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

2. 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)

3. General optimisation method: adaptive dynamics

4. 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

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

6. 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.

7. 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

8. 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

9. 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 D(Z)>0 for Z<z* and D(Z)<0 for Z>z*, true when A>B
yes no
no inv
inv
inv
no inv
Mutant trait y
Mutant trait y
inv
no inv
no inv
inv
Resident trait Z
Resident trait Z

10. Reading PIPs: Mutual Invasibility
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
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

11. stable equilibrium "CONTINUOUSLY STABLE STRATEGY"
Typical PIPs
ATTRACTOR
REPELLOR
no inv
inv
inv
Mutant trait y
Mutant trait y
no inv
no inv
inv
no inv
inv
Resident trait Z
Resident trait Z
stable equilibrium "CONTINUOUSLY STABLE STRATEGY"
unstable equilibrium

12. but not evolutionarily stable "evolutionary branching"
Two interesting PIPs
GARDEN OF EDEN
BRANCHING POINT
inv
no inv
inv
Mutant trait y
Mutant trait y
inv
no 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"

13. Eightfold classification (Geritz et al. 1997)
repellor repellor
"branching point" attractor attractor attractor "garden of eden" repellor
(1) Is a singular phenotype immune to invasions by neighboring phenotypes? (2) When starting from neighboring phenotypes, do successful invaders lie closer to the singular one? (3) Is the singular phenotype capable of invading into all its neighboring types? (4) When considering a pair of neighboring phenotypes to both sides of a singular one, can they invade into each other?
(1) evolutionary stable, (2) convergence stable, (3) invasion potential, (4) mutual invasibility

14. convergence stable A > B
evol. repellors
evol. branching
evolutionary stable, B < 0
The eight possible generic local configurations of the pairwise invasibility plot
and their relation to the second-order derivatives of sy(z). Inside the shaded regions within each separate plot sy(z) is >1, i.e the mutant can invade.
G. Eden
evol. attractors
mutually invasible A > -B
invasion potential, A > 0

15. Application: game theory

16. 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

17. 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)

18. 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 play hawk with probability V/C This is the mixed-strategy ESS.

19. Extension for interactions between relatives: inclusive fitness theory

20. 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"

21. 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

22. 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
ESS occurs when IF effect = 0 z*=(V/C)(1-r)/(1+r)

23. 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

24. r = 1/2 x 1/2 = 1/4

25. r = 1/2 x 1/2 + 1/2 x 1/2 = 1/2

26. Queen Haploid father 1 r = 1/2 x 1/2 + 1 x 1/2 = 3/4
(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

27. 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

28. E.g. reproductive value of males and females in haplodiploidsQ x 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).