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The canonical equation
of adaptive dynamics
a new role for the
effective population sizes
of population genetics
Hans (=JAJ *) Metz
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VEOLIAEcole
Polytechnique
&
Mathematical
Institute,
Leiden
University
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Preamble
Real evolution mostly takes place in higher (even infinite)
dimensional trait spaces.
In higher dimensions the main ideas introduced by means of
the graphical constructions still apply,
but the analysis of the monomorphic dynamics hinges
the ordering properties of the real
line.
To study the dynamics in the higher dimensional case we
ususally fall back on a differential equation,
grandly called the Canonical Equation of adaptive dynamics.
Goal: This talk is given to its derivation, and to an unexpected
recently discovered link with the theory of random genetic drift.
overview
the ecological theatre and the evolutionary play
Long term adaptive evolution proceeds through the
continual filtering of new mutations by selection.
The supply of new phenotypes by mutation depends on
the genotypic architecture and the
genotype to phenotype map.
Selection is a population dynamical process.
Convenient idealisation:
The time scales of the production of novel variation
and of gene substitutions are separated.
the adaptive play
Given a population dynamics one can graft onto it
an adaptive dynamics:
Just assume that individuals are characterised by traits
that may change through mutation,
and that affect their demographic parameters.
The speed of adaptive evolution is proportional to
the population size n .
a second, silent, play: random genetic drift
Given a population dynamical model one can also
graft onto it a random genetic drift.
Just imagine that each individual harbours two alleles
without consequential phenotypic effect,
which in the reproductive process are reassorted
according to Mendel’s laws.
The speed at which variation is lost over time is
inversely proportional to n.
a formal connection between the two plays
Result:
The constants that appear in front of n in the formulas for
the speeds of adaptive evolution and random genetic drift
are the same.
The product of this constant and n is known as the
effective population size ne.
Part I
The ecological theatre
the population dynamical side of adaptive evolution
Adaptive evolution occurs by the repeated
substitution of mutants in largish populations.
Mutants emerge single
in an environment set by a resident population.
How many mutants occur per time unit depends on
the size of the resident birth stream.
The mutant invasion process depends on their phenotype
and on the environment in which they invade.
The assumption that the resident population is large makes
that both the birth stream and the environment can be
calculated from a deterministic community model.
modelling the ecological theatre
The easiest way of accomodating all sort of life history detail
in a single overarching formalism
is to do the population dynamical bookkeeping
from births to births.
(C.f. Lotka’s integral equation.)
By allowing multiple birth states it is moreover possible to
accommodate i.a.
-
parentally mediated differences in offspring size, social status, etc.
spatially distributed populations (use location as birth state component)
cyclic environments (use phase of cycle as birth state component)
genetic sex determination, including haplo-diploid genetics, etc
genetic polymorphisms.
I will proceed as if the number of possible birth states is finite.
modelling the ecological theatre
Odo Diekmann
Mats Gyllenberg
Equilibria of structured populations generally satisfy:
B = L(X|EX) B,
EX = F(I),
I = G(X|EX) B
Diekmann, Gyllenberg & Metz (2003) TPB 63: 309-338
trait vector
(affects the two operators that
describe individual behaviour)
modelling the ecological theatre
Odo Diekmann
Mats Gyllenberg
Equilibria of structured populations generally satisfy:
B = L(X|EX) B,
equilibrium
Birth rate vector
per unit of area
EX = F(I),
environmental input,
i.e., the Environment as
perceived by the individuals
next generation operator
(i.e., lij is the Lifetime number of births
in state i expected from a newborn in state j)
I = G(X|EX) B
per capita lifetime
impingement on
the environment
population output,
i.e., Impingement
on the environment
modelling the ecological theatre
Odo Diekmann
Mats Gyllenberg
Equilibria of structured populations generally satisfy:
B = L(X|EX) B,
equilibrium
Birth rate vector
per unit of area
EX = F(I),
environmental input,
i.e., the Environment as
perceived by the individuals
next generation operator
(i.e., lij is the Lifetime number of births
in state i expected from a newborn in state j)
I = G(X|EX) B
result of the
dynamic
equilibrium
of the
surrounding
community
Part II
The adaptive play
two subsequent limits
(concentrate on traits, and
rescale time appropriately)
individual-based
stochastic process
canonical equation
limit
branching
t
trait value
limit type:
system size  ∞
successful mutations/time  0
mutational step size
0
the canonical equation
The “canonical equation of adaptive dynamics” was derived
to describe the evolution of quantitative traits
in realistic ecologies.
Assumptions: large population sizes, mutation limitation,
small mutational steps*.
dX
 12 n  C
dt
*
 s Y X  


 Y Y  X 
T
and, initially, simple ODE population models
the canonical equation
dX
 12 n  C
dt
 s Y X  


 Y Y  X 
T
X: value of trait vector predominant in the population
n: population size, : mutation probability per birth event
C: mutational covariance matrix, s: invasion fitness, i.e.,
initial relative growth rate of a potential Y mutant population.
history
basic ideas and first derivation (1996)
hard proofs (2003)
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Ulf Dieckmann & Richard Law
Mendelian
discrete
diploids
generations
extensions (2008)
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Nicolas Champagnat & Sylvie Méléard
dX
 122 ne,A  C
dt
hard proof
 s Y X  


 Y Y  X 
Michel Durinx & me
T
general
with
Poisson
life
histories
# offspring
only
rigorousassumptions
for
Underlying
rather
unbiological: individuals reproduce
for pure age
far only
ODE model
dependence
clonally,
have exponentially
distributed lifetimes andsogive
birthfor
at
communitymodel).
equilibria
Tran
ageneral
constant
rate from birth Chi
onwards
(i.e., an ODE population
case
(2006)
non-rigorous
not yet published
an unexpected connection
Part III will argue,
based on conjectured extensions of theorems
that have been proven for simple special cases,
that
ne,A = ne,D
the effective population size for random genetic Drift.
derivation of the canonical equation
For general equilibrium population dynamics the canonical
equation appears first in another form:
dX
2
b 2
dt
e
births per
unit of time
B=bU, 1TU=1
C
probability of mutation
per birth event
evolution is mutation limited
 R0 Y E X  


Y


Y X 
T
mean of
[mutational step  approximation for the
probability that a Y-mutant invades]
mutational steps are small
demographic ingredients 1
2
 2e
 R0 Y E X 



Y


Y X 
* R0(Y | EX)  average life-time offspring number of a mutant allele
producing Y when singly substituted in the resident genotype
( calculated as dominant eigenvalue of a next generation operator L(Y | EX) ).
For the resident:
R0(X | EX) = 1.
U  stable birth state distribution of resident( allele)s ,
normalised dominating right eigenvector of
TU =1,
L(X
|
E
)
,
1
X rate based) reproductive values of newborn resident( allele)s,
V  (birth
co-normalised dominating left eigenvector of L(X | EX), V TU =
1.
demographic ingredients 2
 2e
*
2
 2e
 R0 Y E X 



Y


Y X 


:  Var   vi mi j  u j
 i

j
with mi the lifetime number of offspring born in state i,
begotten by a resident allele born in state j.
For the resident:
R0(X | EX) = 1.
U  stable birth state distribution of resident( allele)s ,
normalised dominating right eigenvector of
TU =1,
L(X
|
E
)
,
1
X rate based) reproductive values of newborn resident( allele)s,
V  (birth
co-normalised dominating left eigenvector of L(X | EX), V TU =
1.
from the theory of branching processes
1000
# individuals
100
10
1
time
0
10
20
In an ergodic environment EX
a population starting from a single individual:
either goes extinct, with probability Q(Y |EX),
or "grows exponentially" at a relative rate s(Y | X).
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For constant EX and small s :
2

P  1  Q   2 ln R0 
 e

Ilan Eshel
dX
2
b 2
dt
e
the average effective step
Z = Y-X
approximation for the probability
that a Y-mutant invades


T 2
Z Z 2
   e

 R0 Y EX  


Y


Y X 
T
C
probability density of
mutational steps

 g(Z )dZ


Brook Taylor
Smooth
on thegenotype
stronger to
phenotype
on the assumption
maps
assumption
thatlead
the to
locally
thatadditive
mutations
genetics.
are 
mutations
distribution
In diploids
the mutational
isunbiased
symmetric
effect
doubles
over a full
around
the resident
substitution
1
2
C
2
 2e
 R0 Y E X  


Y


Y X 
 R0 Y E X  


Y


Y X 
T
mutational covariance matrix
T
aside on genotype to phenotype maps
Some evo-devo:
Genotype to phenotype maps are treated as smooth
maps from some vector space of gene expressions to some
vector space of phenotypes.
Rationale: Most phenotypic evolution is probably regulatory,
and hence quantitative on the level of gene expressions.
regulatory regions
coding region
DNA
reading direction
the genotype to phenotype map: 
Consider a representative locus, to be called A, affecting the
phenotypic trait, with resident allele a and potential mutant A.
Alleles are supposed to affect the phenotype in a quantitative
manner, expressible through an allelic trait Xa resp. XA .
The corresponding phenotypes will be written as Xaa, XaA, XAA:
Xaa = ( ;Xa,Xa; ), XaA= ( ;Xa,XA; ), XAA= ( ;XA,XA; ),
with  subject to the
restriction
( ;Xa,XA; ) = ( ;XA,Xa; )
(  no parental effects).
local approximate additivity
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We assume  to be sufficiently smooth, and express the
smallness of the allelic effect as XA = Xa + Z.
The symmetry restriction implies
∂( ;Xa,XA; )
∂( ;Xa,XA; )
=
: = A´( ;Xa,Xa; )
∂Xa
XA=Xa
∂XA
XA=Xa
Hence
XaA = ( ;Xa,Xa; ) + A´( ;Xa,Xa; ) Z
+ O(2)
XAA= (;Xa,Xa; )+ A´(;Xa,Xa; )Z+ A´(;Xa,Xa ;) Z + O(2)
XAA
= ( ;Xa,Xa; ) + 2 A´( ;Xa,Xa; )Z + O(2).
the mutational covariance matrix
Let, given that a mutation happens, the probability that this mutation
happens at the A-locus be pA,
and let the distribution of the mutational step, Y := XA-Xa,
be fA .
Assume that the total mutational change is not only small but
also unbiased, i.e,
 p  (
A
L
;Xa ,Xa  Y;L ) fA (Y )dY  0
A
then
C   pA  ( L ;Xa , Xa  Y;L ) ( L ;Xa , Xa  Y;L ) fA (Y )dY
T
A
  pA  A ( L ;Xa , Xa ;L )  YY T fA (Y )dY  A T ( L ;Xa , Xa ;L )
A
mutational covariances need not be constant!
In the Mendelian case the derivation from the basic ingredients,
genotype to phenotype map and genotypic mutation structure,
shows that the canonical equation represents but the lowest level
in a moment expansion.
The next level is a differential equation for change of the mutational
covariance matrix, depending on mutational 3rd moments, etc.
recovering population size and fitness
dX
2
 b 2
dt
e
C
 R0 Y E X  


Y


Y X 
T
Tr :
average age at giving birth,
Ts :
average survival time
ne,A
Tr
 2
n
2
Ts e
C
 s Y X  


 Y Y  X 
of the residents.
n  bTs  b  n
Ts


ln  R0 Y E X 
s Y X  
Tr
T
demographic ingredients 3
dX
T
 2 r2 n 
dt
Ts e
C
 s Y X  


Y


Y X 

Ts   a F T (a) U da
*
0
with fi(a) the probability density of the age at death of an individual
born in state i.

Tr   a V T (a) U da
*
0
with (a) composed of the average pro capita birth rates at age a.





0

V (a) U da  V L X | E X U  1. 

T
T
T
aside: robustness of the CE
a dna ™e mi Tk ciuQ
ro s serp m oce d
.erut cip sih t ee s ot ded een era
Géza Meszéna
For small mutational steps
the influence of the mutants on the environment at higher densities
comes in only as a term with a higher order in the mutational step size
than the terms accounted for by the canonical equation.
Whether or not some previous mutants
have not yet gone to fixation
has little influence on the invasion of new mutants.
The applicability of the canonical equation
extends well beyond the case of strict mutation limitation.
Part III
The new result and its ‘proof’
random genetic Drift
Operational definition:
The effective population size for random genetic Drift, ne,D
,
is most easily defined as the parameter occurring in the usual
diffusion approximation for the temporal development of
the probability density  of the frequency p of a neutral gene.
2

p 1  p 

1
1

2 2 ne,D
t
p 2
Interpretation:
The size of a population with non-overlapping generations and
multinomial pro capita off-spring numbers that produces the same
(asymptotic) decay of genetic variability as the focal population.
Proposition
The effective population sizes
ne,A for Adaptive evolution
and
ne,D for random genetic Drift
are equal
whatever the life history or ecological embedding.
idea of the proof
Connect the approximation formula for the invasion probability
derived from branching process theory PB (s) (Eshel’s formula)
with the probability for take-over for a diffusion approximation
including selection, PD  p0 , s%:
P s,n 
Tr
PB s 
lim
 ns,
2 2  lim
 lim
s0
n
s0
ne,D n constant
s
e
s
s0
initial frequency of mutant allele


Ts
1
PD 2n T , Tr s
ne,D
r
 2Ts
n
s
selection coefficient
 Why this particular
proxy for combination
system size of limits ?
 Why this formula for the initial mutant allele frequency p0?
 Why this formula for the selection coefficient ~s ?
idea of the proof
Tr
P s 
2 2  lim B

s0
e
s
lim
ns, s0
ne,D n constant


Ts
1
PD 2n T , Tr s
ne,D
r
 2Ts
n
s
Why this formula for the selection coefficient s~ ?
In the diffusion approximation time is measured in generations.
Hence s = Tr s~ .
idea of the proof
Tr
P s 
2 2  lim B

s0
e
s
lim
ns, s0
ne,D n constant


Ts
1
PD 2n T , Tr s
ne,D
r
 2Ts
n
s
Why this particular combination of limits ?
ne,D n reflects the behavioural laws of resident individuals.
We want to determine the first order term for small positive s
in the Taylor expansion of the invasion probability.
We want to remove the effect of the finiteness of n.
idea of the proof
Tr
P s 
2 2  lim B

s0
e
s
p0
lim
ns, s0
ne,D n constant


Ts
1
PD 2n T , Tr s
ne,D
r
 2Ts
n
s
Why this particular combination of limits ?
Left and right we take subsequent limits in different ways:
(1) from full population dynamics to branching process, followed by
an approximation for the invasion probability,
(2) from full population dynamics to diffusion process, followed by
a two-step approximation for the invasion probability.
To get equality we want the corresponding paths in parameter space
to approach the eventual limit point from the same direction.
idea of the proof
Tr
P s 
2 2  lim B

s0
e
s
lim
ns, s0
ne,D n constant


Ts
1
PD 2n T , Tr s
ne,D
r
 2Ts
n
s
Why this formula for the initial mutant allele frequency p0 ?
belief
Any relevant life history process can be uniformly
approximated by finite state processes.
In continuous time at population dynamical equilibrium (E = EX ) :
B: individual level state transition generator
A: average birth rate operator
Order the states such that the birth states come first. Then the next
generation operator L(X, EX ) can be expressed as
1
L(X, EX )  K AB K
T
with
KT 
1

0
M

 0
0 L
O O
O O
L 0
0
M
0
1
0 L
M
M
0 L
L
L
0

M
M

0
idea of the proof
Tr
PB s 
2 2  lim

s0
e
s
lim
ns, s0
ne,D n constant


Ts
1
PD 2n T , Tr s
ne,D
r
 2Ts
n
s
Why this formula for the initial mutant allele frequency p0 ?
The
consists
N0 = diffusion
N0 = for
N astructured
n the fastpopulation
For genetic
process results
in of a slow
diffusion of the gene frequency 1p, preceded
by a fast process
T
%
p0 
V N0
bringing the population state
2n
V T : co-normalised
to
+B
N p02of
, 2 pA0 (1
 p0 ), (1  p0 )2 
N0, N0 , N0 left eigenvector
N  nU%the stationary population state without genetic differentiation,
with U the normalised eigenvector of A + B.
idea of the proof
Tr
PB s 
2 2  lim

s0
e
s
lim
ns, s0
ne,D n constant


Ts
1
PD 2n T , Tr s
ne,D
r
 2Ts
n
s
Why this formula for the initial mutant allele frequency p0 ?
For N0 = N0 = N  n the fast process results in
1 %T
p0 
2n
V N0
V T : co-normalised left eigenvector of A + B
For a just appeared mutant:
Lemma:
N0  KU
Ts T
V K V
Tr
T
p0 
1 Ts
2 n Tr
Part IV
Finale
a corollary: individual-based calculation of ne,D
Tr
PB s 
2 2  lim

s0
e
s
lim
ns, s0
ne,D n constant
ne,D


Ts
1
PD 2n T , Tr s
ne,D
r
 2Ts
n
s
Tr n

,
2
Ts  e
a result already reached by different means by
William G Hill (1972)
for the simple age-dependent case,
William G Hill
Edward Pollak
and by
Edward Pollak (1979, ...)
for the age dependent case with multiple birth states.
a practical consequence
At equal genetic and developmental architectures
and population dynamical regimes
the speeds of adaptive evolution and random genetic drift
are inversely proportional.
Already for moderately large effective population sizes
adaptive processes dominate, and
neutral substitutions will be largely caused by genetic draft.
The End
a tricky point
The calculation linking the initial condition for the structured mutant
population to the initial condition of the diffusion of the gene frequency
assumes that there is no need to account for extinctions before the
reaching of the slow manifold,
in blatant disagreement with the fact that mutants arrive singly.
The justification lies in a relation linking the invasion probabilities of single
invaders to those of mutants arriving in groups that are
- so small that their influence on the environment can be neglected, yet
- so large that the probability of their extinction before their i-state
distribution has stabilised can be neglected.
a tricky point, cont’d
Consider a mutant population starting from an initial cohort of newborns
 =(1, … , k), i the number in birth state i, drawn from some law .
Let P(,s) be the correspondig probability of invasion.
Ansatz: P(,s) = ()s + o(s), (i) =: i.
Then (under some conditions on the “tail” of  )
k

i 
P( , s)  E 1   1  P( i , s)   o E   o(s)


i 1
Hence


k

i 
k
1   1  P( i , s) 




i 1
 (  )  lim E
 E   i i    TE 
s0
s
 i i

More about the robustness of the
canonical equation
slowing down by within population variation 1
Away from full mutation limitation: the speed of evolution tends in
the case of clonal reproduction to be considerabley smaller than
predicted by the canonical equation, thanks to so-called clonal
interference:
Without mutation limitation the effect of different mutants in no
longer additive as different mutants coming from the same parent
type compete (with ususally the furthest one winning the race).
This effect largely disappears with Mendelian inheritance:
Substitutions occur in parallel, and the local additivity of the
genotype to phenotype map makes that in the deterministic realm
the different loci have little influence on each other’s speed of
substitution.
slowing down by within population variation 2
In the initial stochastic realm evolution is slowed down a bit by
the additional variability in the offspring number of newly
introduced alleles coming from the variability in the partners with
whom they team up to make a body. If this variability would be
fixed
T
 dR 
 dR 
 2e   2e  12  0  Cp  0 
 dX 
 dX 
Cp the covariance matrix of the variation of X in the population.
If  2e would be calculated based on empirically determined components
the additional term is automatically incorporated.
However, the above expression only gives a very partial picture
since Cp is bound to fluctuate in time.
dissecting the limit
lim
ns, s0
ne,D n constant

Ts
1
PD 2n T , Tr s
r
s

why this particular limit?
First hint:
For Karlin models the chosen limit procedure naturally
recovers the result from their branching process limits.
Karlin models (without any population structure)
In Karlin’s conditioned direct product branching process models
n/ne,D = 2 = variance of offspring number
2


1

exp
2
s


1
PD n , s 
1  exp   2ns  2 


lim
ns,
s0

PD n 1, s
2s

1
,
2

which equals the limit result from branching process theory.
why this particular limit?
1  exp   2ne,D s%n 
1  exp  4ne,D s%
First hint:
For Karlin models the chosen limit procedure naturally
recovers the result from their branching process limits.
General heuristics:
-
The effect of finite population size should disappear.
(This effect occurs primarily in the integrals making up the
denominator of the diffusion formula.)  Let ne,D ~s  ∞.
The goal is to obtain the coefficient of the linear term in an
~
~
expansion in s.  Divide by s and let s  ∞.
Eventual “justification”:
A study of the paths in parameter space corresponding to
the two ways of taking subsequent limit shows that in both
paths the limit point is approached in a similar manner.
diffusion approximation versus limit
Population geneticists usually express the diffusion approximation
in the variables relative gene frequency, p, and generations, t.
Mathematicians express the diffusion limit in rescaled time =
t/n, leading to (for the equation with selection)
2

p 1  p 
p 1  p 

1
1

 ns%

2
2 2  ne,D n 

p
p


where the expressions
 ne,D n 
ns 
should be interpreted as single symbols instead of as formulas:
ne,D n  : lim ne,D n
n
lim ns%
ns% : n
 s  ns n 1
the paths in parameter space
~
p0
Bold arrows: branching process limit followed by s~  0.
Thin arrows: each arrow corresponds to a separate diffusion limit;
~,
their angles correspond to the values of [ns]
their endpoints to the values of x0.
The sequence of arrows corresponds to the limits ns~  , s~0.
proof of the lemma
Ts T
V K V
Tr
T
Ts T
V K V
Tr
proof of the lemma
T
S : sij = expected residence time in state i of an individual born in state j
U : normed stationary distribution of h-states (right eigenvector of A+ B)
V : reproductive values of h-states (left eigenvector of A+ B, V TU% 1 )
The upper part of V is proportional to V
L : lij = expected future # kids in state i from parent who is now in state j
The leftmost square block of L equals L.
Ts  1T SU
U  Ts1SU
V T  cV T L%
since vi is the expected long term birth rate if the mutant population is started
with a single individual in state i and vi is the expected long term population
size when the mutant population is started with a single individual in state i.
V U% c V T L%Ts1 SU  1
T

c
Ts
%
= Tr ?
V T LSU
Ts
c T
%
= Tr ?
V LSU
proof of the lemma
Tr is defined as

Tr 
 
T
T
a
V
(a)
U
da

V

  ()d da U
0
0
a
with (a) containing the average pro capita birth rates at age a.
For a finite state process
(a)  K T Ae Ba K
Tr  V T K T AB2 KU.
so that
Similarly

S   e  Ba da   B1 K
0
Therefore indeed

and
L
T
 Ba
T
1
K
Ae
da


K
AB
.

0
%  Tr .
V T LSU
proof of the lemma, discrete time case
Tr is defined as

Tr   aV (a)U  V
T
a1

T

  ()U
a1  a
with (a) containing the average pro capita births at age a.
For a finite state process
(a)  K T ABa K
so that
Tr  V K ABI  B KU.
T
T
2
Similarly

S   B K  B I  B  K
a
a1
Therefore indeed
1
and

L   K T ABa  K T AI  B .
a 0
%  Tr .
V T LSU
1
serendipitous pay off:
quicker and more insightful ways
for calculating ne,D
example: micro- and macrogametes
Assumption: everybody can be considered to be born
(stochastically) equal.

R0 Y E X
*





 1  f Y E X  m Y E X 
2
with
 
m Y EX 
f Y EX
the average lifetime
number of successful
For the resident:

macrogametes
 
microgametes
f X EX  m X EX
  1.
of the mutant
heterozygote
example: micro- and macrogametes
contribution of Mendelian sampling
 2e  14   2f  2(cf,m  1)   2m 
*
with
 2f

2
m
macrogametes
the variance of the lifetime number of successful
microgametes
of the resident, and
cf,m the covariance of these two lifetime numbers.
example: micro- and macrogametes
Denote the average pro capita rate of parenting at age a as (a).
(a)   f (a)   m (a)


  (a)da  f X E   1

0
0
f
X
m
(a)da  m X E X   1

*
Tr 
1
a
 2 (a) da 
0
1
2
T
r,f
 Tr,m 
example: micro- and macrogametes

*
Ts   a f(a) da
0
with f(a) the probability density of the age at death of an individual.
special case: separate sexes
Call the relative frequencies at which the two sexes are born into the
population qf and qm and their average life-lenghts Ts,f and Ts,m ,
then
Ts  qfTs,f  qmTs,m .
example: micro- and macrogametes
Let a subscript + indicate that the quantity in question refers to the
number of offspring of a female or male, then
f Y E X  qf Y E X  f Y E X , m Y E X  qm Y E X m Y E X .
Hence
R0 Y E X  1  qf Y E X  f Y E X   qm Y E X m Y E X  .
2


For the resident f =1 and m =1. Hence
1
1
f X E X  ,
m Y E X 
.
qf
qm
Moreover, the production of macro- and microgametes are mutually
exclusive. Hence
cfm = -1.
on calculating variances
Consider a discrete nonnegative random variable h such that
with probability p:
with probability q = (1-p):
h = 0
h = k
with k another nonnegative random variable,
then Eh = qEk and
2
2
2
Var(h)  Eh 2  Eh   q  Var(k)  Ek    qEk  .
When moreover Eh = 1 so that Ek = q-1:
 2k
Var(k) Ek   1  q
Var(h)  Eh  1  q  Var(k)  q   1 
q
2
2
2
example: separate sexes
 
2
e
1
4
   2(cf,m
2
f
2
2


1

q

 1  qm
f
 1)     f+
 m+
4qf
4qm
2
m
 the coefficient of variation (standard deviation/mean).
Tr n Tr
ne 

2
Ts  e Ts
With the definitions
ne,f
1
2f+  1  qf
2m+  1  qm

4 Ts Ts,f nf 4 Ts Ts,m nm
Tr,m
nm
Tr,f
nf
,

, ne,m 
2
2
Ts,m m+  1  qm
Ts,f f+  1  qf
this gives
Tr,f
f  ,
Tr
Tr,m
m 
,
Tr
4ne,f ne,m
ne 
.
 f ne,f   m ne,m
Generalises the textbook formula for the case Tr,f =Tr,m =1( f =m =1).
example 2: simplest cyclic environment
bj = j-1 nj-1
nj = pj bj
pi probability that a newborn in phase i-1 survives till phase i
bi number of births at phase i
i pro capita number of births at phase i
 0
 p
 1 1
L(X | EX )   0
 M

 0
V
L
0
O
O
L
 bi
i
k
L
0
L
L
O
O
O
0  k 1 pk 1
1 b1
L
 k pk
0
M
M
0
  0
 b b
  2 1
  0
  M
 
  0
1 bk 
L
0
O
O
L
L
0
b1 bk 

L
L
0

O
M


O
O
M


0 bk bk 1 0
 b1 
U  1  M
 bi  
i
 bk 
example 2: simplest cyclic environment
2


bi 




 2  bj 
 2e   Var j   vi mij  u j    i
j
 i



k
b
bi
j

j 
j1



 i
 bi 1 1 2
 bi 1  2j m j
i
 i

b



j
j
k k j b 2j1
k k j b j1
Next use m j  b j1 b j   j p j and write ne, j :
to rewrite this as
b j1
 mj
2
j

 jnj
 mj
2
j
1
1
1
   bi 
.
k j
k j ne, j
2
e
The end result is the textbook harmonic mean expression:
pb
b
Tr n
1
1 1
ne 

 2  ne, j .
2
2
Ts  e
pb b  e
e

nj
p j 2j
The end