Adaptive Dynamics Part 3 - eleves.ens.fr · The basic CEAD derivation assumes unstructured populations. • For physiologically (e.g. age) structured populations see Durinx et al

Régis Ferrière

Antonin Affholder

September 2019

Ecole Normale Supérieure - PSL

Interdisciplinary Master of Life Sciences

M2 Ecology-Evolution

Adaptive DynamicsPart 3

Monday 9/16

• AM - Lecture 1: Introduction to AD: biological motivation. Modeling

approach: ecological model, model of resident-mutant interaction,

derivation of invasion fitness and selection gradient, evolutionary

singularity. In 1D trait space: trait substitution sequence, pairwise

invasibility plot (PIP), stability properties discussed graphically. Simple

example of life-history evolution.

• PM - Tutorial 1: Introduction of project model: one predator-two prey

model, evolution of specialization. Equilibrium analysis of ecological

model, approximation, Python code for PIP construction.

Course content

Tuesday 9/17

• AM - Lecture 2: Evolutionary singularities: review of stability properties and

general classification. Evolutionary dynamics: stochastic process of birth with

mutation – interaction – death. Example of Doebeli-Dieckmann model:

analytical calculations and stochastic simulations, condition for evolutionary

stability vs. branching. Extension of PIP analysis to polymorphic populations.

Evolutionary adaptive responses to (slow) environmental change, evolutionary

suicide, evolutionary rescue.

• PM - Tutorial 2: (2.1) Review of Tutorial 1. Influence of tradeoff shape on ES

stability properties. Stochastic simulations (Gillespie algorithm). Numerical

evidence of evolutionary branching. Simulations under assumptions of TSS.

(2.2) Introduction to individual projects.

Course content

Wednesday 9/18

• AM - Lecture 3: Extension to multiple traits and multiple species: co-

evolutionary dynamics. Canonical equation of adaptive dynamics:

derivation, application to predator-prey coevolution, Red Queen

dynamics. More examples of AD applications in population, community,

and ecosystem ecology

• PM - Tutorial 3: Simulations of canonical equation, comparison with

stochastic simulations. Individual project.

Thursday 9/19 and Friday 9/20

• Tutorials 4 (Thu AM) & 5 (Fri AM): Individual project.

• Project presentations: Friday PM.

Course content

In one species – examples:

• life history: juvenile survival, growth rate, age at maturity, age-dependent fecundity and survival, rate of senescence

• behavior: parental care, aggressiveness, dispersal

In multiple species – examples:

• Investment in mutualistic functions, e.g. pollination rate and seed production

• Host-pathogen co-evolution: investments in virulence, resistance, tolerance

Adaptive dynamics of multiple traits

Canonical equation of adaptive dynamics

Approximation: Adaptive dynamics

Ferrière, Dieckmann, Couvet (2004) Evolutionary Conservation Biology

Champagnat, Ferrière, Méléard 2006 Theor. Pop. Biol.

Mutations rare: Jump process

Trait Substitution Sequence

(Metz et al. 1992, 1996)

Mutations small: Mean path

Canonical Equation

(Dieckmann & Law 1996 JMB)

Jump rate

~ [invasion fitness S(mut, res)]+

S(mut, res) = r(mut, E(res))

Adaptation rate

~ |fitness gradient S|

S direction & strength of

selection

Derive the canonical equation for the Doebeli-Dieckmann model.

Example: Doebeli-Dieckmann model

Ecological dynamics:𝑑𝑛1

𝑑𝑡= 𝑟1 𝑛1, 𝑛2; 𝑥1 𝑛1

𝑑𝑛2

𝑑𝑡= 𝑟2 𝑛1, 𝑛2; 𝑥1 𝑛2

Ecological equilibria:

𝑛1 → ത𝑛1(𝑥1, 𝑥2)

𝑛2 → ത𝑛2(𝑥1, 𝑥2)

Canonical equation of adaptive dynamics:

modeling co-evolution

Example: Predator-prey co-evolution

Canonical equation: strengths

1) Analysis of attractivity of evolutionary

singularities in trait space of dim. > 1.

• Classical tools of local stability

analysis (linearization…) apply.

2) Evolutionary attractors not limited to

steady states (point equilibria)

• Cycles, chaotic attractors…

3) Transient dynamics can be studied.

4) Multiple evolutionary attractors: basins

of attraction can be identified, exit times

can be estimated, order of visits can be

predicted.

5) Emphasize effects of mutation

variance-covariance matrix M on

evolutionary dynamics

• on attractivity, stability, transient

dynamics, basins of attraction.

• M itself may evolve, as well as

mutation rates.

Canonical equation: strengths

Example: Predator-prey co-evolution

Red Queen

coevolutionary dynamics

Because of the assumption of infinitesimal mutations

1) The canonical equation cannot predict evolutionary branching.

• But the canonical equation can be used to describe evolutionary

dynamics before and after branching.

2) The canonical equation cannot be used to study the effect of large

mutations on evolutionary dynamics.

Canonical equation: limitations

The basic CEAD derivation assumes unstructured populations.

• For physiologically (e.g. age) structured populations see Durinx et

al. 2012 J. Math. Biol.

• Main difficulty here is that different classes may produce mutants at

different rates.

Exending the canonical equation

The basic CEAD derivation assumes equilibrium populations.

• For non-equilibrium dynamics, see Dercole et al. 2010 PRSB.

• Main difficulty here is that mutants are produced at different rates as

the size of the reproductive population fluctuates.

Extending the canonical equation

• Can we model adaptive dynamics in variable environments?

Adaptive dynamics Q&A

• How can we study the evolution of phenotypic plasticity using

adaptive dynamics modeling?


• Can we model adaptive dynamics in finite populations?


• How does evolutionary branching relate to speciation?


Adaptive dynamics: Fundamentals

Metz JAJ, Nisbet RM, Geritz SAH (1992) How should we define

‘fitness’ for general ecological scenarios. Trends Ecol. Evol. 7:

198-202.

Metz JAJ, Geritz SAH, Meszéna G, Jacobs FJA, van Heerwaarden

JS (1996) Adaptive dynamics, a geometrical study of the

consequences of nearly faithful reproduction. Pp. 183-231 in SJ

van Strien and SM Verduyn Lunel, eds. Stochastic and spatial

structures of dynamical systems. North Holland, Amsterdam, The

Netherlands.

Dieckmann U, Law R (1996) The dynamical theory of

coevolution: a derivation from stochastic ecological processes.

Journal of Mathematical Biology 34: 579-612.

Geritz SAH, van der Meijden E, Metz JAJ (1997) Evolutionary

dynamics of seed size and seedling competitive ability.

Theoretical Population Biology 55: 324-343.

Geritz SAH, Kisdi E, Meszéna G, Metz JAJ (1998) Evolutionarily

singular strategies and the adaptive growth and branching of

the evolutionary tree. Evolutionnary Ecology 12: 35-37.

Diekmann O (2004) A beginner's guide to adaptive

dynamics. Pp. 47-86 in R. Rudnicki, ed. Mathematical

modelling of population dynamics. Banach Center

Publications vol. 63, Institute of Mathematics, Polish

Academy of Sciences, Warsawa, Poland.

Brännström A, Johansson J, von Festenberg N (2013) The

hitchhiker’s guide to adaptive dynamics. Games 4:304-

328.

http://www.mv.helsinki.fi/home/kisdi/addyn.htm

Adaptive dynamics: more applications

• Explaining the evolutionarily stable diversity of mutualisms

• Evolution of body size and emergence of trophic networks

across temperature gradients

• Microbial adaptation to warming and ecosystem impact on soil

carbon-climate feedback

How does coevolution influence the stability and shape the diversity of mutualisms?

Ecological dynamics of two interacting obligate mutualists

nnymcnxrdt

dnX )1()( 0species X, trait x, density n

mmxndmyrdt

dmY )1()( 0species Y, trait y, density m

2 ,1 ),(001.0)()( 2 dcuuurur YXNumerical example:

x, y : individual investments in production of mutualistic commodities (nectar, pollination time…)

Explaining the evolutionarily stable diversity of

mutualisms

Invasion fitnesses and canonical equations of adaptive dynamics

);,('

)( xyx

x

sxnk

dt

dx XX

);,('

)( yyx

y

symk

dt

dy YY

)()()'(1)()'()';,( ymxnxxyxncxrxyxs XX

)()()'(1)()'()';,( xnymyyxymdyryyxs YY

Functions and • measure competitive ability

to access partners

• depend on relative

investment in mutualism

Explaining the evolutionarily stable diversity of mutualisms

• Both always < 0 !

• Selection is always directional

• Traits x and y evolve toward 0

• Mutualism driven into evolutionary suicide

)(');,('

xrxyx

x

sX

X

)(');,('

yryyx

y

sY

Y

If competition intensity is independent on traits:


Ecological stability Adaptive dynamics

Combination of species X and Y

phenotypes for which X-Y mutualism is

ecologically viable

X-Y mutualism is eroded by natural

selection and breaks down eventually:

‘evolutionary suicide’


If competition intensity is trait-dependent:

’ and ’ measure the degree of competition asymmetry between slightly different phenotypes within each species.

)( )( ')(');,('

ymxnyxrxyx

x

sX

X

)( )( ')(');,('

ymxnxyryyx

y

sY

Y

' '(0)

' '(0)


Worse mutualists(cheaters)

Better mutualists

Competitive ability to access partners

Mechanisms

Passive: difference in phenology…

Active: interference, partner choice, sanction…

Competitionasymmetry

Adaptive dynamics

(a)



Better mutualists

Competitive ability to access commodities

Competition asymmetry too weak• co-evolutionary suicide due to selection eroding mutualism.


Competition asymmetry too strong• co-evolutionary suicide due to runaway selection.


Better mutualists

Competitive ability to access commodities


(a)

(b)

(c)

Co-evolutionary stability occurs for intermediate degrees of competition asymmetry.


Dercole 2005

Poor mutualists (even exploiters) are pervasive in mutualisms.Should we expect them to be ecologically transient, ecologically persistent, or evolutionarily stable?


Better mutualists

Co

mp

etit

ive

abili

ty t

o

acce

ss c

om

mo

dit

ies

Punishingtradeoff

Poor mutualists provide selective background against which better mutualists are favored.

Mu

tual

isti

c in

vest

men

t

Time

With punishing tradeoff, initially weak mutualisms evolve large investments.



Better mutualists

Co

mp

etit

ive

abili

ty t

o

acce

ss c

om

mo

dit

ies

Punishingtradeoff

Rewardingtradeoff

Mu

tual

isti

c in

vest

men

t

Time

Total interspecific trade

• With rewarding asymmetry, selective pressure against cheaters is weak.• Poor mutualists can coexist with strong mutualists.• Adaptive dynamics lead to broad range of mutualist quality.• This correlates with an increase in total mutualism trade.


How diverse should we expect guilds of mutualists to be?

Rew

ard

ingn

ess

Degree of competition asymmetry

Radiation speed Diversity Function (trade)

• Major influence of the ‘rewardedness’ of competition asymmetry


Should we expect the composition of mutualism guilds to be stable?

Mu

tual

ism

tra

it

Time

Branching event

Extinction event• Composition of guilds of mutualists is dynamic.

• Change in guild composition is driven by cycles of evolutionary branching-extinction events.


Should we expect the composition of mutualism guilds to be stable?

Mu

tual

ism

tra

it

Time

Poor mutualists, lower abundance, older lineages, phenotypic stasis

Strong mutualists, higher abundance, younger lineages, ‘taxon cycle’

Branching event

Extinction event• Composition of guilds of mutualists is dynamic.

• Change in guild composition is driven by cycles of evolutionary branching-extinction events.


Conclusions

• Strong mutualism can evolve because of asymmetrical competition for partners and their commodities

• Cheaters can evolve out of a monomorphic mutualism, coexist, and diverge if competitive asymmetry is rewarding

• Worst cheaters are expected to be ancient (evolved early on). They are evolutionarily conserved.

• Evolutionarily young cheaters are predicted to diverge recurrently from strongestmutualist; they are evolutionarily short-lived.


The upper panel shows the trait composition of the community through time, while the lower panel details the different steps of the emergence. The simulation starts with a single species that consumes inorganic nutrient (panel A). Once in a while, mutants appear (here, larger than the resident) and replace their parent (panel B, in which the gray morph goes to extinction). After several replacements, an evolutionary branching happens, leading to coexistence of the mutant and the resident (panel C). A rapid diversification then occurs in which several morphs are able to coexist (panel D). These morphs are then selected to yield differentiated trophic levels (panel E). (Loeuilleand Loreau 2010)

Coevolution driven by interference competition and trophicinteractions: emergence of a size-structured food web

Evolution of simulated size-structured food webs during 10 8 time steps for three values of niche width ( nw ) and competition intensity ( a0 ). Trophic position is determined recursively from the bottom to the top of the food web. The trophic position of a target species is defined as the average trophic position of the species it consumes weighted by the proportion of nutrient these represent in the target species’ diet, plus 1. Since this measure is strongly correlated with body size, similar patterns are obtained using body size. (Loeuille and Loreau 2005)

Documents

Adaptive Dynamics Part 3 - eleves.ens.fr · The basic CEAD derivation assumes unstructured populations. • For physiologically (e.g. age) structured populations see Durinx et al