Adaptive Dynamics
studying the changeof community dynamical parameters
through mutation and selection
Hans (= J A J *) Metz
(formerly ADN) IIASA
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VEOLIA-Ecole Poly-technique
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&Mathematical Institute, Leiden University
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context
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.
functiontrajectories
formtrajectories
genome
development
selection
(darwinian)
(causal)
demography
physics
almost faithful reproduction
ecology
(causal)
fitness
environment
components of the evolutionary mechanism
fitness
functiontrajectories
formtrajectories
genome
development
selection
(darwinian)
(causal)
physics
almost faithful reproduction
ecology
(causal)environment
adaptive dynamics
demography
Stefan Geritz, me & various collaborators
(1992, 1996, 1998, ...)
(pheno)typemorph
strategy
trait vector
(trait value)
point (in trait space)
terminology
corresponding terms:
population genetics:
evolutionary ecology: (meso-evolutionary statics)
adaptive dynamics:(meso-evolutionary dynamics)
(usual perspective)
adaptive dynamics limit
x
adaptive dynamicslimit
individual-basedsimulation
classical largenumber limit
t
, ln() rescale time, only consider traits
rescale numbers to densities
= system size, = mutations / birth
t
from individual dynamics
through
community dynamicsto
adaptive dynamics(AD)
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.
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.)
fitness as dominant transversal eigenvalue
resident population size
i.a. population sizes
mutantpopulationsize
of other species
resident population size
i.a. population sizes
mutantpopulationsize
or, more generally, dominant transversal Lyapunov exponent
fitness as dominant transversal eigenvalue
of other species
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.
implications
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
resident trait value(s) x
evol
utio
nary
tim
e
0
0
0
fitness landscape: (y,E(t))
mutant trait value y
0
0
underlying simplifications
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 mixing4. largish system sizes
5. “good” c(ommunity)-attractors6. interior c-attractors unique
7. fitness smooth in traits8. small mutational steps
essential formost conclusions
essential
essential conceptuallly
meso-evolution proceeds by the
repeated substitution of novel mutations
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.
Invasion of a "good" c-attractor of X leads to a substitution such that this c-attractor is inherited by 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.”
Y and up to O(2),
sY(X) = – sX(Y).
community dynamics: ousting the resident
When an equilibrium point or a limit cycle is invaded, the relative frequency p of Y satisfies
= sX(Y) p(1-p) + O(2),
while the convergence of the dynamics of the total population densities occurs O(1).
dpdt
1
03 4 5 6 7
p
sX(Y) t
Singular strategies X* are defined by sX*(Y) = O(), instead of O().
Proof (sketch):
Near where the mutant trait value y equals the resident trait value x there is a degenerate transcritical bifurcation:
community dynamics: the bifurcation structure
nx →
↑ny
n x→
↑ny
n x →
↑ny
y<xy=xy>x
community dynamics: the bifurcation structure
resident trait value x0
1
mutant trait value y
evolution will be towards increasing x
resident trait value x0
1
mutant trait value y
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
relative frequency of mutant
mutant trait value y
community dynamics: invasion probabilities
probability thatmutant
invades
evolution will be towards increasing x
evolution will be towards decreasing x
The probability that the mutant invades changes as depicted below:
resident trait value x0
1
mutant trait value y
resident trait value x0
1
mutant trait value y
The effective speeds of evolutionary change are proportional to the probabilities that invading mutants survive the initial stochastic phase
graphical tools
+
+
-
y
x
-
fitness contour plotx: residenty: potential mutant
Pairwise Invasibility Plot
trait valuex
x0x1
x1
x2
x
Pairwise Invasibility Plot
PIP
X
X1
2
+
-
+
-
.
Mutual Invasibility Plot
MIP
y
xtrait value
X
x
Mutual Invasibility Plot
+
+
-
-
Pairwise Invasibility Plot
PIP
x1
protection boundary
?
?
substitution boundary
X
X1
2
Mutual Invasibility Plot
MIP
y
xtrait value
X
x
Mutual Invasibility Plot
+
+
-
-
Pairwise Invasibility Plot
PIP
x2
X
X1
2X2
X1
trait valuex
Trait Evolution Plot
TEP
x2
Trait Evolution Plot
y
x
+
+
-
-
Pairwise Invasibility Plot
PIP
evolutionarily singular strategies
definition
x*
x0
+
+
x*
x* is a singular point iff
dy
dsx(y) = 0y=x=x*
(x* is an extremum in the y-direction)
x*
x*
+
+
y
x* x
v=y-x*
u=x-x*
su(v) = a + b1u+b0v + c11u2+2c10uv+c00v2
+ h.o.t
b1=b0=0
a=0b1+b0=0c11+2c10+c00=0
neutrality of resident
x* is an extremum in y
s0(0)=0
su(u)=0su(v) = c11u2−(c11+c) +uv cv
+ . .h o t
(monomorphic) linearisation around y = x = x*
c11+2c10+c00=0
a=0b1+b0=0
neutrality of resident
su(u)= 0
b1=b0=0
x* is an extremum in y
s0(0)=0
su(v) = c11u2−(c11+c) +uv cv
+ . .h o t
monomorphicconvergenceto x0
yes
no
c00
noyes
noyes
c11c11
c00
x0 uninvadable
local PIP classification
the associated local MIPs
c00
c11c11
c00
dimorphismsnono
yesyes
dimorphisms
dimorphic linearisation around y = x1 = x2 = x*
Local coordinates: v = y-x* mutant u1 = x1-x*, u2 = x2-x*
residents
Only directional derivatives (!):
u1 = uw1, u2 = uw2
Only directional derivatives (!)
n1 →
n1→^
↑
n2^
↑n2
A
B
B
A
community state space parameter space
parameter paths attractor paths
A
B
A
B
community dynamics: non-genericity strikes
dimorphic linearisation around y = x1 = x2 = x*
su ,u (v) = +
1(w1,w2u0 v
11(w1,w2u210(w1,w2uv00 V2
h.o.t.
1 2 (*)
Local coordinates: v = y-x* mutant u1 = x1-x*, u2 = x2-x*
residents
Only directional derivatives (!):
u1 = uw1, u2 = uw2
Only directional derivatives (!) :
u1=uw1, u2=uw2
dimorphic linearisation around y = x1 = x2 = x*
s00 (v) = s0(v)
su ,u (u1) = 0 =1 2
su ,u (u2)1 2
neutrality of residents
su ,u (v) =1 2
su ,u (v)2 1
symmetry
if u1=u2=0 we are back in themonomorphic resident case
su ,u (v) =1 2
expansion formula (*)
(v-u1) (v-u2) [c00+ h.o.t]
local dimorphic evolution
0
c00>0
V0
u1 u2
Su ,u (v)1 2
Su ,u (v) = (v-u1) (v-u2) [c00+ h.o.t]1 2
c00<0
Su ,u (v)1 2
V
u1 u2
local TEP classification
monomorphicconvergenceto x0
dimorphicconvergenceto x0
yes
no
evolutionary"branching"evolutionary"branching"
Evolutionary AttractorsEvolutionary Attractors
Evolutionary RepellersEvolutionary Repellers c00
noyes
noyes
c11c11
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more about adaptive branching
t r a i t v a lu e
x
evol
utio
nary
tim
e
t i m e t r a
i t
fitne
ssfitness
minimum
population
. Summary
Ecological Character Simulation
beyond clonality: thwarting the Mendelian mixer
asso
rtativ
enes
s
extensions
a toy example
____ = 1 - Σa(xi,xj)nj/k(xi)dninidt j
k(x)=
a(xi,xj)=e-(xi-xj) xi-xj→xi-xj→
↑a↑a
1/√1/√
Lotka-Volterra competition among individualsdifferentiated according to a one-dimensional trait x.
with
and
population equations:
1-x2 if -1<x<10 elsewhere{
↑k
-1 x → 1
↑k
-1 x → 1
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.
a toy example
____ = 1 - Σa(xi,xj)nj/k(xi)dninidt j
k(x)=
a(xi,xj)=e-(xi-xj) xi-xj→xi-xj→
↑a↑a
1/√1/√
Lotka-Volterra competition among individualsdifferentiated according to a one-dimensional trait x.
with
and
population equations:
1-x2 if -1<x<10 elsewhere{
↑k
-1 x → 1
↑k
-1 x → 1
viable range
competition kernelcompetition kernel
carrying capacity carrying capacity
widthwidth 1
––––––––––––
√2
matryoshka galore
x1
x
x2
Exploring parameter space
=1/3: =: =3:
isoclines correspond to loci of monomorphic singular points.
interrupted: branching prone ( trimorphically repelling)
x1
x
x2
Exploring parameter space
=1/3: =: =3:
of two lines about to merge one goes extinct
more consistency conditions
There also exist various global consistency relations.
x2
x1
y
x
+
-
+
-
Use that on the boundaries of the coexistence set one type is extinct.
a more realistic example
.
Seed size 2
Seed size 1
seed size evolution: TEPs
a potential difficulty: heteroclinic loops
1 2
3
1 2
3
1 2
3
1 2
3
1 2 1 2
3 3
?
a potential difficulty: heteroclinic loops
4
3
1
2
4
23
1
?
The larger the number of types, the larger the fraction ofheteroclinic loops among the possible attractor structures !
things that remain to be done
(Many partial results are available, e.g. Dercole & Rinaldi 2008.)
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.
(Some recent results by Odo Diekmann and Barbara Boldin.)
(Some results in next lecture.)
The end
Stefan Geritz Ulf Dieckmann
in next lecture:
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
subsequent levels of abstraction
time1
10
100
1000
10 200
# individuals
population dynamics: branching process results
or "grow exponentially” either go extinct, mutant populations starting from single individuals
In an a priori given ergodic environment E :
(with a probability that to first order in | Y – X | is proportional
to ((E,Y))+, and with (E,Y) as rate parameter).
matryoshka galore polymorphisms are invariant under permutation of indices
X2
the six purple
volumesshould
be identified
!
adjacent purple volumes are mirror symmetric around a diagonal plane
X1
X3
matryoshka galore the sets of trimorphisms connect to the isoclines of the dimorphisms
X1
X2
X3
(x2 = x3) (x2 = x1)
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