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

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

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

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

8. from individual dynamics through community dynamics to 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. fitness landscape perspective
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 e s i d n t a v l u ( ) x
evolutionary time f c p : y , E m

15. 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
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

16. repeated substitution
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))
• Y has a positive probability to invade into a C community iff sC(Y) > 0.
• After invasion, Xi can be ousted by Y only if sX1,..,Y,.., Xk(Xi) ≤ 0.
• For small mutational steps Y takes over, except near so-called “ess”es.

18. “For small mutational steps invasion implies substitution.”
community dynamics: ousting the resident
Proposition:
Let e = | Y – X | be sufficiently small,
and let X not be close to an “evolutionarily singular strategy”, or to a c(ommunity)-dynamical bifurcation point.
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(e2),
sY(X) = – sX(Y).
“For small mutational steps invasion implies substitution.”

19. community dynamics: ousting the resident
Proof (sketch):
When an equilibrium point or a limit cycle is invaded,
the relative frequency p of Y satisfies
= sX(Y) p(1-p) + O(e2),
while the convergence of the dynamics of the total population densities occurs O(1). dp dt
Singular strategies X* are defined by sX*(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. 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
decreasing x
mutant trait value y
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
increasing x
evolution will be towards
decreasing 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
fitness contour plot
x: resident
y: potential mutant
Pairwise Invasibility Plot
PIP + - y x x1 x x x0 x1 x2 t r a i t v a l u e

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

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

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

28. evolutionarily singular strategies

29. definition

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

31.  local PIP classification

32. the associated local MIPs

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

34. community dynamics: non-genericity strikes

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

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

37. local dimorphic evolution

38. local TEP classification

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

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

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

45. of two lines about to merge one goes extinct

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

47. a more realistic example

48. a potential difficulty: heteroclinic loops1 2 3 ?

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 stay intact for Mendelian populations.
(Many partial results are available, e.g. Dercole & Rinaldi 2008.)
(Some recent results by Odo Diekmann and Barbara Boldin.)
(Some results in next lecture.)

51. The end
Stefan Geritz
Ulf Dieckmann
in next lecture:

53. subsequent levels of abstraction
The different spaces that play a role in adaptive dynamics:
the physical space inhabited by the organisms
the state space of their i(ndividual)-dynamics
the state space of their p(opulation)-dynamics
the space of the influences that they undergo
(fluctuations in light, temperature, food, enemies, conspecifics):
their ‘environment’
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 parameter spaces of families of adaptive dynamics

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

55. adjacent purple volumes are mirror symmetric around a diagonal plane
matryoshka galore
polymorphisms are invariant under permutation of indices X3
the six purple volumes
should be identified ! X2 X1
adjacent purple volumes are mirror symmetric around a diagonal plane

56. matryoshka galore
the sets of trimorphisms connect to the isoclines of the dimorphisms X3 X1 X2
(x2 = x3)
(x2 = x1)