2 n̄B nAA − nAA ) − Θ(∆2 ) > 0.
(4.143)
It is left to show that at the lower bound,
n̄B ṅAA
ṅAB ≥ √
− 2ṅAA 1 +
n̄B nAA
For this we calculate a majorising process on AA:
bAA ≤
f
n (n
Σ5 AA AA
dAA ≥ f nAA ,
+ nAB ) +
f
n2
4Σ5 AB
∆
cn̄B
.
(4.144)
+ Θ(δ),
(4.145)
(4.146)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
ṅAA ≤ −nAA f −
f
(n
Σ5 AA
+ nAB ) +
f
n2
4Σ5 AB
+ Θ(δ).
If we now insert the lower bound and use Proposition 4.6 we get
√
ṅAA ≤ − Σf5 cn̄∆B nAA ( n̄B nAA − nAA ) + Θ(∆2 ) < 0.
26
(4.147)
(4.148)
Thus (4.144) is fulfilled.
(2) First, observe that the upper bound is fullfiled at t = T 1 . We then have to estimate a
majorising process on AB:
bAB
dAB
nAA + 21 nAB
+ Θ(δ),
≤ f (2nAA + nAB ) − f (2nAA + nAB )
Σ5
≥ nAB (D − ∆ + cn̄B − n̄∆B nAA − Θ(∆2 nAA ))
≥ nAB ( f −
∆
n
n̄B AA
− Θ(∆2 nAA )),
(4.149)
(4.150)
(4.151)
ṅAB ≤ − 2n̄f B n2AB − nAB 2 fn̄−∆
nAA + 2 f nAA −
B
2f 2
n
n̄B AA
+ Θ(∆2 nAA ).
(4.152)
As before we calculate the slope of this majorising process if it would reach the upper
bound:
ṅAB ≤ − 2∆
n2 + Θ(∆2 nAA ) < 0.
n̄B AA
(4.153)
By Lemma 4.2 we have to show that
!
ṅAB ≤ ṅAA √
n̄B (1+∆/ f )
n̄B nAA (1+∆/ f )
−2 .
For this we calculate the slope of a minorising process on AA given by
ṅAA ≥ −nAA f − Σf5 (nAA + nAB ) + ∆ + Θ(δ2 ) + 4Σf 5 n2AB .
(4.154)
(4.155)
At the upper bound AA would start to increases:
ṅAA ≥
∆ 2
n
n̄B AA
− Θ(δ2 ) > 0.
(4.156)
Thus we get
!
ṅAA √
n̄B (1+∆/ f )
n̄B nAA (1+∆/ f )
− 2 − ṅAB ≥ √ ∆(1+∆/ f )
n̄B nAA (1+∆/ f )
n2AA − Θ(∆2 nAA ) > 0.
(4.157)
This finishes the proof of (2).
The following Proposition is a statement for the 3-system (AA, AB, BB) but it holds also
true until T 2 in the 6-system (aa, aA, AA, aB, AB, BB) for δ < ∆.
Proposition 4.13. The maximal value nmax
AB of nAB in [T 1 , T 2 ] is bounded by
max
Moreover, let T AB
bounded by
n̄B
n̄B
− Θ(∆) ≤ nmax
+ Θ(∆).
(4.158)
AB ≤
2
2
be the time when nAB takes on its maximum, then nAA and nBB are
n̄B
max
− Θ(∆) ≤ nAA (T AB
)≤
4
n̄B
max
− Θ(∆) ≤ nBB (T AB
)≤
4
n̄B
+ Θ(∆),
4
n̄B
+ Θ(∆).
4
(4.159)
(4.160)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
Proof. From Lemma 4.12 (1) we get that
√
nAB ≥ 2 n̄B nAA − 2nAA 1 +
∆
cn̄B
,
27
(4.161)
We look for the value of AA where the expression on the right hand side takes on its
minimum, thus we have to derivate nAA and set it to zero:
n̄B
− 2 + cn̄∆B = 0
(4.162)
√
n̄B nAA
(4.163)
n̄2B = 4 − 4 cn̄∆B + Θ(∆2 ) n̄B nAA
n̄B
− Θ(∆) = nAA .
(4.164)
4
If we insert this in nAB we get the lower bound:
nAB ≥
n̄B
2
+ Θ(∆).
(4.165)
For the upper bound on nAB we proceed similarly. Form Lemma 4.12 (2) we get
q
nAB ≤ 2 n̄B nAA 1 + ∆f − 2nAA .
(4.166)
Setting the derivation of the rhs to zero gives:
0=
∆
n̄B 1+
f
r
∆
n̄B nAA 1+
f
nAA =
n̄B
+ Θ(∆).
4
−2
(4.167)
(4.168)
Finally we get
nAB ≤
n̄B
2
− Θ(∆)
and
nAA =
n̄B
4
− Θ(∆).
(4.169)
Remark. Note that nAA = nBB ± Θ(∆) =
value.
n̄B
4
± Θ(∆) as soon as nAB reaches its maximal
Proposition 4.14. For all t ∈ [T 1 , T 2 ],
naA ≤ Θ(ε) ∨ naB .
(4.170)
Proof. For t ≤ T = this follows from Proposition 4.9. For t > T = we show this by constructing a majorising process on naA (t):
(naA + naB )(2nAA + nAB + Θ(δ))
baA ≤ f
(4.171)
2Σ5
f (nAA − nBB )
(naA + naB ) +
(naA + naB ),
(4.172)
≤ f +Θ(δ)
2
2n̄A
daA ≥naA D + cn̄B − n̄∆B nAA − ηnBB − Θ(∆2 nAA )
(4.173)
ṅaA
≥naA ( f − ηnBB ),
f
BB )
≤ − naA 2f − f (nAA2n̄−n
−
ηn
−
Θ(δ)
+
n
+
BB
aB
2
A
(4.174)
f (nAA −nBB +Θ(δ))
2n̄A
.
(4.175)
By Lemma 4.2, it is left to show that ṅaA ≤ ṅaB whenever naA = naB . At this upper bound
we have ṅaA ≤ naB ( n̄fA (nAA − nBB ) + ηnBB + Θ(δ)). We now calculate a minorising process
on naB :
baB ≥
f
(n
2Σ5 aA
+ naB )(naB + nAB + 2nBB ),
(4.176)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
daB ≤ naB (D − ∆ + cn̄B ) = f naB ,
ṅaB ≥
f
n (n
2Σ5 aA aB
28
(4.177)
+ nAB + 2nBB ) −
f
n (2nAA
2Σ5 aB
+ 2naA − naB − nAB ).
Thus ṅaB ≥ Σf5 naB (nBB −nAA +nAB ) whenever naA = naB , and hence ṅaB − ṅaA ≥
2nAA + ηnBB − Θ(∆)) > 0 by Proposition 4.13. This finishes the proof.
(4.178)
f
n (2nBB −
n̄A aB
Now we show that the time T aA=δAA is finite and prove that it is smaller than or equal to
T aB=δBB . To estimate the order of magnitude of the time T 2 we need bounds on naA which
depends on ΣaA,aB .
Lemma 4.15. For all t ∈ [T 1 , T 2 ] the aA-population is bounded by
f (nAB + 2nAA )
f (nAB + 2nAA )
ΣaA,aB ≤ naA ≤
ΣaA,aB .
4n̄B ( f + ∆)
n̄A (D − 2∆)
(4.179)
Proof.
(1) We start with the upper bound. First observe that it holds at t = T 1 . By Lemma 4.2 it is
enough to show that if naA would reach the upper bound it would decrease faster than the
bound. Using Proposition 4.6 and that η < c a majorising process on aA is given by
baA ≤
f
Σ
(n
2Σ5 aA,aB AB
daA ≥ naA
+ 2nAA + Θ(δ)),
(4.180)
D + cn̄B − n̄∆B nAA − ηnBB − Θ(∆2 nAA ) ≥ naA (D − 2∆),
(4.181)
f (2nAA + nAB + Θ(δ))
ΣaA,aB − naA (D − 2∆).
(4.182)
2Σ5
We calculate the slope of the majorising process at the upper bound:
ṅaA ≤ f (2nAA + nAB )ΣaA,aB 2Σ1 5 − n̄1A + Θ(δ) ≤ − 2n̄f A (2nAA + nAB + Θ(δ))ΣaA,aB . (4.183)
ṅaA ≤
We have to show that at the upper bound,
f (ṅAB + 2ṅAA )
f (nAB + 2nAA )
ṅaA ≤
ΣaA,aB +
Σ̇aA,aB .
n̄A (D − 2∆)
n̄A (D − 2∆)
To do this we calculate minorising processes on nAB and nAA :
bAB ≥ Σf5 nAB 12 nAB + nAA + nBB + 2Σ5f nAA (nBB − Θ(δ2 )),
(4.184)
(4.185)
dAB ≤ nAB f,
(4.186)
ṅAB ≥ − 2Σf 5 n2AB + 2Σ5f nAA (nBB − Θ(δ2 )),
bAA ≥ Σf5 nAA nAB + nAA − Θ(δ2 ) + 4Σf 5 n2AB ,
(4.187)
dAA ≤ nAA ( f + ∆ + Θ(δ2 )),
ṅAA ≥ −nAA Σf5 nBB + ∆ + Θ(δ2 ) +
(4.189)
f
n2 .
4Σ5 AB
(4.188)
(4.190)
Hence we get that
ṅAB + 2ṅAA ≥ nAA
2f
n −
Σ5 BB
2f
n
Σ5 BB
− 2∆ − Θ(δ2 ) = −2(∆ + Θ(δ2 ))nAA .
(4.191)
By Lemma 4.7, we know that Σ̇aA,aB ≥ −∆ΣaA,aB . Thus the right-hand side minus the
left-hand side of (4.184) is lower-bounded by
−
2 f (∆ + Θ(δ2 ))nAA ΣaA,aB f ∆(nAB + 2nAA )ΣaA,aB f (nAB + 2nAA + Θ(δ))ΣaA,aB
−
+
n̄A (D − 2∆)
n̄A (D − 2∆)
2n̄A
(4.192)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
!
!
f nAB ΣaA,aB
f nAA ΣaA,aB
4∆
2∆
+
+ Θ(δ2 ) > 0.
≥
1−
1−
n̄A
D − 2∆
2n̄A
D − 2∆
29
(4.193)
This finishes the proof of (1).
(2) For the lower bound we proceed similarly (using Lemma 4.2). This time we show that
if naA would reach the lower bound it would start to increase faster than the bound. Using
Proposition 4.6 a minorising process on naA is given by
baA ≥
f
Σ
(2nAA
2Σ5 aA,aB
+ nAB − Θ(δ)),
(4.194)
daA ≤ naA ( f + ∆ + Θ(δ2 )),
(4.195)
f (2nAA + nAB − Θ(δ))
ṅaA ≥
ΣaA,aB − naA ( f + ∆).
(4.196)
2n̄B
We calculate the slope of the minorising process at the lower bound:
f (2nAA + nAB − Θ(δ))
f (2nAA + nAB )
ṅaA ≥
ΣaA,aB −
ΣaA,aB
(4.197)
2n̄B
4n̄B
f (2nAA + nAB − Θ(δ))
=
ΣaA,aB > 0.
(4.198)
4n̄B
Thus the minorising process on naA would increase when the aA-population would reach
the lower bound. To ensure this lower bound we have to show
f (ṅAB + 2ṅAA )
f (nAB + 2nAA )
ṅaA ≥
ΣaA,aB +
Σ̇aA,aB
(4.199)
4n̄B ( f + ∆)
4n̄B ( f + ∆)
For this we consider a majorising process on ΣaA,aB given by:
Σ̇aA,aB ≤
∆
n Σ
n̄B AA aA,aB
− naA (∆ − ηnBB ) + Θ(∆2 nAA ).
(4.200)
Using that η < c, the slope of this process if naA reaches the lower bound is estimated by
f (2nAA + nAB )
(∆ − ηnBB )ΣaA,aB + Θ(∆2 nAA )
(4.201)
Σ̇aA,aB ≤ n̄∆B nAA ΣaA,aB −
4n̄B ( f + ∆)
f (2nAA + nAB ) f − D
≤
ΣaA,aB + n̄∆B nAA ΣaA,aB + Θ(∆2 nAA ).
(4.202)
4n̄B
f +∆
Moreover we need majorising processes on AA and AB:
bAB ≤ Σf5 nAB 12 nAB + nAA + nBB + 2Σ5f nAA nBB + Θ(δ),
(4.203)
ṅAB ≤
∆(1+∆)
nAA ),
n̄B
− 2Σf 5 n2AB + 2Σ5f nAA nBB
bAA ≤
f
n
Σ5 AA
dAB ≥ nAB ( f −
dAA ≥
ṅAA ≥
(nAB + nAA ) +
(4.204)
+
∆(1+∆)
nAA nAB
n̄B
f
n2
4Σ5 AB
+ Θ(δ),
+ Θ(δ),
nAA ( f + ∆ − ∆(1+∆)
nAA ),
n̄B
f
−nAA Σ5 nBB + ∆ − ∆(1+∆)
n
AA
n̄B
(4.205)
(4.206)
(4.207)
+
f
n2
4Σ5 AB
+ Θ(δ).
(4.208)
Thus we have
ṅAB + 2ṅAA ≤ −∆nAA 2 −
2nAA +nAB
n̄B
+ Θ(∆2 nAA ) < Θ(∆2 nAA ).
(4.209)
It is enough to show that
ṅaA ≥
f (2nAA + nAB )
Σ̇aA,aB + Θ(∆2 nAA )ΣaA,aB ,
4n̄B ( f + ∆)
(4.210)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
30
using that η < c we have
f 2 (nAB + 2nAA )2 f − D
f (2nAA + nAB − Θ(δ))
ΣaA,aB −
ΣaA,aB
4n̄B
16n̄2B ( f + ∆) f + ∆
f (nAB + 2nAA ) ∆
−
nAA ΣaA,aB − Θ(∆2 nAA )ΣaA,aB
(4.211)
4n̄B ( f + ∆) n̄B
!
f (2nAA + nAB )
2∆(1 + ∆)nAA
f (2nAA + nAB − Θ(δ))
ΣaA,aB −
ΣaA,aB 1+
−Θ(∆2 nAA )ΣaA,aB
≥
4n̄B
8n̄B
n̄B ( f + ∆)
(4.212)
> 0.
(4.213)
This finishes the proof.
Proposition 4.16. For all t ∈ [T 1 , T 2 ] the process ΣaA,aB is bounded by
AA )
(1) Σ̇aA,aB ≤ naA ηnBB − ∆ nABnAB+Θ(∆n
.
+2nAA
(2) Σ̇aA,aB ≥ naA (ηnBB − ∆ − Θ(δ)).
Proof.
(1) We construct a majorising process on ΣaA,aB and use Proposition 4.6 and Lemma 4.8:
bΣaA,aB
dΣaA,aB
f (nAA + 21 naB + nAB + nBB )
f ( 12 naA + nAA + naB + nAB + nBB )
+ naB
+ Θ(Σ2aA,aB )
≤ naA
Σ5
Σ5
2
≤ f ΣaA,aB + Θ(ΣaA,aB ),
(4.214)
≥ ΣaA,aB (D − ∆ + cΣ5 ) + ∆naA − ηnaA nBB
(4.215)
≥ ΣaA,aB ( f −
Σ̇aA,aB ≤
∆(1+∆)
nAA )
n̄B
∆(1+∆)
nAA naB
n̄B
+ ∆naA − ηnaA nBB ,
− naA (∆ −
∆(1+∆)
nAA
n̄B
To bound naB we use Lemma 4.10:

 ∆(1 + ∆)nAA nAB + 2nBB +
Σ̇aA,aB ≤ naA 
nAB + 2nAA
n̄B
(4.216)
− ηnBB ) + Θ(Σ2aA,aB ).
2∆
c
−∆+
∆(1+∆)
nAA
n̄B
(4.217)


+ ηnBB + Θ(δ) ,
(4.218)
!
∆(nAA (nAB + 2nBB ) + nAA (nAB + 2nAA ) − n̄B (nAB + 2nAA )) + Θ(∆2 nAA )
≤ naA ηnBB +
n̄B (nAB + 2nAA )
(4.219)
!
nAB + Θ(∆nAA )
≤ naA ηnBB − ∆
.
(4.220)
nAB + 2nAA
(2) This time we construct a minorising process on ΣaA,aB by using Proposition 4.6 and
Lemma 4.8:
naA ( 21 naA + nAA + naB + nAB + nBB ) + naB (nAA + 21 naB + nAB + nBB )
− Θ(δ2 )
bΣaA,aB ≥ f
naA + nAA + naB + nAB + nBB
(4.221)
≥ f ΣaA,aB − Θ(δ2 ),
(4.222)
dΣaA,aB ≤ ΣaA,aB (D − ∆ + cΣ5 ) − ηnaA nBB + (∆ + Θ(δ )naA
2
≤ f ΣaA,aB − naA (ηnBB − ∆ − Θ(δ2 )),
Σ̇aA,aB ≥ naA (ηnBB − ∆ − Θ(δ)).
(4.223)
(4.224)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
31
From this Proposition we can deduce
Corollary 4.17. There exists a t∗ ∈ [T 1 , T 2 ], such that for all t ∈ [t∗ , T 2 ] and η >
it holds
Σ̇aA,aB (t) > 0.
4∆
n̄B
=: η? ,
(4.225)
Proof. A fine calculation will show that the competition c − η felt by an aA-individual
from a BB-individual allow the sum ΣaA,aB to grow when η is large enough, whereas it
decreases when η = 0. Note that we consider here the sum ΣaA,aB because the influence
of η cannot be seen in the rates of the aB-population alone. Heuristically, the growth of
the aB-population happens due to the indirect influence (source of a-allele) of the less
decaying aA-population. We prove that the minorising process on ΣaA,aB estimated in the
Proposition 4.16 starts to increase:
Σ̇aA,aB ≥ naA (ηnBB − ∆ − Θ(δ)).
(4.226)
As soon as nBB > ∆/η, the sum-process ΣaA,aB starts to increase. From Lemma 4.11 and
Proposition 4.13 we know that, for t ≥ T = , we have nBB ≥ n̄4B − Θ(∆). Hence, if we choose
the sum-process ΣaA,aB increases.
η > 4∆
n̄B
Now we are able to calculate the time T aA=δAA ∧ T aB=δAB and we will see that T aA=δAA ∧
T aB=δAB = T aA=δAA .
Theorem 4.18. The time T 2 = Θ(ε−1/(1+ηn̄B −∆) ).
Proof. From Proposition 4.16 (2) we have a lower bound on Σ̇aA,aB , and with Lemma 4.15
(2) we can bound this further from below by:
Σ̇aA,aB ≥ naA (ηnBB − ∆ − Θ(δ))
(4.227)
f (nAB (t) + 2nAA (t))
≥ (ηnBB (t) − ∆ − Θ(δ))
ΣaA,aB (t)
(4.228)
4n̄B ( f + ∆)
Θ(ηn̄B /4 − ∆)
≥
ΣaA,aB (t).
(4.229)
Θ(1) + Θ(1)t
where the last estimation on nBB and on nAB comes from Proposition 4.13 and from [25]
√
since we know from there that the time until nAB = Θ( nAA ), starts to decrease like 1/t is
of order Θ(1). As ΣaA,aB (T 1 ) = Θ(ε), the solution of the lower-bounding ODE is:
ΣaA,aB (t) ≥ Θ(ε)(Θ(1) + Θ(1)t)Θ(ηn̄B /4−∆)
(4.230)
By using Proposition 4.16 (1), we get the same kind of solution as an upper bound on
ΣaA,aB (note on the last step we can upper bound nBB by n̄B ):
ΣaA,aB (t) ≤ Θ(ε)(Θ(1) + Θ(1)t)Θ(ηn̄B −∆)
(4.231)
Using (4.230) and Lemma 4.15 we get a minorising process on aA:
naA (t) = Θ(nAB ΣaA,aB ) ≥ Θ(ε)(Θ(1) + Θ(1)t)Θ(ηn̄B /4−∆) /(Θ(1) + Θ(1)t).
(4.232)
The corresponding majorising process has an n̄B instead of n̄B /4. By solving naA = δnAA =
Θ(n2AB ) we get the order of magnitude of T aa=δAA :
Θ(ε−1/(1+ηn̄B −∆) ) ≤ T aa=δAA ≤ Θ(ε−1/(1+ηn̄B /4−∆) )
(4.233)
Note that 1 + ηn̄B − ∆ > 0 for ∆ small enough, and thus T aa=δAA diverges with ε and the
order calculations above are justified.
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
32
It is left to ensure that aB does not exceed δnAB in this time. It follows from Lemma 4.15
that during the time interval [T 1 , T 2 ], we have ΣaA,aB = Θ(naB ). Thus, solving naB = δnAB
amounts to solving Θ(ΣaA,aB ) = Θ(1)/(Θ(1) + Θ(1)t) which gives the very same order of
magnitude as for T aA=δAA . Thus the two times are of the same order.
Note that for η = 0, ΣaA,aB (T 2 ) = Θ(ε1+∆/(1−∆) ). This proves point 1 of Theorem 3.1.
Proposition 4.19. T 2 = T aA=δAA
Proof. This follows from Theorem 4.18 and Proposition 4.8.
Proposition 4.20. At time t = T 2 and if f is taken sufficiently large (Assumption C2), naa
starts to grow out of itself: there exists some positive constant cT2 > 0 such that
ṅaa ≥ cT2 · naa .
(4.234)
Proof. We have nAA (T 2 ) = Θ(ε2/(1+ηn̄B −∆) ). Thus, at the end of the second phase,
1
δnAA
2
δ f naa
,
nAA (1 + Θ(δ)) 2(1 + Θ(δ))
daa ≤ naa (D + ∆ + nAA (1 + Θ(δ))) = naa (D + ∆ + Θ(ε2/(1+ηn̄B −∆) ),
baa ≥ f naa
ṅaa ≥
naa ( δ2f
=
− D − ∆ − Θ(ε2/(1+ηn̄B −∆) )).
the right-hand side is positive for f large enough.
(4.235)
(4.236)
(4.237)
4.4. Phase 3: Exponential growth of aa until co-equilibrium with BB. Since aa is
growing now also out of itself it will influence the sum-process Σ5 = naA +nAA +naB +nAB +
nBB and we need new lower bounds on Σ5 in the following steps, the proof of this works
similar to the one of Proposition 4.6 by taking into account all contributing populations.
Let us compute the ODE to which Σ5 is the solution:
Proposition 4.21. The sum-process Σ5 is the solution to
Σ̇5 =Σ5 ( f − D − ∆ − cΣ5 ) − ∆ (naA + nAA ) − cnaa (naA + nAA ) + 2ηnaA nBB
+ Σf3 naa 12 naA + nAA − 4Σf 5 naB (naA + naB ) − 4Σf 6 naA (2naa + naA + naB ) .
(4.238)
Proof. We calculate the birth- and the death-rate of Σ5 under consideration of the aapopulation:
bΣ5 = Σf3 naa 21 naA + nAA + Σf5 (naB + nAB + nBB ) Σ5 − 14 naB (naA + naB )
F IGURE 6. zoom-in when aa recovers, general qualitative behaviour of
{ni (t) , i ∈ G} (lhs) and log-plot (rhs).
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
33
f (naA + nAA ) Σ6 − naA 12 naa + 41 naA + 14 naB
(4.239)
Σ6
= f Σ5 + Σf3 naa 12 naA + nAA − 4Σf 5 naB (naA + naB ) − 4Σf 6 naA (2naa + naA + naB ) , (4.240)
+
dΣ5 =Σ5 (D − ∆ + cΣ5 ) + (cnaa + ∆) (naA + nAA ) − 2ηnaA nBB ,
(4.241)
Σ̇5 =Σ5 ( f − D − ∆ − cΣ5 ) − (cnaa + ∆) (naA + nAA ) + 2ηnaA nBB
+ Σf3 naa 21 naA + nAA − 4Σf 5 naB (naA + naB ) − 4Σf 6 naA (2naa + naA + naB ) .
(4.242)
Let
T 3 := T n̄aaa −εγ/2 = inf{t > T 2 : naa (t) = n̄a − εγ/2 }.
We will need some preliminary bounds on naA and nAA .
(4.243)
Lemma 4.22. For all t ∈ [T 2 , T 3 ],
naA ≤ Θ(max{naa , naA , naB }nAB ) ≤ Θ(nAB ),
(4.244)
nAA ≤ Θ(max{naA , nAA , nAB } ) ≤ Θ(nAB ).
(4.245)
2
Proof. The populations aA and AA always stays smaller than or equal to Θ(nAB ) since they
are produced in majority from recombination of AB with other smaller population.
We divide this phase into steps (see Figure 6):
Step 1: [T 2 , T aa=aA ],
Step 2: [T aa=aA , T aa=AB ],
Step 3: [T aa=AB , T 3 ].
We distinguish two cases in Step 1 (T aA=AA ≥ T aa=aA and T aA=AA ≤ T aa=aA ), as well as in
Step 2 (T aA=AA ≥ T aa=AB and T aA=AA ≤ T aa=AB ) since we cannot prove which one happens
in general. We introduce some notation for the order of magnitude of nAA (T 2 ). We write
nAA (T 2 ) = Θ(εγ ) with
γ := 2/(1 + ηn̄B − ∆).
(4.246)
4.4.1. Step 1: Time interval [T 2 , T aa=aA ].
Proposition 4.23. For all η < 2c , we have T aa=aA < ∞ and for all t ∈ [T 2 , T aa=aA ] it holds
• the aa-population grows exponentially fast,
• nAA (T aa=aA ) = Θ(εγ ),
• nAB (T aa=aA ) = Θ(εγ/2 ).
Proof. In this step we show that aa crosses the aA-population.
(1) Case 1: T aA=AA ≥ T aa=aA .
In this case naa ≤ naA ≤ nAA . First note that the birth-rate of aA, baA , gets an
additional contributing term, namely:
3
+ nAA
max{naA , nAA } 3
2
≤ f naa
= 2 f naa .
(4.247)
f naa
naa + naA + nAA
max{naA , nAA }
Since naA = Θ(naB nAB ), naa ≤ naA and naB ≤ nAB (cf. Proposition 4.3) in this step
the main contribution to the aA-population still comes from matchings of aB and
AB-individual and thus aA increases but stays of the same order. Considering the
birth-rate baa , we see that only a growing AA-population could stop the growth of
aa. Thus we have to ensure that this population stays small enough.
1
n
2 aA
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
34
Lemma 4.24. The AA-population is bounded from above by
1
nAA ≤ n2AB .
(4.248)
n̄A
Proof. Looking at the rates of AA we see that an increasing aa-population has less
influence on the growth of AA since it only raises the pool of possible partners
and increases the competition (cf. Theorem 4.5). Thus the aa-population directly
can only lower the growth of AA and since aA is always smaller than AB, the
AA-population behaves like n2AB as before. More precisely,
2
bAA ≤ Σf5 21 naA + nAA + 12 nAB = Σf5 nAA (naA + nAA + nAB ) + 4Σf 5 (naA + nAB )2 , (4.249)
bAA ≥
f
n
Σ5 AA
(naA + nAA + nAB ) +
+ nAB )2 −
f
(n
4Σ5 aA
dAA ≥ nAA ( f − Θ(∆)) ,
dAA ≤ nAA ( f + Θ(∆) + cnaa )) ,
ṅAA ≤ −nAA Σf5 (nBB + naB ) − Θ(∆) +
ṅAA ≥ −nAA Σf5 (nBB + naB ) + Θ(∆) +
3f
n n n ,
Σ6 Σ5 aa AA AB
(4.250)
(4.251)
(4.252)
f
4Σ5
f
4Σ5
n2AB + 2nAA nAB + n2AA ,
n2AB + 2nAA nAB + n2AA .
(4.253)
(4.254)
We use again Lemma 4.2. The majorising process on AA at the upper bound
decreases
ṅAA ≤ − 3 f (1+Θ(∆))
n2AB .
4Σ5
(4.255)
It is left to show that ṅAA ≤ n̄2A nAB ṅAB . For this we estimate a minorising process
on AB:
(4.256)
bAB ≥ 2Σ6f 12 naA + nAA + 21 nAB 12 naB + 21 nAB + nBB
(4.257)
≥ f nAB − Σf6 nAB naa + 12 naA + 21 naB + 12 nAB + 2Σ6f nAA 12 naB + nBB ,
dAB ≤ nAB ( f + Θ(naA )) ,
(4.258)
ṅAB ≥ − Σf6 nAB (nAB + Θ(naA )) +
2f
1
n
n
Σ6 AA 2 aB
+ nBB .
(4.259)
At the upper bound the process would increase ṅAB ≥
upper bound holds.
f (1−Θ(∆)) 2
nAB
Σ6
and hence the
We now have to find a majorising process on AB.
Lemma 4.25. For the AB-population it holds:
ṅAB ≤ Θ(n2AB )
bAB
(4.260)
Proof.
(4.261)
≤ 2Σ5f 12 naA + nAA + 12 nAB 12 naB + 21 nAB + nBB
= Σf5 nAB 12 naA + nAA + 21 naB + 12 nAB + nBB + 2Σ5f 12 naA + nAA 12 naB + nBB , (4.262)
dAB ≥ nAB ( f − Θ(∆nAB )) ,
ṅAB ≤ −nAB 2Σf 5 (naA + naB + nAB ) − Θ(∆nAB ) +
(4.263)
3f
1
n
n
Σ5 AA 2 aB
+ nBB ≤ Θ(n2AB ),
(4.264)
by Lemma 4.24 Thus
nAB (t) ≤
Θ
Θ(1)
.
− Θ(1)t
ε−γ/2
(4.265)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
35
−γ/2
γ/2
AB
≥
Θ
ε
.
the
AB-population
needs
to
exceed
Θ(ε
)
is
T
The time T εAB
γ/2
εγ/2
AA
2
Since nAA (t) ≤ Θ(nAB ), the time T εγ the AA-population needs to exceed the order
−γ/2
Θ(εγ ) is T εAA
). Until time T εAA
γ ≥ Θ(ε
γ we consider a minorising process on aa:
Θ(δ)nAA
= Θ(δ) f naa ,
Θ(1)nAA
daa ≤ naa (D + ∆ + Θ(1)nAA ),
ṅaa ≥ naa (Θ(δ) f − D − ∆ − Θ(1)nAA ) ,
baa ≥ f naa
(4.266)
(4.267)
(4.268)
Using Lemma 4.95 we get
naa (t) ≥ Θ(ε20 )eΘ(δ)t .
(4.269)
Thus aa grows exponentially fast and reaches Θ(εγ ) in time Θ (ln (ε−γ )). This time
is shorter than T εAA
γ and we are done.
(2) Case 2: T aA=AA ≤ T aa=aA .
In this case nAA ≤ naA and naa ≤ naA . The aA -population has the same additional
term in ist birth rate as in the case before and by the same reasoning naA stays
smaller than Θ(n2AB ). We make this more precise by considering the growth of AB
and calculating upper bounds on aA and AA:
Lemma 4.26. The AA-population is bounded from above by
nAA ≤
2 2
n .
n̄A AB
(4.270)
Proof. The proof is similar to the one of Lemma 4.8. We have to show that at the
upper bound ṅAA ≤ n̄4A ṅAB nAB . Observe that the lower bound on Σ5 also holds here.
We start by estimating a majorising process on AA and by calculating the slope of
it at the upper bound:
2
bAA ≤ Σf5 21 naA + nAA + 21 nAB
=
f
n (n
Σ5 AA aA
+ nAA + nAB ) +
f
(n
4Σ5 aA
+ nAB )2 ,
(4.271)
dAA ≥ nAA ( f − Θ(∆)),
(4.272)
ṅAA ≤ −nAA ( f − Θ(∆)) +
bAB
f 2
n .
Σ5 AB
(4.273)
The slope of this majorising process at the upper bound is estimated by ṅAA ≤
− f −Θ(∆)
n2AB < 0. Thus if we can show that the slope of a minorising process on AB
n̄A
is positive at the upper bound we are done:
≥ f nAB − 2Σf 6 nAB (2naa + naA + naB + nAB ) + 2Σ6f 12 naA + nAA 12 naB + nBB , (4.274)
dAB ≤ nAB ( f + Θ(∆nAB )),
ṅAB ≥
− 3 f +Θ(∆)
n2AB
Σ6
+
(4.275)
3f
1
n
n
Σ6 AA 2 aB
+ nBB .
The slope of this process at the upper bound can be estimated by ṅAB ≥
0. This finishes the proof.
(4.276)
3 f −Θ(∆) 2
nAB
2n̄B
>
We proceed similarly for the upper bound on aA:
Lemma 4.27. For η < 2c , the aA-population is bounded from above by
naA ≤
6f
n2 .
n̄A (D−∆) AB
(4.277)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
36
12 f
Proof. This time we have to show that at the upper bound ṅaA ≤ n̄A (D−∆)
ṅAB nAB .
c
Using Proposition 4.21, Lemma 4.29 and η < 2 we estimate a majorising process
on aA and calculate the slope of it at the upper bound:
baA ≤ 2Σf 3 naa naA +
f
n (n + 2nAA + naB
2Σ5 aA aA
+ 2Σf 5 naB (2nAA + nAB ) + Σf5 naa 12 naA + nAA ,
naA ,
daA ≥naA ( f − Θ(∆) − ηnBB ) ≥ f +D−Θ(∆)
2
5 f +Θ(∆) 2
D−Θ(∆)
ṅaA ≤ − 2 naA + 2n̄A nAB .
f
n n
Σ3 aa AA
+
+ nAB )
(4.278)
(4.279)
(4.280)
The slope of this majorising process at the upper bound is estimated by ṅaA ≤
n2AB < 0. Thus if we can show that the slope of a minorising process on AB
− f −Θ(∆)
2n̄A
would be positive at the upper bound we are done. For this we use the minorising
process of AB from before (cf.(4.274)) and estimate the slope which is given by
ṅAB ≥ 3 f −Θ(∆)
n2AB > 0. This finishes the proof.
2n̄B
Now we can estimate a majorising process on AB and can bound it by ṅAB ≤
Θ(1)
−Θ(n2AB ), hence nAB (t) ≤ Θ(ε−γ/2
)−Θ(1)t . We have to construct a minorising process
on aa. Since now nAA ≤ naA and naa ≤ naA we can estimate:
1
naA
= 6f naa ,
baa ≥ f naa 2
(4.281)
3naA
daa ≤ naa (D + ∆ + Θ(n2AB )),
(4.282)
f
(4.283)
ṅaa ≥ naa 6 − D − Θ(∆2 ) .
))t
Thus the aa-population grows exponentially fast naa (t) ≥ Θ(ε2γ )e( f /6−D−Θ(∆
. The
γ/2
−γ/2
time T εAB
), is T εAB
. Since
γ/2 the AB-population needs to exceed Θ(ε
γ/2 = Θ ε
2
γ
aa
naA ≤ Θ(nAB ) until this time it holds naA ≤ Θ(ε ). The time T εγ , aa needs to reach
aa=aA
Θ(εγ ) is T εaaγ = Θ (ln ε−γ ). Thus T εaaγ ≤ T εAB
< ∞.
γ/2 and T
2
4.4.2. Step 2: Time interval [T aa=aA , T aa=AB ].
Proposition 4.28. For η < 2c , we have T aa=AB < ∞ and for all t ∈ [T aa=aA , T aa=AB ]:
• the aa-population grows exponentially fast,
• nAA (T aa=AB ) = Θ(ε√γ ),
• nAB (T aa=AB ) = Θ( εγ ).
Proof.
(1) Case 1: T aA=AA ≥ T aa=AB .
Since in this case naA ≤ nAA the arguments of Step 1.1 also hold here for the behaviour of the AA- and AB-populations. Since only an increasing AA-population
would stop the growth of aa, we seethat the
minorising process on aa, constructed
in Step 1.1 before needs time Θ ln ε−3γ/2 < T εAA
γ to increase until n AB .
(2) Case 2: T aA=AA ≤ T aa=AB .
In this step nAB ≥ naa ≥ naA > nAA . The different=ce with Step 1.2 is that the
aa-population is already bigger than the aA-population. Thus we can adapt the
proof of Step 1.2 to this step with small changes. We have to ensure the growth
of the aa-population until reaching nAB . An increasing aA-population cannot stop
the exponential growth of naa . We need to show that nAB does not start to grow
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
37
to much. For this we have to estimate bounds on naA and nAA again. The two
following lemmata are similar to the ones in the step before (Lemma 4.26 and
4.27) but taking into account that now naa ≥ naA .
Lemma 4.29. The AA-population is bounded from above by
nAA ≤
2 2
n .
n̄A AB
(4.284)
Proof. The proof is similar to the one of Lemma 4.8. We have to show that at the
upper bound ṅAA ≤ n̄4A ṅAB nAB . Observe that the lower bound on Σ5 also holds here.
We start by estimating a majorising process on AA and by calculating the slope of
it at the upper bound:
2
(4.285)
bAA ≤ Σf5 21 naA + nAA + 21 nAB
=
f
n (n
Σ5 AA aA
+ nAA + nAB ) +
f
(n
4Σ5 aA
+ nAB )2 ,
(4.286)
dAA ≥ nAA ( f − Θ(∆)),
(4.287)
ṅAA ≤ −nAA ( f − Θ(∆)) +
bAB
f 2
n .
Σ5 AB
(4.288)
The slope of this majorising process at the upper bound is estimated by ṅAA ≤
− f −Θ(∆)
n2AB < 0. Thus if we can show that the slope of a minorising process on AB
n̄A
would be positive at the upper bound we are done:
≥ f nAB − 2Σf 6 nAB (2naa + naA + naB + nAB ) + 2Σ6f 12 naA + nAA 12 naB + nBB , (4.289)
dAB ≤ nAB ( f + Θ(∆nAB )),
ṅAB ≥ − 5 f +Θ(∆)
n2AB +
2Σ6
(4.290)
2f
1
n
n
Σ6 AA 2 aB
+ nBB .
The slope of this process at the upper bound can be estimated by ṅAB ≥
0. This finishes the proof.
(4.291)
3 f −Θ(∆) 2
nAB
4n̄B
>
We proceed similarly for the upper bound on aA:
Lemma 4.30. For η < 2c , the aA-population is bounded from above by
naA ≤
6f
n2 .
n̄A (D−∆) AB
(4.292)
12 f
Proof. This time we have to show that at the upper bound ṅaA ≤ n̄A (D−∆)
ṅAB nAB .
We start by estimating a majorising process on aA an by calculating the slope of it
at the upper bound:
baA ≤ 2f naA + f nAA +
f
n (n + 2nAA + naB + nAB )
2Σ5 aA aA
+ 2Σf 5 naB (2nAA + nAB ) + Σf5 naa 21 naA + nAA ,
daA ≥naA ( f − Θ(∆) − ηnBB ) ≥ f +D−Θ(∆)
naA ,
2
(4.293)
5 f +Θ(∆) 2
nAB .
2n̄A
(4.295)
ṅaA ≤ −
D−Θ(∆)
naA
2
+
(4.294)
We used Proposition 4.21, Lemma 4.29 and η < 2c . The slope of this majorising
process at the upper bound is estimated by ṅaA ≤ − f −Θ(∆)
n2AB < 0. Thus if we can
2n̄A
show that the slope of a minorising process on AB would be positive at the upper
bound we are done. We use the minorising process of AB from before (cf.(4.289))
and estimate the slope which is given by ṅAB ≥ 7 f −Θ(∆)
n2AB > 0. This finishes the
4n̄B
proof.
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
38
Now we can estimate a majorising process on AB and can bound it by ṅAB ≤
Θ(1)
−Θ(n2AB ) and nAB (t) ≤ Θ(ε−γ/2
)−Θ(1)t . We have to construct a minorising process on
aa. Since now nAA ≤ naA ≤ naa we can estimate:
baa ≥ 2f naa ,
daa ≤ naa (D + ∆ + 2cnAB ),
ṅaa ≥ naa 2f − D − Θ(∆) .
(4.296)
(4.297)
(4.298)
Thus the aa-population grows exponentially fast naa (t) ≥ Θ(εγ )e( f /2−D−Θ(∆))t
. The
γ/2
−γ/2
AB
and the
time T εγ/2 the AB-population needs to exceed Θ(ε ) is of order Θ ε
aa
γ/2
−γ/2
time T εγ/2 the aa-population needs to reach Θ(ε ) is of order Θ ln ε
. Thus
aa=AB
aa
AB
< ∞.
T εγ/2 ≤ T εγ/2 and T
4.4.3. Step 3: Time interval [T aa=AB , T 3 ].
Proposition 4.31. For η < 2c we have T 3 < ∞ and for all t ∈ [T aa=AB , T 3 ]:
• the aa-population grows exponentially fast,
• nAA (T 3 ) = Θ(ε√γ ),
• nAB (T 3 ) = Θ( εγ ).
Proof. To ensure the exponential growth of aa we have to consider the behaviour of the
other processes as soon as naa ≥ nAB . Observe that the bound calculated in Lemma 4.29
takes over for this step unless if naA > nAA or naA ≤ nAA . We have to check again the upper
bound on aA:
Lemma 4.32. For η < 2c . the aA-population is bounded from above by
naA ≤
8f
n2 .
n̄A (D−∆) AB
(4.299)
Proof. This time the competition of aA with aa contributes to its death-rate. We have to
16 f
ṅAB nAB . We start by estimating a majorising
show that at the upper bound ṅaA ≤ n̄A (D−∆)
process on aA an by calculating the slope of it at the upper bound:
f
n (n + 2nAA + naB +
2Σ5 aA aA
nAB ) + Σf6 naa 12 naA + nAA ,
baA ≤ 2Σf 3 naa naA + f nAA +
+
f
n (2nAA
2Σ5 aB
+
f +D−Θ(∆)
nAB )
(4.300)
+ cnaa naA ,
daA ≥naA ( f − Θ(∆) − ηnBB + cnaa ) ≥
2
D−Θ(∆) c( f −2D) + 2( f −D) naa naA + 5 f +Θ(∆)
n2AB .
ṅaA ≤ −
2
2n̄A
(4.301)
(4.302)
We used Proposition 4.21, Lemma 4.29 and η < The slope of this majorising process at
n2AB < 0. Thus if we can show that the slope
the upper bound is estimated by ṅaA ≤ − f −Θ(∆)
2n̄A
of a minorising process on AB would be positive at the upper bound we are done:
bAB = 2Σ5f 12 naA + nAA + 12 nAB 12 naB + 12 nAB + nBB − Σ5fΣ6 naa 12 naA + nAA 21 naB + 12 nAB + nBB
(4.303)
≥ Σf5 nAB 12 naA + nAA + 12 naB + 21 nAB + nBB + 2Σ5f 21 naA + nAA 12 naB + nBB − Σf6 naa 12 naA + nAA ,
(4.304)
dAB ≤ nAB ( f + Θ(naA + nAA )),
(4.305)
ṅAB ≥ − 4 f +Θ(∆)
n2AB + Σf5 naA 12 naB + nBB − Θ(nAB naA ).
(4.306)
n̄A
c
.
2
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
Estimation of the slope gives ṅAB ≥
2 f −Θ(∆) 2
nAB
n̄A
> 0. This finishes the proof.
39
We also need a majorising process on AB to ensure that it does not grow too much. For
this we use Proposition 4.21, Lemma 4.29 and Lemma 4.32:
bAB = 2Σ5f 12 naA + nAA + 12 nAB 12 naB + 12 nAB + nBB − Σ5fΣ6 naa 12 naA + nAA 21 naB + 12 nAB + nBB
(4.307)
≤ Σf5 nAB 12 naA + nAA + 12 naB + 21 nAB + nBB + 2Σ5f 21 naA + nAA 12 naB + nBB ,
(4.308)
dAB ≥ nAB ( f − Θ(naA + nAA )),
ṅAB ≤
n2AB
− f −Θ(∆)
Σ5
+
10 f 2
n
n̄A AB
(4.309)
≤ Θ(n2AB ),
(4.310)
Thus we see that the aa-population cannot disturb the behaviour of AB much and we can
Θ(1)
estimate ṅAB ≤ −Θ(n2AB ) and nAB (t) ≤ Θ(ε−γ/2
)−Θ(1)t . We show that the aa-population grows
exponentially fast up until an εγ -neighbourhood of its equilibrium n̄a . Again we construct
a minorising process:
2 n
baa ≥ naa f − Θ nAB
,
(4.311)
aa
daa ≤ naa (D + ∆ + cnaa + Θ(n2AB )),
2 n
.
ṅaa ≥ naa f − D − ∆ − cnaa − Θ nAB
aa
(4.312)
(4.313)
This minorising process on aa increases until an εγ -neighbourhood of n̄a . The time
T 3 the aa-population needs to reach the εγ -neighbourhood of its equilibrium is of orγ/2
der Θ (εγ ln εγ ) and the time T εAB
) is of order
γ/2 the AB-population needs to exceed Θ(ε
−γ/2
AB
Θ ε
. Thus T 3 ≤ T εγ/2 < ∞.
4.5. Phase 4: Convergence to paB = (n̄a , 0, 0, 0, 0, n̄B ). The Jacobian matrix of the field
(2.24) at the fixed point paB has the 6 eigenvalues: 0 (double), and −(2 f − D), −( f − D +
∆), −( f −D−∆), −(( f −D)(5 f −4D)+ f ∆)/(4( f −D)+ηn̄B ) which are strictly negative under
Assumptions (C). Because of the zero eigenvalues, paB is a non-hyperbolic equilibrium
point of the system and linearisation fails to determine its stability properties. Instead,
we use the result of the center manifold theory (18, 26) that asserts that the qualitative
behaviour of the dynamical system in a neighbourhood of the non-hyperbolic critical point
paB is determined by its behaviour on the center manifold near paB .
Theorem 4.33 (The Local Center Manifold Theorem 2.12.1 in 26). Let f ∈ C r (E), where
E is an open subset of Rn containing the origin and r ≥ 1. Suppose that f (0) = 0 and
D f (0) has c eigenvalues with zero real parts and s eigenvalues with negative real parts,
where c + s = n. Then the system ż = f (z) can be written in diagonal form
ẋ = Cx + F(x, y)
ẏ = Py + G(x, y),
(4.314)
(4.315)
where z = (x, y) ∈ Rc ×R s , C is a c×c-matrix with c eigenvalues having zero real parts, P is
a s × s-matrix with s eigenvalues with negative real parts, and F(0) = G(0) = 0, DF(0) =
DG(0) = 0. Furthermore, there exists δ > 0 and a function, h ∈ C r (Nδ (0)), where Nδ (0) is
the δ-neighbourhood of 0, that defines the local center manifold and satisfies:
Dh(x)[Cx + F(x, h(x))] − Ph(x) − G(x, h(x)) = 0,
(4.316)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
40
for |x| < δ. The flow on the center manifold W c (0) is defined by the system of differential
equations
ẋ = Cx + F(x, h(x)),
(4.317)
for all x ∈ Rc with |x| < δ.
The Local Center Manifold Theorem shows that the non-hyperbolic critical point paB is
indeed a stable fixed point and that the flow on the center manifold near the critical point
approaches paB with speed 1t . This can be seen as follows:
By the affine transformation (naa , nBB ) 7→ (naa − n̄a , nBB − n̄B ) we get a translated system
F̃(n) which has a critical point at the origin. The two eigenvectors corresponding to 0
eigenvalues of the Jacobian matrix of F̃ at the fixed point (0, 0, 0, 0, 0, 0) are
EV1 = (0, 0, 0, 0, 1, 0, −1)
and
EV2 = (0, 0, 0, 0, −1, 1, 0)
(4.318)
We perform a new change of variable to work in the basis of eigenvectors of F̃(n). Let us
call the new coordinates x1 , . . . , x6 . Let h(x1 , x2 ) be the local center manifold. We shall
look at its local shape near (0, 0) and expand it up to second order:
 2

λ3 x1 + ν3 x1 x2 + µ3 x22 
λ x2 + ν4 x1 x2 + µ4 x22 
 + O(x3 ).
h(x1 , x2 ) =  4 12
(4.319)
λ5 x1 + ν5 x1 x2 + µ5 x22 
λ6 x12 + ν6 x1 x2 + µ6 x22
We then substitute the series expansions into the center manifold equation (4.316) which
gives us 12 equations for the 12 unknowns λ3 , . . . , µ6 . Substitution of the explicit second
order approximation of the center manifold equation into (4.317) yields the flow on the
local center manifold:
C1 2 E1 2
A1
x1 x2 +
x2 +
x1 + O(x3 )
(4.320)
ẋ1 =
B1
D1
F1
C2 2 E2 2
A2
x1 x2 +
x2 +
x1 + O(x3 )
(4.321)
ẋ2 =
B2
D2
F2
where
A1 = 3c2 D f 2 − c2 ∆ f 2 − 3c2 f 3
B1 = (D − ∆ − f ) 4cD2 − 9cD f + c∆ f + 5c f 2 − 4D2 η + 4D∆η + 8Dη f − 4∆η f − 4η f 2
C1 = 12c D f − 4c D ∆ f − 39c D f + 12c D∆ f + 42c D f − c ∆ f − 8c ∆ f
2
3 2
2
2
2
2
2 3
2
3
2
4
2 2 3
2
(4.322)
(4.323)
4
− 15c2 f 5 + 12cD3 η f 2 − 16cD2 ∆η f 2 − 36cD2 η f 3 + 4cD∆2 η f 2 + 32cD∆η f 3
+ 36cDη f 4 − 4c∆2 η f 3 − 16c∆η f 4 − 12cη f 5
(4.324)
D1 = 8(D − 2 f )(D − f )(D − ∆ − f )×
× 4cD2 − 9cD f + c∆ f + 5c f 2 − 4D2 η + 4D∆η + 8Dη f − 4∆η f − 4η f 2
(4.325)
E1 = c f,
(4.326)
F1 = 2(−D + ∆ + f )
and
A2 = 2c2 D2 f − 3c2 D f 2 + c2 f 3 − 2cD2 η f + 2cD∆η f + 4cDη f 2 − 2c∆η f 2 − 2cη f 3
B2 = (D − ∆ − f ) 4cD2 − 9cD f + c∆ f + 5c f 2 − 4D2 η + 4D∆η + 8Dη f − 4∆η f − 4η f 2
(4.327)
C2 = −3cDη f + c∆η f + 3cη f
D2 = 2(D − 2 f ) 4cD2 − 9cD f + c∆ f + 5c f 2 − 4D2 η + 4D∆η + 8Dη f − 4∆η f − 4η f 2
(4.329)
(4.330)
E2 = 0, F2 = 1.
(4.331)
2
2
3
(4.328)
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.0
0.0
0.0
0.5
1.0
1.5
2.0
0.0
0.5
1.0
1.5
41
2.0
F IGURE 7. Flow of the dynamical system in the center manifold of the
fixed point paB , for η = 0.02 (left) and η = 0.6 (right).
It is left to show that the above system flows toward the origin, at least for η smaller than
a certain constant. To do that, we perform another change of variables which allows us to
work in the positive quadrant. We call the new coordinates (on the center manifold) y1 and
y2 , and the new field F̂. Observe that it is sufficient to prove that the scalar product of the
field with the position is negative. We thus consider the function
s(y1 , y2 ) = (F̂(y1 , y2 ), (y1 , y2 ))
(4.332)
which is a quadratic form in y1 and y2 . As the field F̂ is homogeneous of degree 2 in
its variables, it is enough to consider any direction given by y2 = λy1 , and prove that
s(y1 , λy1 ) < 0 for all λ > 0. As the expressions are so ugly, we work perturbatively in
f and consider it as large as needed. Observe that the numerator and the denominator of
s(y1 , λy1 ) are polynomials of degree 5 in f . We thus look at the coefficients in front of f 5 :
cy1 3 c 16λ3 + 7λ2 + 16λ + 40 − 4η 5λ3 + 8λ2 + 8λ + 8
s(y1 , λy1 ) =
f 5 + Θ( f 4 )
64η − 80c
(4.333)
Observe that the denominator is always negative (because by our Assumptions η ≤ c). The
minimal value of the ratio
16λ3 + 7λ2 + 16λ + 40
r(λ):=
(4.334)
4 5λ3 + 8λ2 + 8λ + 8
is rmax ' 0.593644, thus, asymptotically as f → ∞, the field is attractive for η < c · rmax .
Thus we see that paB is a stable fixed point which is approached with speed 1t as long as
η < c · rmax .
R EFERENCES
[1] M. Baar, A. Bovier, and N. Champagnat. From stochastic, individual-based models to the canonical
equation of adaptive dynamics - In one step. Ann. Appl. Probab., online first, 2016.
[2] R. Bürger. The mathematical theory of selection, recombination, and mutation. Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2000.
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
42
[3] N. Champagnat. A microscopic interpretation for adaptive dynamics trait substitution sequence models.
Stochastic Process. Appl., 116(8):1127–1160, 2006.
[4] N. Champagnat, R. Ferrière, and G. Ben Arous. The Canonical Equation of Adaptive Dynamics: A
Mathematical View. Selection, 2:73–83., 2001.
[5] N. Champagnat, R. Ferrière, and S. Méléard. From individual stochastic processes to macroscopic
models in adaptive evolution. Stochastic Models, 24(suppl. 1):2–44, 2008.
[6] N. Champagnat and S. Méléard. Polymorphic evolution sequence and evolutionary branching. Probab.
Theory Related Fields, 151(1-2):45–94, 2011.
[7] P. Collet, S. Méléard, and J. A. J. Metz. A rigorous model study of the adaptive dynamics of Mendelian
diploids. J. Math. Biol., 67(3):569–607, 2013.
[8] C. Coron. Stochastic modeling of density-dependent diploid populations and the extinction vortex. Adv.
in Appl. Probab., 46(2):446–477., 2014.
[9] C. Coron. Slow-fast stochastic diffusion dynamics and quasi-stationarity for diploid populations with
varying size. J. Math. Biol., 72(1-2):171–202., 2016.
[10] C. Coron, S. Méléard, E. Porcher, and A. Robert. Quantifying the mutational meltdown in diploid
populations. The American Naturalist, 181(5):623–36, 2013.
[11] J. F. Crow, M. Kimura, and Others. An introduction to population genetics theory. An introduction to
population genetics theory., 1970.
[12] U. Dieckmann and R. Law. The dynamical theory of coevolution: a derivation from stochastic ecological processes. J. Math. Biol., 34(5-6):579–612., 1996.
[13] W. J. Ewens. Mathematical population genetics. I. Theoretical introduction, volume 27 of Interdisciplinary Applied Mathematics. Springer-Verlag, New York, second edition, 2004.
[14] R. Fisher. The correlation between relatives on the supposition of Mendelian inheritance. Trans. Roy.
Soc. Edinb., 42:399–433., 1918.
[15] N. Fournier and S. Méléard. A microscopic probabilistic description of a locally regulated population
and macroscopic approximations. Ann. Appl. Probab., 14(4):1880–1919, 2004.
[16] J. Haldane. A mathematical theory of natural and artificial selection. Part I. Trans. Camb. Phil. Soc.,
23:19–41., 1924.
[17] J. Haldane. A mathematical theory of natural and artificial selection. Part II. Trans. Camb. Phil. Soc.,
Biol, Sci., 1:158–163., 1924.
[18] M. W. Hirsch, C. C. Pugh, and M. Shub. Invariant manifolds. Lecture Notes in Mathematics, Vol. 583.
Springer-Verlag, Berlin-New York, 1977.
[19] J. Hofbauer and K. Sigmund. Adaptive dynamics and evolutionary stability. Appl Math Lett, 3(4):75–
79, 1990.
[20] E. Kisdi and S. A. H. Geritz. Adaptive dynamics in allele space: Evolution of genetic polymorphism
by small mutations in a heterogeneous environment. Evolution, 53:993–1008, 1999.
[21] P. Marrow, R. Law, and C. Cannings. The Coevolution of Predator–Prey Interactions: ESSS and Red
Queen Dynamics. Proceedings of the Royal Society of London B: Biological Sciences, 250(1328):133–
141., 1992.
[22] J. Metz, R. Nisbet, and S. Geritz. How should we define ”fitness” for general ecological scenarios?
Trends in Ecology and Evolution, 7(6):198–202, 1992.
[23] J. A. J. Metz, S. A. H. Geritz, G. Meszéna, F. J. A. Jacobs, and J. S. van Heerwaarden. Adaptive
dynamics, a geometrical study of the consequences of nearly faithful reproduction. In Stochastic and
spatial structures of dynamical systems (Amsterdam, 1995), Koninklijke Nederlandse Akademie van
Wetenschappen Eerste Reeks, 45, pages 183–231. North-Holland, Amsterdam, 1996.
[24] T. Nagylaki. Introduction to theoretical population genetics, volume 21 of Biomathematics. SpringerVerlag, Berlin, 1992.
[25] R. Neukirch and A. Bovier. Survival of a recessive allele in a Mendelian diploid model. J. Math. Biol.,
pages 1–43, nov 2016.
[26] L. Perko. Differential equations and dynamical systems, volume 7 of Texts in Applied Mathematics.
Springer-Verlag, New York, 1991.
[27] S. Wright. Evolution in Mendelian populations. Genetics, 16:97–157., 1931.
[28] G. U. Yule. On the theory of inheritance of quantitative compound characters on the basis of Mendel’s
laws: a preliminary note. Spottiswoode & Company, Limited, 1907.
THE RECOVERY OF A RECESSIVE ALLELE IN A MENDELIAN DIPLOID MODEL
43
A. B OVIER , I NSTITUT F ÜR A NGEWANDTE M ATHEMATIK , R HEINISCHE F RIEDRICH -W ILHELMS U NIVERSIT ÄT , E NDENICHER A LLEE 60, 53115 B ONN , G ERMANY
E-mail address: [email protected]
L. C OQUILLE , I NSTITUT F OURIER , UMR 5582 DU CNRS, U NIVERSIT É DE G RENOBLE A LPES , 100
M ATH ÉMATIQUES , 38610 G I ÈRES , F RANCE
E-mail address: [email protected]
RUE DES
R. N EUKIRCH , I NSTITUT F ÜR A NGEWANDTE M ATHEMATIK , R HEINISCHE F RIEDRICH -W ILHELMS U NIVERSIT ÄT , E NDENICHER A LLEE 60, 53115 B ONN , G ERMANY
E-mail address: [email protected]

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