adaptive dynamics studying the change of community dynamical parameters through mutation and...

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

c00QuickTime™ and a

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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)