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Topics in evolutionary dynamics

Lecture 1: Adaptation & the concept of fitness

François Massol

3rd summer school on Mathematical Biology

São Paulo, February 2014

Lecture outline

1. Basics of evolution

2. Defining adaptation

3. The concept(s) of fitness

4. Fitness in different guises

BASICS OF EVOLUTION

Basics of evolution

What are the forces shaping the evolution of organisms?

Basics of evolution


tim

e

population

Basics of evolution


mutation

selection

migration

drift

tim

e

population

Basics of evolution


mutation

selection

migration

drift tim

e

population

Basics of evolution


mutation: new random variants

selection: heterogeneous degrees of success

migration: mixing with neighboring populations

drift: randomness takes its toll

Basics of evolution


mutation: new random variants

selection: heterogeneous degrees of success

migration: mixing with neighboring populations

drift: randomness takes its toll

Evolution shapes the distribution of genotypes

Genotype = set of alleles carried at different loci

Genotype × Environment = Phenotype

Basics of evolution

Population genetics formalism (p + q = 1)

Mutation

Selection

Migration

Drift

Basics of evolution


Mutation

Selection

Migration

Drift

1 1t A a t a A tp p q

Basics of evolution


Mutation

Selection

Migration

Drift


1 /t A tp w p w

Basics of evolution


Mutation

Selection

Migration

Drift


1 /t A tp w p w

1 1t t tp m p mP

Basics of evolution


Mutation

Selection

Migration

Drift


1 /t A tp w p w

1 1t t tp m p mP

1k N k

t t t

NkP p p q

kN

Basics of evolution

Mutation / Selection / Migration / Drift

What else?

Basics of evolution

Mutation / Selection / Migration / Drift

What else?

Homogamy vs. panmixia (mating system)

Dominance

Recombination

Epistasis (on selection)

Plasticity (environmental dependence)

Epigenetic effects

Basics of evolution

Conditions for the evolution of traits

Differential success

Standing variance of trait values

Heritability of trait values

Bergstrom & Dugatkin, “Evolution”

DEFINING ADAPTATION

Defining adaptation

Defining adaptation

Evolution

Optimization

Natural selection

Adaptation

Defining adaptation

Evolution

Optimization

Natural selection

Adaptation

process / history

outcome

Defining adaptation

History-based definitions – Features built by natural selection for their current

role (Gould & Vrba)

– A derived character that evolved in response to a specific selective agent (Harvey & Pagel)

– Biological machinery or process shaped by natural selection to help solve one or more problems faced by the organism (Williams & Nesse)

– A is an adaptation for task T in population P iff A became prevalent in P because there was selection for A, where the selective advantage of A was due to the fact that A helped performed task T (Sober)

Defining adaptation

History-based definitions – Features built by natural selection for their current

role (Gould & Vrba)

– A derived character that evolved in response to a specific selective agent (Harvey & Pagel)

– Biological machinery or process shaped by natural selection to help solve one or more problems faced by the organism (Williams & Nesse)

– A is an adaptation for task T in population P iff A became prevalent in P because there was selection for A, where the selective advantage of A was due to the fact that A helped performed task T (Sober)

Character arising through past natural selection

Problem: what character doesn’t?

Defining adaptation

Nonhistorical definitions – A feature of the organism which interacts

operationally with some factor of its environment so that the individual survives and reproduces (Bock)

– Greater ecological-physiological efficiency than is achieved by other members of the populations (Mayr)

– An aspect of the developmental pattern which facilitates the survival and/or reproduction of its carrier in a certain succession of environments (Dobzhansky)

– A strategy that has the highest per capita growth given the conditions (Mitchell & Valone)

Defining adaptation

Nonhistorical definitions – A feature of the organism which interacts

operationally with some factor of its environment so that the individual survives and reproduces (Bock)

– Greater ecological-physiological efficiency than is achieved by other members of the populations (Mayr)

– An aspect of the developmental pattern which facilitates the survival and/or reproduction of its carrier in a certain succession of environments (Dobzhansky)

– A strategy that has the highest per capita growth given the conditions (Mitchell & Valone)

Character conferring the best fitness among available ones

Minor problem: can this happen?

Defining adaptation

A synthesis of definitions (Reeve & Sherman)

A phenotypic variant that results in the highest fitness among a specified set of variants in a given environment

Defining adaptation



“optimization” in some sense

among available variants (contingent on history)

conditional on environmental state

Defining adaptation



“optimization” in some sense

among available variants

conditional on environmental state

Pending question:

what is fitness?

THE CONCEPT(S) OF FITNESS

The concept(s) of fitness

The simplest: growth = fitness

( )dN

r N Ndt

Growth rate

1 ( )t t tN N N

Reproductive ratio


The simplest: growth = fitness

Problem: what if all genotypes grow or decline together?

distinguish absolute from relative growth rate

may depend on context / density / frequency

( )dN

r N Ndt

Growth rate

1 ( )t t tN N N

Reproductive ratio


The simple: selection = fitness

( )dN

r N Ndt

Relative growth rate

( )ii i

dfr N r f

dt

1 ( )t t tN N N

1

i t

it it

t

Nf f

N


What for?

time

frequency

1

3

2

4

1. invasibility

2. persistence

3. fixation

4. coexistence


What for?

time

frequency

1

3

2

4

1. invasibility

2. persistence

3. fixation

4. coexistence

Same problem: what happens with rare genotypes?


The simple: selection = fitness when rare

Invasion fitness

0

( )if

ir N r


The complex: unstable absence = fitness



0

( )if

ir N r



0

( )if

ir N r

( )ii i

dfr N r f

dt



0

( )if

ir N r

0

.

if

df

dtf

J

Jacobian matrix

/i j ijg f J i i ig r fr



More generally:

define invasion fitness as the leading eigenvalue of the Jacobian in the absence of the focal type

0

.

if

df

dtf

J

Jacobian matrix


Even more generally:

define invasion fitness as any criterion that qualifies whether the leading eigenvalue of the Jacobian in the absence of the focal type has negative real part

More than one fitness criterion might exist!

FITNESS IN DIFFERENT GUISES

Fitness in different guises

Population genetics fitness

With selection only

1

i t

it it

t

w Pp

wp

P

*

*

i

i

w P

w P



Divergent selection in habitats x and y

Life cycle: Reproduction, regulation, total migration

Model = soft selection (Levene 1953)

iyixi x y

x y

w

w

wq

wq



Divergent selection in habitats x and y

Life cycle: Reproduction, total migration, regulation

Model = hard selection (Dempster 1955; Ravigné et al. 2004)

x ix

y

iy

i

x x y

yq w q w

q w q w



Divergent selection + environmental variability

Use matrices for reproduction, migration, ... events

…

Sp .ax . .m R E S D

Sp .ax . .m R E D S

Sp .max . E D S

Ravigné’s model

Levene’s model

Dempster’s model

Massol 2013


Stage-structured populations



0 0 0 1 2 0 3

2 2

3

1 1 0

1 1

12 2

3 3

3

(1

(1 0 0

0 (1 0

0 0

)

)

)t t

s f

s

s f q s f s f s

q s q

q s q

q s s

s

N N

0 1 2 3

s0f0+ (1-q1)s1

q1s1 q2s2 q3s3

s∞ (1-q2)s2 (1-q3)s3

s0f3

s0f2 s0f1



1t t N F G N

Fecundity + Survival/change in stage

0 0 0 1 2 0 3

2 2

3

1 1 0

1 1

12 2

3 3

3

(1

(1 0 0

0 (1 0

0 0

)

)

)t t

s f

s

s f q s f s f s

q s q

q s q

q s s

s

N N



1t t N F G N


0 1 2

0

3

0

0 0 0 0

0 0 0

0 0

0

0 0

Ts

ff f f

F e f

1 1

1 1

2

2

3

3 3

3

2

2

(1 0 0 0

(1 0 0

0

)

)

)(1 0

0 0

q

q s q

q s q

q

s

s

s

s

s

G 0 1,0,0,0e

0 0 1 2 3, , ,f f fs ff



Formula for fitness (≠ maximal eigenvalue)

1t t N F G N


1

0. .Tw f I G e 0 1,0,0,0e

0 0 1 2 3, , ,f f fs ff

0TF e f



Formula for fitness (≠ maximal eigenvalue)

1t t N F G N


Stage-wise fecundity

Time passed in each stage

1

0. .Tw f I G e 0 1,0,0,0e

0 0 1 2 3, , ,f f fs ff

0TF e f



1t t N F G N

demonstration, french way



1t t N F G N

1

( )

F G I I G I G F 1 I I



1t t N F G N

1

Sp 1 Sp ( )

I G F 1 IF G

1

( )




1t t N F G N

1

Sp 1 Sp ( )

I G F 1 IF G

1

( )


1( )max Sp

I G F 1 I

is a strictly decreasing function of λ



1t t N F G N

1

Sp 1 Sp ( )

I G F 1 IF G

1

( )


1( )max Sp

I G F 1 I

is a strictly decreasing function of λ

1Sp max Sp 11

I G FF G



1Sp max Sp 11

I G FF G

0TF e f

0 1,0,0,0e

0 0 1 2 3, , ,f f fs ff



1Sp max Sp 11

I G FF G

0TF e f

0 1,0,0,0e

0 0 1 2 3, , ,f f fs ff

1 1

0Sp 0, T

I G F I G ef

Fitness! w



A demographic re-interpretation

0TF e f

0 1,0,0,0e

0 0 1 2 3, , ,f f fs ff

1

0. .Tw f I G e

1

0 00

T T k

k

f I G e f G e

Offspring produced at time k

Lifetime offspring production

Thank you for your attention!

Further reading

Evolution, adaptation Grafen A. (1999). Formal Darwinism, the individual-as-maximizing-agent analogy

and bet-hedging. Proc R Soc Lond B Biol Sci, 266, 799-803.

Maynard Smith J. (1989). Evolutionary Genetics. Oxford University Press, Oxford.

Reeve H.K. & Sherman P.W. (1993). Adaptation and the goals of evolutionary research. Q Rev Biol, 1-32.

Further reading

Fitness Caswell H. (2001). Matrix Population Models: Construction, Analysis, and

Interpretation, Second edition. 2nd edn. Sinauer Associates, Inc. Publishers, Sunderland, Massachussets.

Charlesworth B. (1994). Evolution in age-structured populations, Second edition. 2nd edn. Cambridge University Press, Cambridge.

Diekmann O., Heesterbeek J.A.P. & Metz J.A.J. (1990). On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J Math Biol, 28, 365-382.

Roughgarden J. (1979). Theory of Population Genetics and Evolutionary Ecology: an Introduction. MacMillan publishing Co., Inc.

Further reading

Selection & population genetics de Meeus T., Michalakis Y., Renaud F. & Olivieri I. (1993). Polymorphism in

heterogeneous environments, evolution of habitat selection and sympatric speciation: soft and hard selection models. Evol Ecol, 7, 175-198.

Débarre F. & Gandon S. (2011). Evolution in heterogeneous environments: between soft and hard selection. Am Nat, 177, E84-E97.

Ravigné V., Olivieri I. & Dieckmann U. (2004). Implications of habitat choice for protected polymorphisms. Evol Ecol Res, 6, 125-145.

Wallace B. (1975). Hard and soft selection revisited. Evolution, 29, 465-473.

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