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

1. 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

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 fitnessw(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

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

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

12. 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)

13. Eightfold classification(Geritz et al. 1997) repellorrepellor "branching point"attractorattractorattractor"garden of eden" repellor (1) evolutionary stable, (2) convergence stable, (3) invasion potential, (4) mutual invasibility

14. Application: game theory

15. 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

16. 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)

17. 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/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 isw(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.

18. Extension for interactions between relatives: inclusive fitness theory

19. 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"

20. 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

21. 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)

22. 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

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

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

25. (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

26. 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

27. E.g. reproductive value of males and females in haplodiploids M Q x Q M frequency of allele in queens in next generation pf’=(1/2).pf+(1/2).pmfrequency 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=2/3From 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).