The approach of “adaptive dynamics”

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Table of contents

Introduction
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
Bibliographie
Glossaire des notions biologiques