1. ANALYSIS OF MULTI-SPECIES ECOLOGICAL AND EVOLUTIONARY DYNAMICS
Ecole Normale Supérieure, Paris
December 9-13, 2013 6.
Adaptive Dynamics (AD) and its canonical equation (F. Dercole)
Introduction to evolutionary dynamics with examples within and beyond biology. Modeling approaches to evolutionary dynamics. The AD approach through a representative example: the evolution driven by the competition for resources.
The AD canonical equation.
Further readings
Analysis of Evolutionary Processes, Princeton Univ. Press, 2008, Chaps. 1-3 and Appx. B, C
Technovation (2008) 28:
J. Theor. Biol. (1999) 197:

2. A naive introduction to innovation and competition processes
Innovations and competition
Evolution
- I want to give you a naive introduction to what I like to call ‘innovation and competition processes’, which are the processes responsible for evolution. It is naive in the sense that it is a bit far from the real biology you are used to, but it is close to the dynamical phenomena that we aim at modeling
- It also wants to emphasize that the evolutionary paradigm can be applied outside biology, to describe innovations and competition processes for example in economics and in the social sciences, though here we will always refer to biology
- So, innovation is the appearance of a new type of individual in a population that we assume, for simplicity, to be uniform before the innovation. For us an innovation will be a genetic mutation. But it can also correspond, for example, to the immigration of an alien individual.
- After the innovation, the innovative group can spread and invade the former resident population, or it can disappear, depending on the competition between resident and innovative individuals. And note that even good innovations, giving a competitive advantage to the innovative individuals, might go extinct simply because innovative individuals are just a few at the beginning, so that they could for example die before reaching maturity
- If we have invasion, then the outcome can be either substitution, if the innovative population asymptotically replaces the former resident, or coexistence
- Now, evolution schematically is the result of a sequence of innovation events each followed by a competition phase. We consider the typical situation in which genetic mutations are rare events on the demographic time-scale (the one that I indicate with the time t) and have small phenotypic effect, namely they induce small changes in the macroscopic features of the individual that matter for competition
- Then, if we look at one of these phenotypes, that I will call trait in the following, and if we look at it on a slower time-scale (that I indicate with the time tau given by t times a small epsilon), we can see the evolution of the trait. Because each time we have an innovation replacing the former resident, than the trait under observation made a small step, and evolution is pictured as the accumulations of those steps
- Actually, there are many details that are most naturally described stochastically, such as the time at which we have a mutation and its phenotypic effect. So what we aim at describing deterministically, with ODEs, is the average evolutionary trajectory among all possible realizations

3. Evolutionary attractors
stationary
non-stationary
(Red Queen Dynamics)
multiple
Evolutionary branching
Evolutionary extinction
- Once we have ODEs for the adaptive traits we can expect all kinds of evolutionary attractors… and attractor multiplicity
- But we can even more. What for example if a certain mutation coexists with the resident trait? Well, from that time on we have two resident traits. Here one and two refer to the competition scenarios we have seen yesterday
- and if competition then favors differentiation (this in biology is called disruptive selection), that means that… then we have evolutionary branching
- but we can also loose populations by evolutionary extinction, that is the evolution of the adaptive traits toward the boundary of the evolution set. Recall we defined yesterday the evolution set as the set of the values of the adaptive traits that allow ecological coexistence of the coevolving populations
- So these are all the dynamical phenomena that we want to model: we want to deterministically describe the expected evolutionary trajectory and being able to add coevolving populations through evolutionary branching and removing those that undergo evolutionary extinction

4. Evolution in biology
- Here I have pictures showing the phenomena I told you about in biology, for example the number of marine families over evolutionary time, increasing over certain phases and decreasing in others
- the dynamics, through pictures or measurements, of various human traits
- a very nice example of evolutionary branching in dapnia
- and here a possible case of oscillating evolutionary dynamics in a species of sardine

5. Evolution outside biology
- But I like to stress that analogous phenomena can be observed even outside biology
- for example invasion of new technology replacing others, or coexisting with others for a while but then driven to extinction
- or branching in engineering, from a first type of aircraft to many other types that still coexist… or the branching from fixed phone to mobile phones…
- or even here the oscillating evolutionary dynamics of fashion…
(see f.r. 1 Chap. 1)

6. Modeling approaches to innovation and competition processes
- There are many modeling approaches to innovation and competition processes, some of them developed by biologists, some other by economists and sociologists, like game theory
- I don’t want to enter in any detail, but just recall you that AD, that is the one I am working on, and the one we see in this course, is just one of the possible approaches
- it certainly has limitations, that I will discuss in a moment, but it is the only approach that can describe all the dynamical phenomena I have been talking about
(see f.r. 1 Chap. 2)

7. Adaptive Dynamics – Basic assumptions
Each individual is characterized by 0, 1, or more inheritable traits (phenotypes/strategies)
Traits are quantitative characteristics
described as continuous variables (symbol ), possibly through a scaling
Reproduction is clonal (asexual)
thus offspring are either characterized by the trait of the parent or are mutants
Mutations in different traits of the same individual are independent
Mutations are rare on the ecological time scale
Mutations are small
The coexistence of populations is stationary
The (abiotic) environment is isolated, uniform, and invariant
- Let’s now see the basic assumptions of AD, most of which can be relaxed but a couple of them
- 3: This allows us to avoid the genetic description!!! Because offspring are…
- 4: this also assumes a genetic simplification…
- 5 e 6 are the fundamental assumptions
- 8: as almost all models considered in this course…
- As I said all this limitations but 5 and 6 can be relaxed, even the assumption on sex, because if you have sexual reproduction you can apply AD not at the level of the individual, but at the level of the single allele considered as a continuous variable…
- But in this course I keep all this assumptions to introduce you the simplest case
- All I say is taken from our book, that basically contains the work of my Ph.D. This is not an advertisement to sell the book, actually you have the full pdf of the book in the further readings
See f.r. 1. See also the original contributions
Metz et al. (in Stochastic and Spatial Structures of Dynamical Systems, Elsevier 1999)
Geritz et al. (Phys. Rev. Lett. 78, , 1997; Evol. Ecol. 12, 35-57, 1998)
Dieckmann & Law (J. Math. Biol. 34, , 1996)

8. The AD working scheme
- This is a picture from the book which shows the working scheme of AD
- I will come back several time to this scheme, for the time being just note that we have the two time-scales…, while the balls here are the different models we have to play with
- In particular this is the evolutionary model that describes the dynamics of the adaptive traits on the evolutionary time-scale, and it is called the canonical equation of AD

9. The AD canonical equation through a simple example
Question: Does the competition for resources optimize a morphological phenotype,
e.g. body size, or promote genetic diversity? (see f.r. 2 and 3)
Let’s start with a single resident population: the resident model is the logistic one!
The resident (ecological) equilibrium
- I now want to derive the canonical equation in the context of a representative example, that is the case of competition for a limited resourse
- The interesting question here is…
- We start from a single resident population, ruled on the ecological time-scale by the logistic model, and we assume that the carrying capacity depends on the trait x according to a bell shaped function like this
- It means that depending on the body size, a different density can be sustained by the environment, and in particular there is a particular body size x0 that is best fitted to the environment, while far from that size individuals have difficulties in accessing food for example or are weaker competitors
- and we know that the stable equilibrium of the logistic model is the carrying capacity and we indicate it with ‘nbar of x’

10. The AD working scheme
- Back to the scheme, this is the resident model

11. The carrying capacity
- and this is the analytical expression of the function K
- it is not important, but just remember that x0 is the trait value at which it is maximum and sigma_K is the width of the bell

12. The AD working scheme
- Now we introduce a mutant and build the resident-mutant model

13. The AD canonical equation through a simple example
Question: Does the competition for resources optimize a morphological phenotype,
e.g. body size, or promote genetic diversity? (see f.r. 2 and 3)
Let’s start with a single resident population: the resident model is the logistic one!
The resident (ecological) equilibrium
The resident-mutant model
- The resident-mutant model is the L-V model we have seen yesterday, where n’… x’…
- alpha here is the competition function that in general depends of the traits of both competitors
- note that the competition function between residents and mutants is the only new ingredient to be added when going from the resident to the resident-mutant model
- (tornare indietro allo schema) Actually the direction is reversed in the scheme, because the r-m model degenerates into the r-m be removing the mutants, but the info in the r-m is typically not enough to write the r-m model, exactly because we shoud add a piece of information that is the competition function
- But in practice we always start from a r-m and we extend it to r-m model
- This is the competition function…
The competition function

14. The competition function
The carrying capacity
The competition function
- here you see the analytical expression, that again is not important in this lecture, but note it is a function of the ratio x’/x, it is always 1 when x’=x as it should be, and it is bell shaped with a width controlled by sigma_alpha
- so we assume that as soon as two individuals are too different is size, they do not compete anymore, like if they then search food in different places, e.g. in holes of different sizes in the environment
- beta tells us if competition is symmetric or not. If beta = 1 …
symmetric competition
asymmetric competition
The model parameters :

15. The AD canonical equation through a simple example
Question: Does the competition for resources optimize a morphological phenotype,
e.g. body size, or promote genetic diversity? (see f.r. 2 and 3)
Let’s start with a single resident population: the resident model is the logistic one!
The resident (ecological) equilibrium
The resident-mutant model
- Back to the r-m model, let’s call f the per-capita growth rate of the resident population and note that the one of the mutant population is given by the same f but with switched resident and mutant arguments
- so remember that f gives the per-capita growth rate of the population whose density is placed at first argument
The competition function

16. The AD working scheme
- We are now almost ready to derive the AD canonical equation (we have two of the three elements needed)

17. The mutant invasion fitness
It is the initial per-capita rate of growth of the mutant population
Technically, it is the eigenvalue determining invasion
The pairwise invasibility plot: the sign of the fitness
if then
and have opposite sign
moreover, invasion implies substitution
(see f.r. 1 Appx. B)
- The most important concept here is the so-called mutant invasion fitness, that you might have already encountered, for example if you have taken Davis Classen course
- it is…
- and I am sure you have also seen the so-called PIP
The selection derivative :
We expect to to have the same sign of

18. The AD canonical equation
(see f.r. 1 Chap. 3 and Appx. C)
where is the probability of a mutation at birth, is the standard deviation of mutations,
and in the limit of extremely rare and small mutations

19. The AD working scheme

20. The AD canonical equation
(see f.r. 1 Chap. 3 and Appx. C)
where is the probability of a mutation at birth, is the standard deviation of mutations,
and in the limit of extremely rare and small mutations
The evolutionary equilibrium
such that It results
Stability via linearization
eigenvalue
is stable for all parameter settings, so there are no bifurcations

21. ? At , , so that invasion does not necessarily imply
substitution. Can residents and mutants coexist and undergo
evolutionary branching?
And what if we have a large mutation (or, most likely, the introduction of an alien species)?
The resident-mutant model (or a suitable resident model) gives the resulting ecological attractor
(an equilibrium?) for which we can derive the corresponding canonical equation