Applied Mathematics, 2011, 2, 11-22
doi:10.4236/am.2011.21002 Published Online January 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
Discrete Evolutionary Genetics: Multiplicative Fitnesses
and the Mutation-Fitness Balance
Thierry Huillet1, Servet Martinez2
1Laboratoire de Ph ysique Théorique et Modélisation, CNRS-UMR 8089, Université de Cergy-Pontoise,
Cergy-Pontoise, France
2Departamento de Ingeniería Matemática, Centro Modelamiento Ma temático, UMI 2807,
UCHILE-CN RS, Santiag o, Chile
E-mail: Thierry.Huillet@u-cergy.fr, smartine@dim.uchile.cl
Received June 18, 2010; revi sed October 16, 2010; accepted October 20, 2010
Abstract
We revisit the multi-allelic mutation-fitness balance problem especially when fitnesses are multiplicative.
Using ideas arising from quasi-stationary distributions, we analyze the qualitative differences between the
fitness-first and mutation-first models, under various schemes of the mutation pattern. We give some sto-
chastic domination relations between the equilibrium states resulting from these models.
Keywords: Evolutionary Genetics, Fitness Landscape, Selection, Mutation, Stochastic Models,
Quasi-Stationarity
1. Introduction and Outline
Population genetics aims at elucidating the fate of the
allelic population composition when various dr iving ‘for-
ces’ such as selection or mutation are at stake in the gene
pool. This requires to identify first the updatin g mechan-
isms responsible of the gene frequency-distributions evo-
lution over time. In this note, we shall briefly revisit the
basics of the deterministic dynamics arising in discrete-
time asexual evolutionary genetics when the origin of
motion is either the fitness or the mutations or both. We
start with the multi-allelic haploid case before dealing
with the diploid case. First, we consider general fitness
mechanisms, then general mutation mechanisms and then
we shall combine the two.
The general purpose of the Sections 2-3 is to introduce
separately the marginal allelic dyna mics driv en by fitness
and then the one driven by mutations. These issues are
of course part of the standard models discussed for exa-
mple in [1-4].
In Section 4, we stress that there are two different
ways to combine the fitness and the mutation effects.
One (fitness-first), which is classical, consists in app-
lying first the fitness mapping and then let mutation act
on the result. The other (mutation-first) consists in rev-
ersing the order. Stochastic models pertaining to the
mutation/selection combination are numerous. See [5,6]
(and the References therein) for the relation of a muta-
tion/selection model with ancestral branching processes.
A recent discussion on a Markov chain evolution to
study the probability that a new mutant beco mes fixed in
a Moran type model can be found in [7]. A work descri-
bing phenotypic variation and natural selection by mod-
eling population as a Mark ov point process can be found
in [8].
In Section 5, we focus on a model with multiplicative
fitnesses and general mutation pattern and we analyze
both the fitness-first and mutation-first dynamics. Starting
with the fitness-first dynamics, we observe that it has the
structure of a discrete-time nonlinear master equation of
some Markov process whose construction we give. In
this stochastic interpretation, the polymorphic equilibr-
ium state interprets as a quasi-stationary distribution of
the Markov process conditioned to be currently alive. It
is the left eigenvector of some sub-stochastic matrix
A
associated to its spectral radius. The corresponding right
survival eigenvector makes sense in this in terpretation. A
similar interpretation can be given when dealing with the
mutation-first dynamics driven now by some sub-stocha-
stic matrix B with its own left and right Perron-Frobe-
nius eigenvectors. The matrices
A
and B are diago-
nally similar. Using these stochastic tools, we observe
that the mean fitness at equilibrium of the model B is
larger than the on e of model
A
, together with some sto-
T. HUILLET ET AL.
Copyright © 2011 SciRes. AM
12
chastic domination properties between both the left and
right Perron-Frobenius eigenvectors of the models
A
and B. If we specify the structure of the mutation ma-
trix to be reversible, then the right and left Perron-
Frobenius eigenvectors of each model can be related to
one another by using an appropriate Schur product. Some
simplifications also occur if we deal with symmetric
mutations because the right (left) eigenvector of
A
coincides with the left (right) eigenvector of B.
Section 6 particularizes the study of Section 5 when a
house of cards condition holds for the mutation matrix.
Because this mutation model is quite restrictive, some
simplifications occur and the shapes of the polymorphic
equilibrium states can be made more explicit.
The interpretation of the fitness-first and mutation-first
dynamics in terms of a stochastic process conditioned on
not being currently absorbed in some coffin state sugg-
ests that related conditional models for the evolutionary
dynamics involving multiplicative fitness and mutations
could also be relevant. In Section 7 we suggest to con-
dition the process on its non-extinction either locally
(stepwise) or to condition it globally on not getting ex-
tinct in the remote future. Models
A
and B lead to
different conditional dynamics.
2. Evolution under Fitness: The
Deterministic Point of View
We briefly describe the frequency distribution dynamics
when fitness only drives the process. We start with the
haploid case before moving to the diploid case.
2.1. Single Locus: Haploid Population with K
Alleles
Consider K alleles k
A
, =1, ,kK attached to a
single locus. Suppose the current time-t allelic fre-
quency distribution is given by the column vector
:k
x
x, =1, ,kK1. We therefore have

=1
=::= =1
K
Kk
k
Sx
xx0x the
K
simplex. Let
:= >0,
k
ww =1,,kK denote the absolute fitnesses
of the alleles. Let

*
:= =
ll
l
wwx
xwx (1)
be the mean fitness of the population at time t. The
variance in absolute fitness

2
x and the variance in
relative fitness
2
x are given respectively by
 

 


2
2
=1
22
2
2
=1
=;
=1=.
K
kk
k
Kk
k
k
xw w
w
xww




xx
x
xxx
(2)
2.1.1. Dynamics
The discrete-time update of the allele frequency distri-
bution on th e simplex
K
S is given by2:
 
=:=, =1,,.
'kk
kk
xw
x
pkK
w
xx
(3)
As required, the vector
 
:= k
ppxx , =1, ,kK,
maps
K
S into
K
S. In vector form, with :=Dx diag
, =1,,
k
x
kK, the nonlinear deterministic dynamics
reads3:
  
11
== ,
'DD
ww
wx
xpxx= w
xx
or, with := '
xxx
, the increment of x

1
=DI
w





w
xx.
x
Without loss of generality, we can assume that
1
0< =1.
K
ww
Thus that allele
K
A
has largest
fitness.
Let ,:
kl KK
SS
be the involution exchanging the
coordinates k and l. When =
kl
ww, we have that
,,
=()
'
'
kl kl

xx, and so the evolution is symmetric
under ,kl
. In that case, the alleles k and l can be
merged into a single one.
For
K
S
x, let support


=:>0,
k
kxx. Let
=:=1
k
kw
be the set of alleles with maximal
fitness. Any
K
S
x such that support
x is
called an equilibrium state. A vector
==0,,0,1,0, ,0
k
xe  with k
is called a pure
(or monomorphic) equilibrium state.
2.1.2. Mean Fitness Increase
According to the dynamical system (3), unless its equili-
brium state is attained, the absolute mean fitness
wx
increases. Indeed, with


='
wwwxxx
:
 
 
2
== 1
=0,
k
kk kk
kk
kk
k
w
wwxwx
w
wx
w
w







xx
x
x
(4)
1In the sequel, a boldface variable, say x, will represent a col-
umn-vector so that its transpose, say *
x, will be a line-vector. Simi-
larly, *
A
will stand for the transpose of some matrix
A
.
2The symbol ' is a common and useful notation to denote the up-
dated frequency.
3
D
xw clearly is the Schur product of x and w. See [3] page 238
for a sim ilar notational convenience.
T. HUILLET ET AL.
Copyright © 2011 SciRes. AM
13
and it is >0 except when support

x.
The mean fitness is maximal at equilibrium. The rate
of increase of

wx is:

 


22
2
=1==.
k
k
k
kk
k
x
ww
x
ww x





xx
xx (5)
These last two facts are sometimes termed the 1930s
Fisher fundamental theorem of natural selection (FTNS).
Then, if there is an allele whose fitness is strictly larger
than the ones of the others starting from any initial state
of
K
S which is not an extremal point, the haploid traje-
ctories will converge to this fittest state.
2.2. Single Locus: Diploid Population with K
Alleles
We now run into similar considerations but with diploid
populations.
2.2.1. Joint Evolutionary Dynamics
Let ,0
kl
w, ,=1, ,kl K stand for the absolute fit-
ness of the genotypes kl
A
A attached to a single locus.
Assume ,,
=
kl lk
ww (,kl
w being proportional to the pro-
bability of an kl
A
A surviving to maturity, it is natural to
take ,,
=
kl lk
ww
). Let then W be the symmetric fitness
matrix with ,klentry ,kl
w.
Assume the current frequency distribution at time t
of the genotypes kl
A
A is given by ,kl
x
. Let
X
be the
frequencies array with ,klentry ,kl
x
. The joint evolu-
tionary dynamics in the diploid case is given by the
updating:
,
,, ,,
,
= where ()=.
()
kl
'
klklkl kl
kl
w
x
xXxw
X
(6)
The relative fitness of the genotype kl
A
A is
,kl
wX
. The joint dynamics takes the matrix form:
 
11
==
'
X
XW WX
XX


where stands for the (commutative) Hadamard pro-
duct of matrices.
Let
J
be the KK matrix whose entries are all 1
(the identity for ). T h en


 
1
:= =
1
=.
'
X
XX XJW
X
WXJ
X
 
Let


2
2,,
,=1
22
,
2,2
,=1
()= ();
()
=1=
() ()
K
kl kl
kl
Kkl
kl
kl
XxwX
X
Xx
X
X




(7)
stand respectively the genotypic variance in absolute fit-
ness and the diploid variance in relative fitness. The in-
crease of the mean fitness is given by

2,
,, ,,
,,
2
==
()
=()()0,
kl
kl klklkl
kl kl
w
Xxwxw
X
XX






 (8)
which vanishes only at the equilibrium states maximizing
lk
w,, with a relative rate of increase:
2
=wX wXX
. This is the diploid version of
the FTNS.
2.2.2. Marginal Allelic Dynamics
Assuming a Hardy-Weinberg equilibrium, the frequency
distribution at time t, say ,kl
x
, of the genotypes kl
A
A
is given by: ,=
klkl
x
xx where ,
=
kkl
l
x
x
is the mar-
ginal frequency of allele k
A
in the whole genotypic
population. The frequency information is =
X
x1 (1 is
the unit K-vector) and the mean fitness is given by the
quadratic form:
*
,
,
:= =
kl kl
kl
x
xw W
xxx
. Let
 

 


2
2,
,=1
22
,
22
,=1
=;
=1=
K
kl kl
kl
Kkl
kl
kl
xx w
w
xx





xx
x
xxx
(9)
be respectively the genotypic varian ce in ab solute fitness
and the diploid variance in relative fitness.
If we first define the frequency-dependent marginal
fitness of k
A
by
 
,
=:=
kkll
l
k
wW wx
xx , the mar-
ginal dynamics is given as in (3) by:
  
1
== =:,=1,,.
k
'
kkk k
k
w
x
xxWpkK

xxx
xx
(10)
In vector form (10) reads

11
===:,
'W
DW D

xx
xxxpx
xx
where p maps
K
S into
K
S. Iterating, the time-t fre-
quency distribution
tx is the ttimes composition
of p applied to some
0x.
In the diploid case, assuming fitnesses to be multi-
plicative, say with ,=
klk l
www
, then selection acts on
the gametes rather than on the genotypes. Observing
T. HUILLET ET AL.
Copyright © 2011 SciRes. AM
14

*=
kk
ll
l
ww
wx
W
x
xx , the dynamics (10) boils down to (3).
However, the mean fitness in this case is


2
=ll
lwx
x and not

=ll
lwx
x as in the
haploid case.
2.2.3. Increase of Mean Fitness
Again, the mean fitness

x, as a Lyapunov function,
increases as time passes by. We indeed have



  
,,
2,,
=
1
=0,
'
kk klllkkll
kl kl
xwwxwxwx




xxx
xx
x
and vanishes only when the process has reached equili-
brium.
Its partial rate of increase due to frequency shifts only
is
 
:= kk
kxw

xx. It satisfies






22
2
1
=1==,
2
k
k
kA
kk
k
x
w
xx







xx x
xx
(11)
where

2
A
x is the allelic variance in relative fitness
 

2
2
=1
:=21 .
Kk
Ak
k
w
x




x
xx (12)
2.2.4. An Alternative Representation of the Allelic
Dynamics
There is an alternative vectorial representation of the
dynamics (10) emphasizing its gradient-like character.
Define the matrix

*
=GD
x
xxx. It is symmetric,
positive semi-positive whose quadratic form vanishes
only for the constants.

Gx is partially invertible on
the space orthogonal to the constants with left-inv e rse

11
1
=.GIJD
K



x
x
Note
 
1
GGxxδδ. Looking for a left-inverse in
the weaker sense of the quadratic form, that is satisfying
 
1
**
=GG I
δxxδδδ
for all δ with =0δ, every

11
=GIJD
K



x
x
would do the job for any R
. In particular =0
.
Introduce the quan ti ty
 
1
=log
2
W
V
xx. Then, (10)
may be recast as the gradient-like dynamics:
  
1
==,
W
GWG V
xxxxx
x (13)
with *
==0x1x as a result of

**
=G1x0. Note

**
=0.
WWW
VVGV
 xxx xx
Based on [9,10], the dynamics (13) is of gradient-type
with respect to th e Shashahani-Svirezhev distance metric
given by




1/2
1/2 2
1
*1
=1
,== .
K
'
Gkk
k
dG xx

 


xxx xx
Its trajectories are perpendicular to the level surfaces
of W
V with respect to this metric. From (11) and (12),
'
G
dx, x, which is the length of x, is also the square-
root of half the allelic variance (the standard deviation)
in relative fitness.
3. The Mutation Mapping
We now briefly describe the frequency distribution dyna-
mics when mutation is the only driving source of motion.
Assume alleles mutate according to the scheme:
kl
A
Awith probability
,0,1
kl
satisfying ,=0
kk
and ,
0< 1
kl
lk
for all k. Let ,
:= kl
M
be
the mutation pattern matrix; we shall assume that the
non-negative matrix
M
is irreducible. We first con-
sider the deterministic diploid model involving muta-
tions.
3.1. Only Mutations
Considering first an updating mechanism of the freq-
uencies where only mutations operate, we get
,,
=,=1,,.
'
kk lklkkl
lk lk
x
xxxkK



(14)
In matrix form, with *
M
the transpose of
M

*
==:=:,
'MM
MD
1
xx xxMxpx (15)
and the update of the frequencies with mutations is given
by the affine transformation
*
:= .
M
I
DM
1
M
We have 0M and *
=
M
M if and only if
M
is
stochastic, =
M
11. Also **
=1M1 and then M maps
K
S into
K
S because if *=11x, then

*
*** *
== ==1
'
1x1MxM1 x1x. The matrix *
M is
stochastic and irreducible and so, by Perron-Frobenius
theorem, it has a unique strictly positive probability left-
eigenvector associated to the real dominant eigenvalue
1. Let *
eq
x be this line-vector, then ***
=
eq eq
xxM, or
T. HUILLET ET AL.
Copyright © 2011 SciRes. AM
15
=
eq eq
xMx. Under the irreducibility assumption on
M
,
the frequencies d ynamics involving only mutations has a
unique polymorphic equilibrium fixed point >
eq
x0.
When M is primitive then *
=
lim t
teq
Mx1. This
shows that
 
*
:==00 =,
teq eq
t
t
xx Mxx1xx
regardless of the initial condition

0x belonging to
K
S.
Note finally that from (15):
 
==:,IV
M
xM xx (16)
where
 
*
1
=2
VI
MxxMx is the quadratic mutation
potential. The probability right-eigenvector eq
x of M
uniquely solves

=0VMx with
=0
eq
VMx maxi-
mal.
3.2. Remarks and Special Cases
1) Reversible mutations: Let eq
x solve ***
=
eq eq
xxM.
Define
*1
=.
eq eq
DD
xx
MM

We have *1
==
eq
eq
D
x
M1Mx 1
, so *
M
 is the sto-
chastic matrix of the time-reversed process at equili-
brium with invariant measure *
eq
x. If **
=MM
, then
the mutation pattern is said to be time-reversible. In this
case
,
,,
,
=.
eq k
kllk eql
x
x

2) If *
=
M
M, then *
=MM and M is doubly sto -
chastic. In that case,

*
1
=1,,1=:
eq b
K
xx. A model
with symmetric mutations by assuming for instance mul-
tiplicative mutations: ,=
klkl

. In this case, with μ
the column vector of the k
s,
*
=.
I
Dμ
Mμμ μ
Alternatively, assuming ,1
=0,
1
kl K



for all
kl leads to

=1
J
KI
M which is also
symmetric.
It is not necessary that *
=
M
M in order to have M
doubly stochastic. It suffices to impose *
=
M
M11
. In
that case although *
MM
, the overall input-output
mutation probabilities attached to any state coincide and
the equilibrium state again matches with the barycenter
b
x of
K
S.
3) (Kingman house of cards, [4]). Assume the muta-
tion probabilities only depend on the terminal state, that
is: ,=
kll

for all kl
, still with ,=0
kk
. Let
*1
=,..., K
μ. Then, *
=
M
Dμ
1μ, =M
1μ1μ
where *
min<:=< 1max
kk
μμ1,
*
=1
I
Mμ1μ and

*
===1 .
'MDM
x
xMxxx1μμx (17)
The equilibrium state is =
eq
xμμ
. Note that
1
11
K

μ. This model is reversible. In this model
the coordinates are decoupled:
=1
'
kk k
x
x
μ,
depends only on k
x
.
4) Assume the mutation probabilities only depend on
the initial state, that is: ,=
kl k
for all lk. Then
*
=.IKD
μ
M1μ
This mutation model is also reversible and the equi-
librium state is
,1/
=.
1/k
eq kl
l
x
5) (Random walk). In this case,
M
is tri-diagonal
with ,,1
=
kl kk
u
and eq
x is a truncated geometric
distribution with common ratio =uu

:
1
,1
=1
=,=1,,.
k
eq kKl
l
x
kK
This model is reversible.
6) (Cyclic mutation pattern). Here, ,,1
=
klk lk

,
=1, ,1kK
and ,,1
=
K
lKl

. This model is not
reversible and
,1
=.
1k
eq kl
l
x
4. Combining Fitness and Mutations
Let us now consider the dyna mics driven both by fitness
and mutation. There are two ways to combine the fitness
and mutation effects. One (fitness-first), which is clas-
sical, consists in applying first the fitness mapping and
then let mutation act on the result. The other (mutation-
first), which seems to be less popular, consists in rever-
sing the order.
4.1. Fitness-First Dynamics
It is typically obtained by applying first the fitness ope-
rator and then the mutation one to give the ‘fitness-first’
T. HUILLET ET AL.
Copyright © 2011 SciRes. AM
16
dynamics [11]:
**
11
==,
'W
DDW
WW
xx
xMxMx
xxxx F (18)
defining a new nonlinear transformation. Alternatively,

=
'
xpx where

*
1
=DW
Wx
pxM x
xx is the new
mapping from
K
S to
K
S to consider. Component-wise,
this is also as required
 
,,
1
=,
=1 ,,.
'
kkklkllkkkl
lk lk
xxw wxxw
kK







xxx
x
(19)
We have:

*
,1 ,1,,1 ,
()= ,,,1,,,
kkkkkl kkkKK
lk S



pe 
and so the extremal states k
e are not invariant under p
and from the fixed-point theorem, there exists some
equilibrium state in
K
S. Using the representation (13)
and (15):
 
 
=
=.
W
W
IGV
VGV
 
 
M
xM xMxx
xMx x
(20)
This is not a gradient-like dynamics in general b ecau se
there is a competition between the mutation and fitness
potentials VM and W
V.
When =
I
M (no mutatio n) (20) boils down into (13)
and when JW= (no selection), (20) boils down into
(16). When both =
I
M (no mutation) and =WJ (no
selection), =0xwith corresponding neutral
=pxx
.
4.2. Mutation-First
Because W was assumed symmetric
*
=WW, there
is another way to combine the mutation-selection effects.
It is obtained by applying first the mutation operator and
then the fitness operator to give the `mutation-first’ dyn-
amics:
 
'1
=: =,
W
D
Mx
xpxMx
Mx (21)
where

*
:= W
MxMxMx . We have:
 
*
*==1W
1p xMxMxMx if and only if
*
=WW and under this condition, this n ew ()px again
maps the
K
simplex
K
S onto itself. The dynamics of
:=yMx is

1
=,
'W
D
y
yMy
y
which is of the form (18) and 1
=
xMy may be reco-
vered as an output from y only if M is invertible.
Component-wise, each component

k
px may be
read from

,,
=1, =1,,.
'k
kkkllkl
lk lk
W
x
xxkK










Mx
Mx
(22)
Equation (21) may also be recast as:


 
1
=
=,
W
IGW
VG V

 
M
xM xMxMx
Mx
xMx Mx
(23)
where, as before,

=log 2
W
V
xx. The mean fit-
ness function appearing in (23) is

***
:= ,WW
MxMxMx=x MMx
or else, the new fitness matrix to consider is
*
:= ,WW
MMMwhich is itself symmetric because
*
=WW. Just like (20), (23) neither is a gradient-like
dynamics.
Although natural, this alternative `mutation-first' way
to combine mutations and fitness effects seems to have
been less studied in the literature.
5. Multiplicative Fitness
We now focus on the multiplicative fitness model. Fol-
lowing the previous observations, we shall distinguish
two cases.
5.1. Fitness-First
In the haploid case or in the diploid case when fitnesses
are multiplicative, ,=
klk l
www, with

=ll
l
wwx
x
,,
==,
=1,,,
'
llkk klk llk kkl
llk lk
wwxwxxwwxxw
kK




xx
or
 
11
==,
'DD
ww
xw
xMwMx
xx
(24)
where w is the constant colu mn-vector of the k
ws and
*
=wxwx.
When dealing with multiplicative fitn esses models, we
shall assume >0
minkk
w, =1
maxkk
w and the second
largest <1
k
w.
The image of the extremal states =l
xe by the trans-
formation '
xx reduces to l
Me which belongs to
the interior of
K
S. In that case, there exists a unique,
globally stable polymorphic equilibrium state which is
the fixed-point of (24). This follows from the Perron-
Frobenius theorem commented in the forthcoming para-
graph. Recall that in the absence of mutations, the multi-
T. HUILLET ET AL.
Copyright © 2011 SciRes. AM
17
plicative fitness model cannot have a polymorphic equili-
brium state.
5.1.1. Polymorphic Equilibrium and Steady Mean
Fitness
Let *
=AD
wM, the latter ‘selection-first’ recurrence
may be recast as
  
**
*
1
1= .ttA
tA
xx
x1 (25)
Under our assumptions on w, =A1w and therefore
A
is sub-stochastic. By iteration
 
**
*
1
=(0).
0
t
t
tA
A
xx
x1
When M is primitive, so is 0A which has Per-
ron-Frobenius left and right probability eigenvectors
*>0
A
x and >0
A
y associated to its largest eigenvalue
1> >0
A
.
Then, *
,,
11
=
lim
t
AA
tAAkAk
k
Axy




yx showing that,
 
0:=> ,
lim A
tt

xxx0
which is the required limiting polymorphic state. The
value of A
(respectively 2
A
) is the limiting haploid
(diploid) mean fitness because:

**
:====
AA AAA
ww A
xwxx1 . When looking at the
equivalent reformulation (25) of (24), A
x can be inter-
preted as a quasi-stationary distribution as developed
now. For the precise definition, see [12].
5.1.2. A Stochastic Interpretation of the Deterministic
Dynamics (25)
A vector x of
K
S can be thought of as a probability
vector. The dynamical Equation (24), as a nonlinear
update mapping from
K
S to
K
S may be viewed as the
discrete-time nonlinear master equation of some Markov
process whose construction we now give. We shall need
to introduce an extra state, say

=0 which will be
absorbing for the process we shall now construct. It will
be useful to extend the matrix
A
to in the follow-
ing way:
,0, 0,,0
=1
=1, =.
K
kklll
l
AAA
Let then

Lt be the random labels distribution of an
individual at ti m e t, with enlarged state-space

0,1, ,
K
. Let

:=1,2
t
Ut be an i.i.d. driving
sequence of uniformly distributed random variables on
[0,1]. Consider the random evolution equation
 





1,
1= ,==<=,
't'
Lt l
LtlLt lUALt l
111
where
,0,1,,
'
ll K.
From this construction


:Ls s t is measurable
with respect to
:
s
Ust
and we ge t
 

,
1== .
''
L
tl
LtlLt A (26)
Let
be the first time that

Lt hits the absorbing
state
=0. Using the extinction time (26) may be
recast as
 


,
1=, > 1=>.
Ltk
LtktLtAt

 1
Putting
:== ,>
k
ztLt kt
, we get an un -nor-
malized version of (25):





*
,
1=>=,{1,, }.
kLtkk
ztAttA kK
1z
We clearly have

,
,,
>= ,
lim Ak
t
Ak
tAk Ak
k
y
txy


(27)
and so
1> >0
t
A
t

geometrically fast. From
the last expression, the right-hand-side may be inter-
preted as the propensity of a type-k allele to survive to
its fate: the eventual extinction. If ,,
>
Ak Al
yy indeed,
the extinction time of the process started at k is larger
than the one started at l (has larger survival asymptotic
tails). We shall call A
y the survival probability vector.
Defining the normalized conditional probabilities

 

=1
===>,
k
kK
k
k
zt
x
tLtkt
zt
we obtain the normalized haploid dynamics (25)


*
*
=1
=,1,,.
'k
kK
k
k
A
x
kK
A
x
x
It may now be viewed as the nonlinear master equa-
tion of some stochastic Markovian process. In view of
this construction, the vector A
x is the quasi-stationary
distribution of
Lt given>t
.
We note that the appeal to the coffin state
was a
necessary step to understand the normalization kk
zx,
and the stochastic interpretation of (25) allows to give
sense to the right eigenvector A
y of
A
.
Clearly the above construction can be done for N
particles, in particular =2N in the diploid case.
5.2. Mutation-First
If instead of (24) the dynamics is of the type ‘mutation-
first’
T. HUILLET ET AL.
Copyright © 2011 SciRes. AM
18
*
1
=,
'Dw
xMx
wMx (28)
because the mutation operator was applied first in the
composition of the fitnesses and mutation effects, the
latter recurrence may be recast as:
 
**
*
1
1= (),ttB
tB
xx
x1 (29)
where now *
=BD
w
M. By iteration
 
**
*
1
=(0).
0
t
t
tB
B
xx
x1
5.2.1. Equilibrium and Mean Fitness at Equilibrium
Let >
B
x0 and >
B
y0 be now the left and right
probability Perron-Frobenius eigenvector of B asso-
ciated to its largest eigenvalue >0
B
. We have
*
,,
11
=
lim
t
B
B
tBBkBk
k
Bxy




yx showing that
 
0:=> .
lim B
tt

xxx0
We have 1
=BDAD
ww
and so B is diagonally similar
to
A
. Therefore =
B
A

and =
B
AA
Dw
w
xx toge-
ther with 11
=
B
AA
DD

ww
yyy.
The limiting equilibrium mean fitness is now
**
:==
BBB
wwxx w (or 2
B
w in the diploid case). Re-
calling =
B
AA
Dw
w
xx, we have
2,
,
=.
kAk
k
BkAk
k
wx
wwx
Since ,
==
AkAkA
k
wwx
, under the multiplicative
fitness hypothesis we obtain:
Proposition 1. =<<1.
AAB
ww
Remark: The quantity

2
2,
=1
k
AAk
kA
w
x
w



x is
the variance in relative fitness at equilibrium for the
model
A
We therefore have
 

22
=1or=1 .
B
ABAA
A
www
w

xx
The equilibrium fitness of the second model is larger
than the one of the first. Without mutations, only the
fittest state, say
K
under our hypotheses, will survive,
leading to an equilibrium mean fitness equal to

*
0,0, ,0,1==1
K
ww. Therefore, both mutation mo-
dels lead to a decrease of the equilibrium mean fitness,
when compared to the one without mutations. However,
the first model involves mutations which are more dele-
terious than the ones relative to the second one where
mutations appear more advantageously.
Note finally that
**1* *1*
=====
B
BB BBBBA
DB D


ww
wMx wxwx1x
and, since *
=BD
w
M then 1*
=BD
wM so that
1=B
w1 if

*
111
1
:=,, K
ww

w is the reciprocal fit-
ness vector. As a result, *1*1
==1
BBB
B

xwxw so that
,
*1 1
:== .
Bk
B
B
kk
x
w
xw
5.2.2. A Stochastic Interpretation of the Deterministic
Dynamics (29)
We can repeat the above construction substituting B
for
A
and we are done.
5.2.3. The Stochastic Dominations xx
B
st A
and
yy
AstB
For two K
dimensional probability vectors a and b,
we put st
ab if for each l
=1=1 .
ll
kk
kk
ab
Proposition 2. We have
B
st A
xx.
Proof: =
B
AA
Dw
w
xx and therefore
,
,
,
=1
=.
kAk
Bk K
kAk
k
wx
x
wx
With ,
=1
:= l
lAk
k
x
, we have

,,
,
=1=1= 1
,,
=1= 1
=1 .
Kl K
Ak Ak
kAkl klk
lK
kk kl
Ak Ak
kkl
xx
wx ww
xx


 

Since

,1
=1
,
=1
,
lAk
kl
l
k
Ak
k
x
www
x
and

,1
=1
,
=1
,,
KAk
klK
K
kl
Ak
kl
x
www
x
,
,
=1=1 ,
=1
,
Kl
Ak
kAkk l
kk
Ak
k
x
wx w
x

and therefore
,
=1
,,
=1 =1
,
=1
=,
l
kAk
ll
k
Ak Bk
K
kk
kAk
k
wx
x
x
wx

which means
B
st A
xx.
T. HUILLET ET AL.
Copyright © 2011 SciRes. AM
19
We point out that we used the order
1
0< =1
K
ww on .wWould we have consi-
dered the reverse order, we would get the opposite
dominat ion relati onship.
We also have the following stochastic domination pro-
perty between the two survival probability vectors:
Corollary 3. AstB
yy.
Proof: Because B and
A
are diagonally similar,
we also have =
ABB
DD
ww
yyy and the same argu-
ment applies substituting

,
AB
yy for

,
B
A
xx in the
previous pro of .
5.3. Symmetric Mutations
When *
=MM, mutations are symmetric and *
=BA.
Therefore =
B
A
xy and =
B
A
yx. The left (right) pro-
bability eigenvector of B matches with the right (left)
probability eigenvector of
A
.s In this case, there is a
stochastic domination property between the left and right
eigenvectors of both models, namely
Proposition 4. If mutations are symmetric AstA
yx
and
B
st B
xy.
5.4. Reversible Mutations
When dealing with reversible mutations with equilib rium
distribution eq
x, we show now that the r ight eigenvector
of
A
(or B) can be computed from the left eigen-
vector by using an appropriate Schur product involving
w and eq
x
Let
A
be an irreducible non-negative matrix. Let

,
AA
xy be the left and right probability eigenvectors
of
A
, associated to the spectral radius A
of
A
.
If there exists a positive vector η such that
,,
=
kkl llk
A
A

for all , kl
,
A
is said to be reversible
with respect to η.
Consider the stochastic matrix
11
A
A
A
A
DAD

yy
(30)
Its left probability invariant measure is easily seen to
be ,,
A
y
AAkAk
Ak
Dxy
xx
, which is the normalized
Schur product of A
x and A
y We have
Lemma 5. If
A
is reversible with respect to η, then
A
is reversible with respect to 2
A
y
Dη the Schur
product of A
y, A
y and η
Proof:
2
,,,
11
,,,,
2
,,
,
.
AkAk Ak
kk
lkA lkAklkl
AllAl lAl
yyy
AAA A
yy
y




 

As an illustration, we shall consider the fitness-first
dynamics for which *
=AD
wM. This
A
will be rever-
sible with respect to η if and only if *
M itself is rev-
ersible. Indeed, η must satisfy
,,
=,
kkkl lllk
ww
 
(31)
and if this is the case *
M must be reversible with
respect to eq
x, leading to ,
=.
keqkk
x
w
In case *
M is reversible,
A
is reversible with respect
to 2
A
Dyη with entries proportional to

2
,,eqkAkk
x
yw.
But this must be the invariant measure of
A
which, up
to a normalizing constant, is A
A
Dyx with un-normalized
entries ,,AkAk
x
y. For this point, see also [13]. We
conclude:
Proposition 6. If the mutation matrix is reversible
with respect to eq
x, then
A
is reversible with respect
to 21 12
=
weq weq
AA
DD DD

yy
xx and
1
1
=.
wA
eq
A
wA
eq
DD
DD
x
x
y
x
y
Example: If *
M is symmetric, it is reversible with
respect to the uniform measure ,=1 .
eq k
x
K Therefore
11
=.
AwAwA
DD

xyy Up to a normalization constant,
we have
22 2
,,,, ,,
===,
eqk AkkAkkkAkAk Ak
x
ywywwxxy
the Schur product of A
x and A
y.
Substituting B for
A
and

,
B
B
xy for
,
AA
xy ,
the same holds true for *
=BD
w
M and w e ge t
If the mutation matrix is reversible with respect to
eq
x, then 11
B
B
B
BDBD

yy
is reversible with respect to
22
=
weq weq
BB
DD DD
yy
xx
and
,,
,,,
= or =.
wB
eqkeq kB k
BBk
keqkBk
wB
eq k
DD wx y
xwx y
DD
x
x
y
x
y
6. Multiplicative Fitness and the House of
Cards Condition
We shall again distinguish two cases.
6.1. Fitness-First
Assume the house of cards condition holds, leading to:
*
=AD
wM, with

**
=1
I
M1μμ. In this case, the
computations become more explicit. Since for all k,
,==1
kll k
lk lk


μ, we have
1
and 1.
11
K
K
K

μμ
T. HUILLET ET AL.
Copyright © 2011 SciRes. AM
20
Under the multiplicative fitness and th e house of cards
conditions, Equation (19) reads

*
1
=
1
=.
'
kkkkllkkl
lk lk
kkk
xxwwxxw
w
wx





x
μ
xw
The equilibrium frequency distribution is therefo re the
solution to the equ a tion
*
1
=,
AA
A
D
w
μ
xxμ
xw
which, s i n ce *
=
AA
xw, is exactly seen to be:

,=,=1,,,
11 /
k
Ak
kA
x
kK
w
μ
(32)
where

0,1
A
is such that ,=1
Ak
kx
. See [4]
where these results appear first. Alleles k
A
with largest
frequencies are those for which both

,
kk
w
are large.
The equilibrium mean fitness is

*=,
11 /
kk
AA
kkA
w
w

xw= μ (33)
the spectral radius of
A
and *
=AD
wM satisfying
=A1w. Because >
A
x0, we have:

=>1 .
max
AAk
k
ww
μ (34)
If =1,μ,=,
Ak k
x
=1, ,kK and =
Akk
kw

.
6.2. Mutation-First
When fitnesses are multiplicative: *
=Www is symme-
tric, (21) is also

*
1
=: =.
'Dw
xpx Mx
wMx With

*
1
=,, ,
k
www this simplifies to give:
,,
*
=1 ,=1,,.
'k
kkkllkl
lk lk
w
x
xxkK










wMx
(35)
When the house of cards condition holds, *
=BD
w
M,
with

**
=1
I
M1μμ. Equation (35) further simpli-
fies to:
 

**
=1,=1,,.
1
'k
kkk
w
x
xkK

 μ
wμμwx
(36)
From Equation (36), the equilibrium frequency distri-
bution is the solu tion to the equations
 

,,
**
=1,=1,,,
1
k
Bkk Bk
B
w
x
xkK

 μ
wμμxw
which is exactly seen to be:


,*
=,=1,,,
1
kk
Bk
Bk
w
x
kK
ww
 wμμ
(37)
where
*
=0,1
BB
wxw is such that ,=1
Bk
kx
.
Alleles k
A
with largest frequencies are those for
which the product kk
w
is largest. Because A and B
are diagonally similar, we have =
B
AA
Dw
w
xx where
==
AAB
w

. From the expression (32) of ,Ak
x
, we get
the alternative expression

,=,=1,,.
1
kk
Bk
Bk
w
x
kK
w
μ
(38)
We also have

2
*=,
1
kk
B
B
kBk
ww
w

xw= μ (39)
the equilibrium mean fitness under *
=BD
w
M .
Comparing the tw o expressions (3 7) and (38) o f kB
x,,
this suggests that

*
=1
B
B
w
wμμ. Thus
*
=== 1
AAB B
ww

wμμ and
*
=>0.
BA B
ww wμwμ (40)
As a result
*
1
>
B
wwμ
μ (41)
gives a lower bound for B
w in terms of the average of
w with respect to the mutation equilibrium probability
measure =.
eq
xμμ
Because >
B
x0, from (37), we also have:

*
1<1,.
kB
wwk
μwμμ
If 1>0μ, this means
*
>1
maxk
Bk
wwwμ/μ whereas if 1<0μ this
means

*
<1.
mink
Bk
ww
wμ/μ When =1μ,
2
=
B
kk kk
kk
www

is explicit, together with
,=,=1,,.
kk
Bk kk
k
w
x
kK
w
Recalling that when =1:μ==
AA kk
k
ww

, we
can check in this particular case that: >
B
A
ww.
7. Alternative Conditional Models for the
Evolutionary Dynamics
The interpretation of (24) and of (29) in terms of a sto-
chastic process conditioned on not being currently ab-
T. HUILLET ET AL.
Copyright © 2011 SciRes. AM
21
sorbed in suggests that other conditional models for
the evolutionary dynamics involving multiplicative fit-
ness and mutations could also be worth investigating.
Consider first the fitness-first model (24) driven by
*
=AD
wM. Let
1
1A
A
DA
and consider the updating
dynamics on the simplex
 
**
1ttAxx (42)
Because =A1w, we have
*
A
M which is the
pure mutation stochastic matrix. Using the terms of the
stochastic inter preta t i o n of ( 2 4) , we have:



,011
kl Lk
APL l
 which is the transition ma-
trix of a one-step conditioned process. Therefore condi-
tioning locally

Lt on non-extinction brings one back
to the pure underlying mutation model with
eq
A
xx.
Let us consider a similar conditioning but for the
mutation-first dynamics. With now*
=,BD
w
Mlet
1
1
=B
BDB
and consider the dynamics
 
**
1ttBxx (43)
We have
*
11*
1
=B
BDBDD

w
Mw
M which is stochastic
but cannot be reduced to mutation effects in general.
This is an additional illustration of the differences bet-
ween the two models based on
A
or B. Note that
B
and
A
are not diagonally similar.
Let now
11
yA
A
Ay
A
DAD

and consider the dyna-
mics on the simplex
 
**
1ttAxx
(44)
We have:



,0
lim 1
kl sLk
A
PLls

 which
is the transition matrix of a process conditioned on not
getting extinct in the remote future (see [14]). Therefore
conditioning globally ()Lt on non-extinction in the far
future brings one back to a standard (linear) Chapman-
Kolmogorov evolution equation. This conditioning being
more stringent than the one involved in (24), one expects
its limiting frequ enc y distribution (which is
,,
AAAkAk
Ak
Dxy
x
xy
) to stay away more signifi-
cantly from the origin {0}. For this model, the mean
fitness at equilibrium will be:
,,
*
,,
.
A
kAkAk
k
A
AAk Ak
k
wxy
wD xy

x
wy
When mutations are reversible,
12
,,
12
,,,
.
keqkAk
Ak keqkAk
k
wx x
xwxx
Similar conclusions can be drawn if we define
11
:
B
B
B
BDAD

yy
.
The main interest is that in both conditioning (either
local or global), the deterministic updating mechanisms
are now linear in sharp contrast with (24) and (29) in-
volving rational updating mechanisms.
7. Acknowledgments
The authors are indebted for support of the Basal Conicyt
project and S. M. to the Guggenheim fellowship.
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