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Theoretical Modelling in Biology (G0G41A )
Pt I. Analytical Models
IV. Optimisation and inclusive fitness models
Tom Wenseleers
Dept. of Biology, K.U.Leuven
28 October 2008
Aims
• last week we showed how to do exact genetic
models
• aim of this lesson: show how under some limiting
cases the results of such models can also be
obtained using simpler optimisation methods
(adaptive dynamics)
• discuss the relationship with evolutionary game
theory (ESS)
• plus extend these optimisation methods to deal
with interactions between relatives
(inclusive fitness theory / kin selection)
General optimisation
method: adaptive
dynamics
Optimisation methods
• in limiting case where selection is weak
(mutations have small effect) the equilibria in
genetic models can also be calculated using
optimisation methods (adaptive dynamics)
• first step: write down invasion fitness w(y,Z) =
fitness rare mutant
(phenotype y)
fitness of resident type
(phenotype Z)
• if invasion fitness > 1 then
fitness mutant > fitness resident
and mutant can spread
• evolutionary dynamics can be investigated using
pairwise invasibility plots
Pairwise invasibility plots
= contour plot of invasion fitness
Mutant trait y
invasion possible
fitness rare mutant >
fitness resident type
invasion impossible
fitness rare mutant >
fitness resident type
one trait substitution
evolutionary singular strategy
("equilibrium")
Resident trait Z
Evolutionary singular strategy
• Selection for a slight increase in phenotype
is determined by the selection gradient
w( y, Z )
D( Z ) 
y
y Z
• A phenotype z* for which the selection
differential is zero we call an evolutionary
singular strategy. This represents a
candidate equilibrium.
Reading PIPs: Evolutionary Stability
is a singular strategy immune to invasions by neighbouring
phenotypes? yes → evolutionarily stable strategy (ESS)
i.e. equilibrium is stable
 2 w( y, Z )
true when B 
(local fitness maximum)
2
yes
no inv
no inv
Resident trait z
no
Mutant trait y
Mutant trait y
y
inv
inv
Resident trait z
y  Z  z*
0
Reading PIPs: Invasion Potential
yes
no inv
inv
inv
no inv
Resident trait Z
 2 w( y, Z )
true when A 
Z 2
no
Mutant trait y
Mutant trait y
is the singular strategy capable of invading into all its
neighbouring types?
inv
no inv
no inv
inv
Resident trait Z
0
y  Z  z*
Reading PIPs: Convergence Stability
when starting from neighbouring phenotypes, do successful
invaders lie closer to the singular strategy?
i.e. is the singular strategy attracting or attainable
inv
inv
no inv
Resident trait Z
Mutant trait y
Mutant trait y
D(Z)>0 for Z<z* and D(Z)<0 for Z>z*, true when A>B
yes
no
inv
no inv
no inv
no inv
inv
Resident trait Z
Reading PIPs: Mutual Invasibility
yes
no inv
inv
inv
no inv
Resident trait Z
no
Mutant trait y
Mutant trait y
can a pair of neighbouring phenotypes on either side
of a singular one invade each other?
w(y1,y2)>0 and w(y2,y1)>0, true when A>-B
inv
no inv
no inv
inv
Resident trait Z
Typical PIPs
no inv
inv
inv
no inv
REPELLOR
Mutant trait y
Mutant trait y
ATTRACTOR
inv
no inv no inv
inv
Resident trait Z
Resident trait Z
stable equilibrium
"CONTINUOUSLY STABLE STRATEGY"
unstable equilibrium
Two interesting PIPs
inv
no inv
no inv
inv
BRANCHING POINT
Mutant trait y
Mutant trait y
GARDEN OF EDEN
inv
inv
Resident trait z
Resident trait z
evolutionarily stable,
but not convergence stable
(i.e. there is a steady state
but not an attracting one)
convergence stable,
but not evolutionarily stable
"evolutionary branching"
Eightfold classification
(Geritz et al. 1997)
repellor
repellor
"branching point"
attractor
attractor
attractor
"garden of eden"
repellor
(1) evolutionary stable, (2) convergence stable, (3) invasion potential, (4) mutual invasibility
Application:
game theory
Game theory
• "game theory": study of optimal strategic
behaviour, developed by Maynard Smith
• extension of economic game theory, but with
evolutionary logic and without assuming that
individuals act rationally
• fitness consequences summarized in payoff matrix
hawk-dove game
Two types of equilibria
• evolutionarily stable state:
equilibrium mix between different strategies
attained when fitness strategy A=fitness strategy B
• evolutionarily stable strategy (ESS):
strategy that is immune to invasion by any other
phenotype
- continuously-stable ESS: individuals express a
continuous phenotype
- mixed-strategy ESS: individuals express strategies
with a certain probability (special case of a continuous
phenotype)
Calculating ESSs
• e.g. hawk-dove game
earlier we calculated that evolutionarily stable state consist of an
equilibrium prop. of V/C hawks
• what if individuals play mixed strategies?
assume individual 1 plays hawk with prob. y1 and social interactant plays
hawk with prob. y2, fitness of individual 1 is then
w1(y1, y2)=w0+(1-y1).(1- y2).V/2+y1.(1- y2).V+y1. y2.(V-C)/2
• invasion fitness, i.e. fitness of individual playing hawk with prob. y in
pop. where individuals play hawk with prob. Z is w(y,Z)=w1(y,Z)/w1(Z,Z)
• ESS occurs when
D( Z ) 
w( y, Z )
y
0
y Z
• true when z*=V/C, i.e. individuals play
hawk with probability V/C
This is the mixed-strategy ESS.
Extension for interactions
between relatives:
inclusive fitness theory
Problem
• in the previous slide the evolutionarily stable
strategy that we found is the one that maximised
personal reproduction
• but is it ever possible that animals do not strictly
maximise their personal reproduction?
• William Hamilton: yes, if interactions occur between
relatives. In that case we need to take into account
that relatives contain copies of one's own genes.
Can select for altruism (helping another at a cost to
oneself) = inclusive fitness theory or "kin selection"
Inclusive fitness theory
• condition for gene spread is given by
inclusive fitness effect = effect on own fitness
+ effect on someone else's fitness.relatedness
• relatedness = probability that a copy of a rare
gene is also present in the recipient
• e.g. gene for altruism selected for when
B.r > C = Hamilton's rule
Calculating costs & benefits
in Hamilton's rule
• e.g. hawk-dove game
assume individual 1 plays hawk with prob. y1 and social
interactant plays hawk with prob. y2,
fitness of individual 1 is then
w1(y1, y2)=w0+(1-y1).(1- y2).V/2+y1.(1- y2).V+y1. y2.(V-C)/2
and similarly fitness of individual 2 is given by
w2(y1, y2)=w0+(1-y1).(1- y2).V/2+y2.(1- y1).V+y1. y2.(V-C)/2
• inclusive fitness effect of increasing one's probability of
playing hawk
w1 ( y1 , y2 ) w2 ( y1 , y2 )

.r  0
y1
y1
• ESS occurs when IF effect = 0
z*=(V/C)(1-r)/(1+r)
Calculating relatedness
• Need a pedigree to calculate r that includes
both the actor and recipient and that shows
all possible direct routes of connection
between the two
• Then follow the paths and multiply the
relatedness coefficients within one path,
sum across paths
r = 1/2 x 1/2 = 1/4
r = 1/2 x 1/2 + 1/2 x 1/2 = 1/2
(c) Full-sister in haplodiploid social insects
Queen
AB
AC
Haploid father
1
C
AC, BC
r = 1/2 x 1/2 + 1 x 1/2 = 3/4
Class-structured populations
• sometimes a trait affects different classes of
individuals (e.g. age classes, sexes)
• not all classes of individuals make the same
genetic contribution to future generations
• e.g. a young individual in the prime of its life will
make a larger contribution than an individual that is
about to die
• taken into account in concept of reproductive
value. In Hamilton's rule we will use life-for-life
relatedness = reproduce value x regression
relatednesss
E.g. reproductive value of
males and females in haplodiploids
M
x
Q
Q
M
frequency of allele in queens in next generation pf’=(1/2).pf+(1/2).pm
frequency of allele in males in next generation pm’=pf
if we introduce a gene in all males in the first generation then we initially have
pm=1, pf=0; after 100 generations we get pm=pf=1/3
if we introduce a gene in all queens in the first generation then we initially have
pm=0, pf=1; after 100 generations we get pm=pf=2/3
From this one can see that males contribute half as many genes to the future gene pool as
queens. Hence their relative reproductive value is 1/2. Regression relatedness between a
queen and a son e.g. is 1, but life-fore-life relatedness = 1 x 1/2 = 1/2
Formally reproductive value is given by the dominant left eigenvector of the gene
transmission matrix A (=dominant right eigenvector of transpose of A).