A microscopic interpretation for adaptive dynamics trait substitution sequence models 

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Biological approaches of natural selection

The principle of natural selection is known since Charles Darwin: amongseveral individuals of a species, the ones that are better adapted to their natural environment transmit their characteristics to a larger number of descendants than the ones that are less adapted. This implies, on large time scale, the disappearance of the disadvantaged individual characteristics and the persistence of the adapted characteristics.
Since then, our understanding of this phenomenon has progressed mainly in the details of the molecular and physiological mechanisms. DNA was identified as the chemical support of heredity. The genetic code, translating genes into proteins, has been deciphered, and we know that proteins are responsible for almost every chemical reactions in a living organism (syntheses, enzymes, regulations, cycles, transport, etc.). But we still barely understand the other main mecanisms: how the genotype of an individual translates into its phenotype (i.e. the set of its global characteristics, such as reproductive efficiency, rate of food intake, height, etc.) which determines its adaptation to its environment? how to quantify this adaptation from the individual phenotypes? how do the differences between individual lead to the selection of the best adapted ones, in large time scales? Out of these three problems, it is probably the last one for which mathematics can be the most useful. Evolution, as understood nowadays, can be represented as in figure I.1 below.

The approach of “adaptive dynamics”

The recent alternative approach of the theory of adaptive dynamics consists in simplifying the reproductive scheme by considering an asexual population (clonal reproduction, as for bacteria and some plants) and in paying more attention to the ecology of the system. The phenotypic variability then only comes from mutations. This allows to skip the notion of genotype, considering that the mutations directly act at the level of phenotypes. Figure I.2 sums up the main outlines of the modelling of evolution in the framework of adaptive dynamics.

Evolution by successive invasions of mutants : the adaptive dynamics jump processes

The approach that we present in this section refers to the first works in the theory of adaptives dynamics. The three fundamental papers are Hofbauer and Sigmund [41], Marrow et al. [57] and Metz et al. [61], but the most detailled presentation of the basis and main consequences of this approach can be found in Metz et al. [59]. From the mathematical point of view, no rigourous and systematic work had been yet engaged to support the theoretical framework sketched in these articles. The main purpose of chapters II and III of this thesis is to provide a mathematical basis for some adaptive dynamics models.
Let us consider one or several phenotypic traits characterizing individuals in a single asexual population (we restict ourselves to a single species in order to simplify notations). Let X be the space of possible trait values (i.e. a finite number of phenotypic characteristics). For example, X can be a subset of Rd. Let us begin our study with the case of a monomorphic population with trait x (i.e. every individual of the population has the same trait value x ∈ X). Under (HB1), the only possible evolution for this monomorphic population is the appearence of a unique mutant trait y ∈ X, followed by the death or the survival of traits x or y (this mutant trait cannot be helped by another mutant trait, since the mutations are rare). x is called the resident trait, as opposed to the mutant trait y. We will sometimes call “resident (resp. mutant) population x” the resident (resp. mutant) population with trait x. When the mutant trait y do not get immediately extinct, we will say that the invasion of the resident population x by the mutant trait y has occured. This invasion can be defined as the increase of the mutant population’s size from a single individual to a “significative” size with respect to the resident population’s size (this notion will be described in more details below). Finally, once the invasion of y has occured, the competition between x and y may drive both, one or none of them to extinction. We will say that fixation has occured when only y survives.

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Adaptive dynamics jump process: general case

This section is inspired from [59], but its contents are mainly original. We can now predict the fate (invasion, fixation, coexistence) of a mutant trait appearing in a monomorphic population. Under (HB1), the next mutant trait will appear only once invasion, fixation or coexistence of the first one has been decided, and its fate can be analyzed in the same way. The global evolution of the population in a large time scale results from the succession of these invasions. This evolution may be a gradual process (cf. Rand and Wilson [66]) when each mutant gets fixed in the population, or a diversification process if coexistence of many traits takes place.
To go further in our study, we have to generalize the results of the previous sections to the case when the initial population is polymorphic. Assumption (HB4) has to be modified as follows:
Biological hypothesis (HB4’) Under (HB2), the dynamics of a k-morphic population with traits x1, . . . ,xk ∈ X is governed by a system of density-dependent ODEs, with a unique non-null globally asymptotically stable steady state.
Note that this is a very strong assumption and there is no general method known to verify it for particular dynamics. However, it is a very convenient tool to construct the biological model. It is possible to weaken this assumption and to describe a more general biological model by allowing the existence of several stable steady states (see section 3 of the introduction of this thesis).

Table of contents :

1 Mod´elisation individuelle : syst`eme de particules en interaction 
1.1 Construction du mod`ele individuel
1.2 Limite des grandes populations
1.3 Simulations num´eriques
2 R´esum´e des chapitres I `a 
2.1 Chapitre I : la biologie des dynamiques adaptatives
2.2 Chapitre II : justification du processus de saut monomorphique des dynamiques adaptatives `a partir du mod` ele individu-centr´e
2.3 Chapitre III : l’´equation canonique et le mod`ele de diffusion
2.4 Chapitre IV : ´etude du mod`ele de diffusion, grandes d´eviations et probl`eme de sortie de domaine
3 Conclusions et perspectives
I An introduction to adaptive dynamics 
1 Biological approaches of natural selection
1.1 Themechanisms of evolution
1.2 The approach of “adaptive dynamics”
2 Evolution by successive invasions of mutants : the adaptive dynamics jump processes
2.1 Invasion and fitness
2.2 Fixation or coexistence?
2.3 Adaptive dynamics jump process: general case
2.4 Adaptive dynamics jump process: monomorphic case .
2.5 Adaptive dynamics jump process: k-morphic case
3 Small jumps assumption: biological consequences
3.1 Classification of the evolutionary singularities in dimension
3.2 The canonical equation of adaptive dynamics
4 Conclusion
II A microscopic interpretation for adaptive dynamics trait substitution sequence models 
1 Introduction
2 Birth and death processes
2.1 Comparison results
2.2 Problemof exit froma domain
2.3 Some results on branching processes
3 Proof of Theorem 1.1
III Convergence of adaptive dynamics jump processes to the canonical equation and degenerate diffusion approximation 
1 Introduction
2 n-morphic trait substitution sequence
3 Polymorphic canonical equation of adaptive dynamics
3.1 Rescaled process
3.2 Convergence result
3.3 Proof of Theorem 3.1
4 Diffusionmodel of adaptive dynamics
4.1 Diffusion generator
4.2 Regularity of the coefficients a, b and ˜b
4.3 Concluding remarks and comments
IVExistence, uniqueness, strong Markov property and large deviations for degenerate diffusion models of evolution 
1 Introduction
2 Description of themodel
2.1 The fitness function
2.2 Themutation law
2.3 The diffusion approximation
3 Weak existence, uniqueness in law and Markov property
3.1 Construction of a particular solution to (1)
3.2 Uniqueness and strongMarkov property
3.3 Study of (22): the dimension 1 case
3.4 Study of (22): the dimension d ≥ 2 case
4 Large deviations for Xε as ε → 0
4.1 Statement of the result
4.2 Proof of Theorem 4.1
5 Application to the problem of exit froma domain
A Analyse du syst`eme de comp´etition logistique dimorphique 
B Code C de simulation du mod`ele individu-centr´e 
Glossaire des notions biologiques


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