Theoretical Modelling in Biology (G0G41A )

Pt I. Analytical Models

IV. Optimisation and inclusive fitness models

Tom WenseleersDept. 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 thenfitness mutant > fitness resident and mutant can spread

• evolutionary dynamics can be investigated using pairwise invasibility plots

Pairwise invasibility plots= contour plot of invasion fitness

Resident trait Z

Mut

ant t

rait

y

invasion possible fitness rare mutant > fitness resident type

invasion impossible fitness rare mutant > fitness resident type

one trait substitution

evolutionary singular strategy ("equilibrium")

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.

ZyyZywZD

),()(

Reading PIPs: Evolutionary Stabilityis a singular strategy immune to invasions by neighbouring phenotypes? yes → evolutionarily stable strategy (ESS)i.e. equilibrium is stable(local fitness maximum)

Resident trait z

Mut

ant t

rait

y

yes

Resident trait z

Mut

ant t

rait

y

no

invinv

no inv

no inv

0),( when true*

2

2

zZyyZywB

Reading PIPs: Invasion Potentialis the singular strategy capable of invading into all its neighbouring types?

Resident trait Z

Mut

ant t

rait

y

yes

Resident trait Z

Mut

ant t

rait

y

nono inv

no inv

invinv

inv

inv

no invno inv

0),( when true*

2

2

zZyZ

ZywA

Reading PIPs: Convergence Stabilitywhen starting from neighbouring phenotypes, do successful invaders lie closer to the singular strategy?i.e. is the singular strategy attracting or attainable

D(Z)>0 for Z<z* and D(Z)<0 for Z>z*, true when A>B

Resident trait Z

Mut

ant t

rait

y

yes

Resident trait Z

Mut

ant t

rait

y

nono inv

no inv

invinv

inv

inv

no invno inv

Reading PIPs: Mutual Invasibilitycan 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

Resident trait Z

Mut

ant t

rait

y

yes

Resident trait Z

Mut

ant t

rait

y

nono inv

no inv

invinv

inv

inv

no invno inv

Typical PIPs

Resident trait Z

Mut

ant t

rait

y

ATTRACTORno inv

no inv

invinv

Resident trait Z

Mut

ant t

rait

y inv

inv

no inv no inv

REPELLOR

stable equilibrium "CONTINUOUSLY STABLE STRATEGY"

unstable equilibrium

Two interesting PIPsGARDEN OF EDEN BRANCHING POINT

evolutionarily stable,but not convergence stable(i.e. there is a steady statebut not an attracting one)

convergence stable,but not evolutionarily stable

"evolutionary branching"

Resident trait z

Mut

ant t

rait

y inv

inv

no invno inv

Resident trait z

Mut

ant t

rait

y

invinv

(1) evolutionary stable, (2) convergence stable, (3) invasion potential, (4) mutual invasibility

repellorrepellor"branching point"attractorattractorattractor"garden of eden" repellor

Eightfold classification(Geritz et al. 1997)

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 matrixhawk-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

• true when z*=V/C, i.e. individuals playhawk with probability V/CThis is the mixed-strategy ESS. 0),()(

Zyy

ZywZD

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 whenB.r > C = Hamilton's rule

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/2+y1.(1- y2).V+y1. y2.(V-C)/2and similarly fitness of individual 2 is given byw2(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

• ESS occurs when IF effect = 0z*=(V/C)(1-r)/(1+r) 0.),(),(

1

212

1

211

r

yyyw

yyyw

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

r = 1/2 x 1/2 + 1 x 1/2 = 3/4

(c) Full-sister in haplodiploid social insects

Queen Haploid father

1AB C

AC, BCAC

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 Qx

MQ

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=1/3if 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).