Adaptive Dynamics

1. & Mathematical Institute, Leiden University (formerly ADN) IIASA VEOLIA- Ecole Poly- technique Adaptive Dynamics studying the change of community dynamical parameters through mutation and selection Hans (=JAJ*) Metz

2. context

3. evolutionary scales • micro-evolution: changes in gene frequencies on a population dynamical time scale, • meso-evolution: evolutionary changes in the values of traits of representative individuals and concomitant patterns of taxonomic diversification (as result of multiple mutant substitutions), • macro-evolution: changes, like anatomical innovations, that cannot be described in terms of a fixed set of traits. Goal: get a mathematical grip on meso-evolution.

4. components of the evolutionary mechanism ecology environment (causal) ( causal ) demography function trajectories physics form selection fitness trajectories development ( darwinian ) genome almost faithful reproduction

5. Stefan Geritz, me & various collaborators (1992, 1996, 1998, ...) adaptive dynamics ecology environment (causal) ( causal ) demography function trajectories physics form selection fitness trajectories development ( darwinian ) genome almost faithful reproduction

6. terminology corresponding terms: (pheno)type morph strategy trait vector (traitvalue) point (in trait space) population genetics: evolutionary ecology: (meso-evolutionary statics) adaptive dynamics: (meso-evolutionary dynamics)

7. t adaptive dynamics limit    rescale numbers to densities  , ln() rescale time, only consider traits (usual perspective)  = system size,  = mutations / birth

8. from individual dynamicsthroughcommunity dynamicsto adaptive dynamics(AD)

9. community dynamics: residents • Populations are represented as frequency distributions (measures) over a space of i(ndividual)-states (e.g. spanned by age and size). • Environments (E) are delimited such that given their environment individuals are independent, • and hence their mean numbers have linear dynamics. • Resident populations are assumed to be so large that we can approximate their dynamics deterministically. • These resident populations influence the environment so that they do not grow out of bounds. • Therefore the community dynamics have attractors, which are assumed to produce ergodic environments.

10. community dynamics: mutants • Mutations are rare. •  They enter the population singly. Hence, initially their impact on the environment can be neglected. •  The initial growth of a mutant population can be approximated with a branching process. • Invasion fitness is the (generalised) Malthusian parameter (= averaged long term exponential growth rate of the mean) of this proces:  (Existence guaranteed by the multiplicative ergodic theorem.)

11. fitness as dominant transversal eigenvalue mutant population size i.a. population sizes of other species resident population size

12. fitness as dominant transversal eigenvalue or, more generally, dominant transversal Lyapunov exponent mutant population size i.a. population sizes of other species resident population size

13. implications • Fitnesses are not given quantities, but depend on (1) the traits of the individuals, X, Y, (2) the environment in which they live, E:(Y,E) | (Y | E) with E set by the resident community:E = Eattr(C), C={X1,...,Xk) • Residents have fitness zero.

14. Evolution proceeds through uphill movements in a fitness landscape that keeps changing so as to keep the fitness of the resident types at exactly zero. • Evolution proceeds through uphill movements in a fitness landscape r ( y , E ( t ) ) f i t n e s s l a n d s c a p e : 0 0 evolutionary time 0 0 0 m u t a n t t r a i t v a l u e y r e s i d e n t t r a i t v a l u e ( s ) x fitness landscape perspective

15. essential conceptuallly essential essential for most conclusions i.e., separated population dynamical and mutational time scales: the population dynamics relaxes before the next mutant comes underlying simplifications 1. mutation limited evolution 2. clonal reproduction 3. good local mixing 4. largish system sizes 5. “good” c(ommunity)-attractors 6. interior c-attractors unique 7. fitness smooth in traits 8. small mutational steps

16. meso-evolution proceeds by the repeated substitution of novel mutations

17. fate of novel mutations C := {X1,..,Xk}: trait values of the residents Environment: Eattr(C) Y: trait value of mutant Fitness (rate of exponential growth in numbers) of mutant sC(Y):=(Y|Eattr(C)) • Yhas a positive probability to invade into a C community iff sC(Y)>0. • Afterinvasion,Xicanbeoustedby Yonly if sX1,..,Y,..,Xk(Xi) ≤ 0. • For small mutational steps Y takes over, except near so-called “ess”es.

18. community dynamics:ousting the resident Proposition: Let e = |Y– X|be sufficiently small, and letXnot be closeto an “evolutionarilysingularstrategy”, or to ac(ommunity)-dynamical bifurcationpoint. Invasion of a "good" c-attractor ofXleads to a substitution such that this c-attractor is inherited byY Y and up to O(e2), sY(X)= – sX(Y). “For small mutational steps invasion implies substitution.”

19. dp dt community dynamics:ousting the resident Proof (sketch): When an equilibrium point or a limit cycle is invaded, the relative frequency p ofYsatisfies = sX(Y)p(1-p) + O(e2), while the convergence of the dynamics of the total population densities occurs O(1). Singular strategiesX* are defined bysX*(Y)= O(e2),instead of O(e).

20. community dynamics:the bifurcation structure Near where the mutant trait value y equals the resident trait value x there is a degenerate transcritical bifurcation:

21. mutanttraitvalue y evolution will be towards decreasing x community dynamics:the bifurcation structure Near where the mutant trait value y equals the resident trait value x there is a degenerate transcritical bifurcation:  relative frequency of mutant evolution will be towards increasing x The effective speeds of evolutionary change are proportional to the probabilities that invading mutants survive the initial stochastic phase

22. community dynamics: invasion probabilities The probability that the mutant invades changes as depicted below: probability that mutant invades evolution will be towards decreasing x evolution will be towards increasing x The effective speeds of evolutionary change are proportional to the probabilities that invading mutants survive the initial stochastic phase

23. graphical tools

24. Pairwise Invasibility Plot - + y + - x fitness contour plot x: resident y: potential mutant Pairwise Invasibility Plot PIP x1 x x x0 x1 x2 t r a i t v a l u e

25. ? protection boundary X 2 substitution boundary ? X 1 Mutual Invasibility Plot Pairwise Invasibility Plot PIP Mutual Invasibility Plot MIP - + y X + - x1 x x t r a i t v a l u e

26. X 2 X 1 Mutual Invasibility Plot Pairwise Invasibility Plot PIP Mutual Invasibility Plot MIP - + x2 y X + - x x t r a i t v a l u e

27. - + y X X 2 2 + - X X x 1 1 Trait Evolution Plot Pairwise Invasibility Plot PIP Trait Evolution Plot TEP x2 x t r a i t v a l u e

28. evolutionarily singular strategies

29. definition

30. neutrality of resident a=0 s (u)=0 b +b =0 u 1 0 c + 2 c + c = 0 1 1 1 0 0 0 (monomorphic) linearisation around y = x = x*

31.  local PIP classification

32. the associated local MIPs

33. dimorphic linearisation around y = x1 = x2 = x* Only directional derivatives (!)

34. community dynamics: non-genericity strikes

35. a s (v) = + u ,u ( ) 1 2 * ) b + b + (w w u v , 1 1 2 0 + g g ) + g ) + 2 2 (w w u (w w uv , , V 11 1 2 10 1 2 00 h.o.t. dimorphic linearisation around y = x1 = x2 = x* : u1=uw1, u2=uw2 Only directional derivatives (!)

36. dimorphic linearisation around y = x1 = x2 = x*

37. local dimorphic evolution

38. local TEP classification

39. population fitness fitness minimum t r a i t t i m e more about adaptive branching evolutionary time

40. beyond clonality: thwarting the Mendelian mixer assortativeness

41. extensions

42. a toy example Lotka-Volterra  all per capita growth rates are linear functions of the population densities Lotka-Volterra  all per capita growth rates are linear functions of the population densities LV models are unrealistic, but useful since they have explicit expressions for the invasion fitnesses.

43. width 1 –––––––––––– √2 width competition kernel competition kernel carrying capacity carrying capacity a toy example viable range

44. isoclines correspond to loci of monomorphic singular points. matryoshka galore interrupted:branching prone ( trimorphically repelling)

45. of two lines about to merge one goes extinct

46. more consistency conditions Use that on the boundaries of the coexistence set one type is extinct. • There also exist various global consistency relations.

47. a more realistic example

48. 3 3 2 2 1 1 a potential difficulty: heteroclinic loops ?

49. a potential difficulty: heteroclinic loops ? The larger the number of types, the larger the fraction of heteroclinic loops among the possible attractor structures!

50. things that remain to be done • Analyse how to deal with the heteroclinic loop problem. • Classify the geometries of the fitness landscapes, and coexistence sets near singular points in higher dimensions. • Extend the collection of known global geometrical results. • Develop a fullfledged bifurcation theory for AD. • Develop analogous theories for less than fully smooth s-functions. • Delineate to what extent, and in which manner, AD results stayintact for Mendelianpopulations. (Many partial results are available, e.g. Dercole & Rinaldi 2008.) (Some recent results by Odo Diekmann and Barbara Boldin.) (Some results in next lecture.)