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.

Zyy

ZywZD

),(

)(

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

trai

t y

yes

Resident trait z

Mut

ant

trai

t y

no

inv

inv

no inv

no inv

0),(

when true*

2

2

zZyy

ZywB

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

Resident trait Z

Mut

ant

trai

t y

yes

Resident trait Z

Mut

ant

trai

t y

no

no inv

no inv

invinv

inv

inv

no inv

no 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

trai

t y

yes

Resident trait Z

Mut

ant

trai

t y

no

no 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

trai

t y

yes

Resident trait Z

Mut

ant

trai

t y

no

no inv

no inv

invinv

inv

inv

no inv

no inv

Typical PIPs

Resident trait Z

Mut

ant

trai

t y

ATTRACTOR

no inv

no inv

invinv

Resident trait Z

Mut

ant

trai

t 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

trai

t y inv

inv

no inv

no inv

Resident trait z

Mut

ant

trai

t y

inv

inv

(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 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

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

),()(

Zy

y

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 when

B.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

ry

yyw

y

yyw

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).