Theoretical Modelling in Biology (G0G41A ) Pt I. Analytical Models IV. Optimisation and inclusive fitness models

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Theoretical Modelling in Biology (G0G41A ) Pt
I. Analytical ModelsIV. Optimisation and
inclusive fitness models

• Tom Wenseleers
• Dept. of Biology, K.U.Leuven

28 October 2008
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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)

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General optimisation method adaptive dynamics
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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 gt 1 thenfitness mutant gt
  fitness resident and mutant can spread
• evolutionary dynamics can be investigated using
  pairwise invasibility plots

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Pairwise invasibility plots contour plot of
invasion fitness
invasion possible fitness rare
mutant gt fitness resident type
invasion impossible fitness rare mutant gt
fitness resident type one trait
substitution evolutionary singular
strategy ("equilibrium")
Mutant trait y
Resident trait Z
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Evolutionary singular strategy

• Selection for a slight increase in phenotype is
  determined by the selection gradient
• A phenotype z for which the selection
  differential is zero we call an evolutionary
  singular strategy. This represents a candidate
  equilibrium.

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Reading PIPs Evolutionary Stability

• is a singular strategy immune to invasions by
  neighbouring phenotypes? yes ? evolutionarily
  stable strategy (ESS)i.e. equilibrium is
  stable(local fitness maximum)

yes
no
no inv
inv
Mutant trait y
Mutant trait y
inv
no inv
Resident trait z
Resident trait z
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Reading PIPs Invasion Potential

• is the singular strategy capable of invading into
  all its neighbouring types?

yes
no
no inv
inv
inv
no inv
Mutant trait y
Mutant trait y
inv
no inv
inv
no inv
Resident trait Z
Resident trait Z
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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 attainableD(Z)gt0 for Zltz and
  D(Z)lt0 for Zgtz, true when AgtB

yes
no
inv
no inv
inv
no inv
Mutant trait y
Mutant trait y
no inv
inv
inv
no inv
Resident trait Z
Resident trait Z
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Reading PIPs Mutual Invasibility

• can a pair of neighbouring phenotypes on either
  side of a singular one invade each other?
• w(y1,y2)gt0 and w(y2,y1)gt0, true when Agt-B

yes
no
no inv
inv
inv
no inv
Mutant trait y
Mutant trait y
inv
no inv
inv
no inv
Resident trait Z
Resident trait Z
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Typical PIPs
ATTRACTOR
REPELLOR
no inv
inv
inv
no inv
no inv
Mutant trait y
Mutant trait y
inv
no inv
inv
Resident trait Z
Resident trait Z
unstable equilibrium
stable equilibrium "CONTINUOUSLY STABLE STRATEGY"
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Two interesting PIPs
BRANCHING POINT
GARDEN OF EDEN
inv
inv
no inv
Mutant trait y
Mutant trait y
inv
no inv
inv
Resident trait z
Resident trait z
convergence stable, but not evolutionarily
stable"evolutionary branching"
evolutionarily stable,but not convergence
stable(i.e. there is a steady statebut not an
attracting one)
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Eightfold classification(Geritz et al. 1997)
repellorrepellor "branching point"attractorattr
actorattractor"garden of eden" repellor
(1) evolutionary stable, (2) convergence stable,
(3) invasion potential, (4) mutual invasibility
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convergence stableA gt B
evol. repellors
evol. branching
evolutionary stable, B lt 0
G. Eden
evol. attractors
mutually invasibleA gt -B
invasion potential, A gt 0
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Application game theory
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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
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Two types of equilibria

• evolutionarily stable state equilibrium mix
  between different strategies attained when
  fitness strategy Afitness 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)

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Calculating ESSs

• e.g. hawk-dove gameearlier 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/2y1.(1- y2).Vy1. 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
• true when zV/C, i.e. individuals playhawk with
  probability V/CThis is the mixed-strategy ESS.

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Extension for interactions between relatives
inclusive fitness theory
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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"

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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 gt C Hamilton's rule

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Calculating costs benefits in Hamilton's rule

• e.g. hawk-dove gameassume 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/2y1.(1-
  y2).Vy1. y2.(V-C)/2and similarly fitness of
  individual 2 is given byw2(y1, y2)w0(1-y1).(1-
  y2).V/2y2.(1- y1).Vy1. y2.(V-C)/2
• inclusive fitness effect of increasing one's
  probability of playing hawk
• ESS occurs when IF effect 0z(V/C)(1-r)/(1r)

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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

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r 1/2 x 1/2 1/4
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r 1/2 x 1/2 1/2 x 1/2 1/2
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(c) Full-sister in haplodiploid social insects
Queen
Haploid father
AB
C
1
AC
AC, BC
r 1/2 x 1/2 1 x 1/2 3/4
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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

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E.g. reproductive value of males and females in
haplodiploids
M
Q
x
M
Q
frequency of allele in queens in next generation
pf(1/2).pf(1/2).pm frequency of allele in
males in next generation pmpf if we introduce
a gene in all males in the first generation then
we initially have pm1, pf0 after 100
generations we get pmpf1/3if we introduce a
gene in all queens in the first generation then
we initially have pm0, pf1 after 100
generations we get pmpf2/3From 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).