Kalle Parvinen. Department of Mathematics FIN University of Turku, Finland

Transcription

1 Adaptive dynamics: on the origin of species by sympatric speciation, and species extinction by evolutionary suicide. With an application to the evolution of public goods cooperation. Department of Mathematics FIN University of Turku, Finland

2 Outline 1 Cooperation, game theory and adaptive dynamics Cooperation Adaptive dynamics Public goods games 2 3 Summary and conclusions References

3 Different types of cooperation Cooperation Adaptive dynamics Public goods games Pairwise interactions A donor decides whether or not to help a recipient Give coins to street musicians, beggars, etc Help a person fallen into fragile ice Public goods Each individual decides whether or not to contribute to a common resource Public transport Environment preservation Donation of blood Cooperation is costly temptation of defection (free-riding)

4 Methods Cooperation Adaptive dynamics Public goods games Game-theoretical approach Small number of possible strategies (to help or not) Individuals can observe the performance of others and imitate the best ones Replicator equation of populations Prisoner s dilemma player 2 Payoff player 1 matrix Cooperate Defect Cooperate R T Defect S P Temptation > Reward > Punishment > Sucker

5 Methods Cooperation Adaptive dynamics Public goods games Adaptive dynamics approach (evolutionary biology) Strategies are real numbers (how much do I help) Ecological dynamics: those performing better than others in the current environment get more offspring than others population sizes change environment changes Evolutionary dynamics: poor strategies are outcompeted, new strategies appear by mutation.

6 Evolutionary analysis Cooperation Adaptive dynamics Public goods games Adaptive dynamics is a mathematical framework for modelling (phenotypical) evolutionary change by natural selection in complex nonlinear ecological systems. References Metz, J. A. J., R. M. Nisbet, and S. A. H. Geritz (1992). How should we define "fitness" for general ecological scenarios? Trends in Ecology & Evolution 7, Geritz, S. A. H., É. Kisdi, G. Meszéna, and J. A. J. Metz (1998). Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evolutionary Ecology 12,

7 Adaptive dynamics Cooperation Adaptive dynamics Public goods games Assumptions Clonal reproduction (can be relaxed) Small mutational steps (not infinitesimally small) Mutations are rare the resident population is on a population-dynamical attractor when a mutation appears (not strict in practice)

8 Invasion fitness Cooperation Adaptive dynamics Public goods games Definition long-term exponential growth rate r of a mutant or expected number of offspring R produced by a mutant during its entire life in an environment E res set by the resident Adaptive dynamics A mutant can invade, and possibly replace the resident if r(s mut, E res ) > 0, (or equivalently R(s mut, E res ) > 1). Mutation-invasion events result in the change of the strategy of the individuals constituting the population. See for example Geritz et al. (1998)

9 Possible outcomes Cooperation Adaptive dynamics Public goods games r(sres, Emut) < 0 r(s mut, E res ) > 0 r(s mut, E res ) < 0 Mutant invades and replaces the resident Coexistence Mutant cannot invade when rare Mutant cannot invade r(sres, Emut) > 0

10 Cooperation Adaptive dynamics Public goods games Selection gradient and singular strategies Selection gradient (fitness gradient) D(s) = s mut r(s mut, E res ) smut=s res=s is the expected direction of evolution with small steps Singular strategies: selection gradient vanishes Possible endpoints of evolution D(s) = 0

11 Cooperation Adaptive dynamics Public goods games Evolutionarily stable strategy (ESS) Unbeatable strategy: r(s, E res ) < 0 for all s s A (local) fitness maximum r(s smut 2 mut, E res ) smut=s res=s < 0 4 Fitness Strategy

12 Cooperation Adaptive dynamics Public goods games Branching point: first step towards speciation Singular strategy which can be invaded r(s, E res ) > 0 for all s s A (local) fitness minimum 2 r(s smut 2 mut, E res ) smut=s res=s > Fitness Strategy

13 Cooperation Adaptive dynamics Public goods games Branching point: first step towards speciation Disruptive selection evolutionary branching Fitness Strategy

14 Cooperation Adaptive dynamics Public goods games Allopatric speciation reproductive isolation by a geographical barrier Sympatric speciation reproductive isolation evolves without geographical isolation.

15 Public goods games Cooperation Adaptive dynamics Public goods games Individuals can choose how much they contribute to a common pool Resources in the common pool are multiplied with factor ρ and shared equally among everybody (below ρ = 2) A A B Common pool + B C + = C D nothing D

16 Public goods games Cooperation Adaptive dynamics Public goods games Individuals can choose how much they contribute to a common pool Resources in the common pool are multiplied with factor ρ and shared equally among everybody s: individual investment N: number of players Payoff from own investment ρ N s s = ( ρ N 1 ) s Cooperation favoured only if ρ > N Question: What else is needed for cooperation to be favoured when 1 < ρ < N?

17 Cooperation Adaptive dynamics Public goods games Mechanisms promoting cooperation Direct reciprocity I scratch your back and you scratch mine. Assortment Cooperators likely to be surrounded by cooperators direct benefit Surrounded by relatives indirect benefit (kin selection) Indirect reciprocity I scratch your back and someone will scratch mine. Mechanisms for assortment Spatial setting Recognizable tags (green beard), vulnerable to mimicry

18 Model by Hauert, Holmes and Doebeli (2006) Cooperators and defectors live in sites In order to get players, N sites are randomly chosen Empty sites the actual amount of players may be < N Payoffs from the public goods game affect the birth rate Reproduction possible to empty sites only Death rate δ is constant d dt x = x(zf C δ) cooperators, s = 1 d dt y = y(zf D δ) defectors, s = 0 z = 1 x y empty space f C and f D are average payoffs of a cooperator and defector in the current environment

19 Hauert et al. (2006): Phase-plane plots Proc. R. Soc. B , Proc. R. Soc. B , Extinction Cooperators only in an equilibrium Coexistence of cooperators and defectors in an equilibrium or in a limit cycle

20 Hauert et al. (2006): Results by Hauert et al. Rich ecological dynamics Ecological viability of cooperators possible even with ρ < N Also coexistence of cooperators and defectors is possible My work: adaptive dynamics of cooperation Do the conclusions above change if mutations are allowed?

21 Details of the ecological model x i : density of individuals with strategy s i z: proportion of empty space, 0 k i=1 x i = 1 z 1. N: maximal group size ρ: payoff multiplication factor Generalization: Functional response in receiving benefits [ ] total investment payoff = h ρ my own investment size of group h is positive and strictly increasing function with h(0) = 0.

22 Resident fitness Chance that an individual finds itself in a group of size S N ( ) N 1 p S (N, z) = (1 z) S 1 z N S S 1 Thus S 1 is binomially distributed, S Bin(N 1, 1 z) Resident equilibria x = 1 z are solved from r(s res, N, z) = (h(ρs res ) s res ) }{{} (1 z N 1) }{{} payoff when game played probability not to be alone z δ = 0

23 Resident equilibria with different functional response Linear: h(y) = y Population size xres ρ = 4. ρ = Resident strategy s res Holling II: h(y) = y/(1 + αy) Population size xres α = 0. α = 0.1. α = Resident strategy s res Viability condition (Parvinen, 2010) s > s bif = δ N N/(N 1) ρ 1 (N 1), (h(ρs res ) s res ) (N 1)N N/(1 N) δ > 0 linear case in general

24 Fitness of a mutant Invasion fitness r(s mut, s res, N, z) = z N S=2 Derivative z = z N S=2 N S=2 [ ( p S (N, z) h ρ s ) ] mut + (S 1)s res s mut δ. S s mut r(s mut, s res, N, z) = [ ( ρ p S (N, z) S h ρ s ) mut + (S 1)s res 1] S s mut=s res ρ ] p S (N, z)[ S h (ρs res ) 1 = D(s res )

25 For linear functional response Derivative s mut r(s mut, s res, N, z) = N [ ( ρ z p S (N, z) S h ρ s ) mut + (S 1)s ] res 1. S S=2 }{{} =1 Mutant strategy smut Invasion fitness is a linear function of s mut Any singular strategy is neutral against mutant invasions Resident strategy sres Strategies Evolutionary time ρ = 4, N = 5, δ = 0.6

26 Functional response determines the type of singularity Second derivative z S=2 2 r(s smut 2 mut, s res, N, z) = N [ ( ρ ) ( 2 p S (N, z) h ρ s )] mut + (S 1)s res S S s mut=s res = zρ 2 h (ρs res ) N p S (N, z) 1 S 2 S=2 Concave h: h (ρs res ) < 0 Evolutionarily stable strategy Convex h: h (ρs res ) > 0 Branching point (if convergence stable)

27 Nonlinear functional responses Concave (Holling II) h(y) = Mutant strategy smut y 1+αy Resident strategy sres ρ = 4, N = 5, δ = 0.6, α = 0.05 Locally convex (Holling III) h(y) = y 2 Mutant strategy smut βy Resident strategy sres ρ = 2.4, N = 5, δ = 0.2, β = 0.25

28 so far Original model of Hauert et al. (linear h) Ecological coexistence of cooperators and defectors will not emerge with small mutational steps (no evolutionary branching) is not structurally stable Nonlinear functional response h Ecological coexistence of cooperators and defectors can emerge with small mutational steps (evolutionary branching) for locally convex h But this is not the whole story!

29 The selection gradient may be negative for all viable strategies Strategies will reach the lower boundary of viability Invasion by a nonviable mutant (kamikaze-mutant) causes the extinction of the whole population Mutant strategy smut Mutant Resident strategy sres Parameters: ρ = 2.3, α = 0, N = 5, δ = 0.6. Resident

30 Mutant strategy smut Strategies Resident strategy sres Evolutionary time Mutant 0.70 Population size Parameters: ρ = 2.3, α = 0, N = 5, δ = 0.6. Resident Evolutionary time

31 Evolution acts on the level of individuals: individual benefit versus common good Frequency-dependent selection: population size is not maximized Extreme example: selection-driven extinction Related to the tragedy of the commons Several example models with realistic ecological settings Empirical testing is difficult

32 Classification of evolutionary scenarios, theory For linear or strictly concave functional responses: Theorem (Parvinen, 2010) 1 The fitness gradient is negative for all viable s res. Therefore evolutionary suicide occurs 2 There exists a unique globally attracting singular strategy. It is uninvadable, if the functional response function h is strictly concave. If it is linear h(y) = y, the singular strategy is neutral against mutant invasions. Case 2 occurs, if at the lower bound of viability s bif Otherwise case 1 occurs. ρh (ρs bif ) > N N/(N 1) N.

33 Parameter dependence Linear: α = 0, any δ > 0 Concave: α = 0.05, δ = 0.6 Factor ρ 6 5 ρ > N ESS, neutral against invasions Extinction, ρ < Maximal group size N Factor ρ 6 5 ρ > N uninvadable ESS Extinction Maximal group size N

34 Evolutionary branching Mutant strategy smut Resident strategy sres Resident strategy s res Resident strategy s res1

35 Evolutionary branching can result in extinction The dimorphic population will reach the boundary of coexistence A Hopf bifurcation near the boundary a limit cycle appears. At the boundary of coexistence the limit cycle disappears 0 extinction Resident strategy sres Resident strategy s res1 A dimorphic population can also experience evolutionary suicide!

36 Evolutionary branching can result in extinction 0.8 Resident strategy sres Resident strategy s res1 Strategies Population size Evolutionary time Evolutionary time

37 Evolution acts on the level of individuals: individual benefit versus common good Frequency-dependent selection: population size is not maximized Extreme example: selection-driven extinction Related to the tragedy of the commons Several example models with realistic ecological settings Empirical testing is difficult Review article: K. Parvinen:. Acta Biotheoretica 53, (2005)

38 Summary and conclusions Summary and conclusions References For small values of ρ, evolutionary suicide occurs For larger values of ρ, convergence stable singular strategy - Neutral, if linear functional response - ESS, if concave functional response - Branching point, if convex functional response Ecological coexistence of cooperators and defectors not structurally stable for linear trade-off may result in the extinction of a viable population, also after evolutionary branching Evolution of continuous strategies may reveal otherwise hidden, interesting phenomena

39 References Summary and conclusions References Cooperation C. Hauert, M. Holmes and M. Doebeli (2006) Evolutionary games and population dynamics: maintenance of cooperation in public goods games Proc. Royal Soc. London B 273, K. Parvinen, (2010) may prevent the coexistence of defectors and cooperators and even cause extinction Proc. Royal Soc. London B, 277, R. Ferrière (2000), Adaptive responses to environmental threats: evolutionary suicide, insurance, and rescue, Options, IIASA K. Parvinen (2005) Acta Biotheoretica 53,