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

2. preamble: some terminology  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 (through multiple mutant substitutions),  macro-evolution : changes where one cannot even speak in terms of a fixed set of traits, like anatomical innovations. Goal: get a mathematical grip on meso-evolution.

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

4. fitness function trajectories form trajectories genome development selection (darwinian) (causal) physics almost faithful reproduction ecology (causal) environment Adaptive Dynamics demography

5. AD’s basis in Commmunity Dynamics  Populations are considered as measures over a space ot 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.  The resulting dynamical systems therefore have attractors, which are assumed to produce ergodic environments.

6. AD’s basis in Commmunity Dynamics  Mutants enter the population singly.  Therefore, 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.)   Residents have fitness zero.

7. AD’s basis in Commmunity Dynamics resident population size population sizes of other species mutant population size fitness as dominant transversal eigenvalue

8. resident population size population sizes of other species mutant population size or, more generally, dominant transversal Lyapunov exponent AD’s basis in Commmunity Dynamics

9.  Fitnesses are not given quantities, but depend on (1) the traits of the individuals, X, Y, (2) the environment in which they live:  (Y | E)  The latter is set by the resident community : E = E attr (C), C={X 1,...,X k ) biological implications Evolutionary progress is almost exclusively determined by the fitnesses of potential mutants.

10. AD: fitness landscapes change with evolution  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 resident trait value(s) x evolutionary time 0 0 0 fitness landscape:  (y,E(t)) mutant trait value y 0 0

11. type morph strategy trait vector point (in trait space) type morph strategy trait vector ( trait value) point (in trait space) effective synonyms

12. The different spaces that play a role in adaptive dynamics: the trait space in which their evolution takes place ( = parameter space of their i- and therefore of their p-dynamics ) = the ‘state space’ of their adaptive dynamics the physical space inhabited by the organisms the state space of their i(ndividual)-dynamics the space of the influences that they undergo ( fluctuations in light, temperature, food, enemies, conspecifics ): their ‘environment’ the parameter spaces of families of adaptive dynamics the state space of their p(opulation)-dynamics scaling up from organisms to trait evolution

13. the simplifications underlying AD essential for most conclusions i.e., separated population dynamical and mutational time scales: the population dynamics relaxes before the next mutant comes 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 essential conceptuallly essential

14. from CD to AD: nature of the limits   ,  rescale time, only consider traits    rescale numbers to densities  = system size,  = mutations / birth t

15. the mechanism of meso-evolution C : = {X 1,..,X k } : trait values of the residents E nvironment: E attr (C) Y : trait value of mutant Fitness (rate of exponential growth in numbers) of mutant s C (Y) : =  ( E attr (C), Y) * Y has a positive probability to invade into a C community iff s C (Y) > 0. * After invasion, X i can be ousted by Y only if s X 1,.., Y,.., X k ( X i ) ≤ 0. * For small mutational steps Y takes over, except near so-called “ess”es.

16. population dynamics: branching process results or "grow exponentially” either go extinct, mutant populations starting from single individuals In an a priori given ergodic environment: (with a probability that to first order in | Y – X | is proportional to their (fitness) +, and with their fitness as rate parameter).

17. Invasion of a "good" c-attractor of X leads to a substitution such that this c-attractor is inherited by Y Y and up to O(  2 ), s Y (X) = – s X (Y). community dynamics: ousting the resident? Proposition: Let  = | Y – X | be sufficiently small, and let X not be close to an “evolutionarily singular strategy”, or to a c(ommunity)-dynamical bifurcation point. “ For small mutational steps invasion implies substitution.”

18. community dynamics: sketch of the proof When an equilibrium point or a limit cycle is invaded, the relative frequency p of Y satisfies = s X (Y) p(1-p) + O(  2 ), while the convergence of the dynamics of the total population densities occurs O(1). dp dt Singular strategies X* are defined by s X* (Y) = O (    ), instead of O(  ).

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

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

21. population dynamics: the bifurcation structure probability that mutant invades evolution will be towards increasing x evolution will be towards decreasing x Near where the mutant equals the resident, the probability that the mutant invades changes as depicted below: The effective speeds of evolutionary change are proportional to the probabilities that invading mutants survive the initial stochastic phase

22. + + - y x - fitness contour plot x: resident y: potential mutant the graphical tools of AD trait value x x0x0 x1x1 x1x1 x2x2 x

23. X X 1 2 Mutual Invasibility Plot MIP y x trait value X x the graphical tools of AD + + - - Pairwise Invasibility Plot PIP

24. when does invasion imply substitution? protection boundary

25. trait value x Trait Evolution Plot TEP x2x2 the graphical tools of AD y x + + - - Pairwise Invasibility Plot PIP

26. evolutionarily singular strategies

27. (monomorphic) linearisation around y = x = x* c 11 +2c 10 +c 00 =0 a=0 b 1 +b 0 =0 neutrality of resident s u (u)= 0

28.  local PIP classification

29.  dimorphisms

30. dimorphic linearisation around y = x 1 = x 2 = x* Only directional derivatives (!)

31. population dynamics: non-genericity strikes

32. dimorphic linearisation around y = x 1 = x 2 = x* Only directional derivatives (!) : u 1 =uw 1, u 2 =uw 2

33. dimorphic linearisation around y = x 1 = x 2 = x*

34. local types of dimorphic evolution

35. local TEP classification

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

37. beyond clonality: thwarting the Mendelian mixer assortativeness

38. 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.

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

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

41. of two lines about to merge one goes extinct

42. matryoshka galore polymorphisms are invariant under permutation of indices X2X2 the six purple volumes should be identified ! adjacent purple volumes are mirror symmetric around a diagonal plane X1X1 X3X3

43. matryoshka galore the sets of trimorphisms connect to the isoclines of the dimorphisms isocline of species 1 isocline of species 3 X1X1 X2X2 X3X3 ( x 2 = x 3 )( x 2 = x 1 )

44. more consistency conditions!  There also exist various global consistency relations! The classification of the singular points was based on just a smoothness assumption and some ecologically reasonable consistency conditions. is extinct. the coexistence set one type. Use that on the boundaries of

45. a more complicated example

46. a potential difficulty: heteroclinic loops 1 2 1 2 33 ?

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

48. much remains to be done! (Many partial results are floating around.)  Classify the geometries of the fitness landscapes, and coexistence sets near singular points in higher dimensions.  Develop a fullfledged bifurcation theory for AD.  Develop (more) global geometrical results.  Delineate to what extent, and in which manner, AD results stay intact for Mendelian populations.  Develop analogous theories for not fully smooth s-functions.  Analyse how to deal with the heteroclinic loop problem. (Some recent results by Odo and Barbara Boldin.)

49. The end