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Table of contents
Introduction
A Introduction Générale
A.1 Bref historique
A.1.1 Premiers modèles discrets
A.1.2 Premiers modèles continus
A.2 Équations de réaction-diffusion
A.2.1 Équations différentielles ordinaires
A.2.2 Modèles avec espace
A.2.3 Conditions au bord
A.2.4 Équations de réaction-diffusion
A.2.5 Objet de ce manuscrit
B Introduction à la partie I
B.1 Introduction
B.1.1 Présentation du problème
B.1.2 Sujets connexes et état de l’art
B.2 Présentation des résultats
B.2.1 Critère pour la non-existence de patterns
B.2.2 Valeur propre principale généralisée
B.2.3 Non-existence de patterns en domaine non-borné
B.2.4 Autres extensions
C Introduction à la partie II
C.1 Introduction
C.1.1 L’approche Hamilton-Jacobi sur un exemple simple
C.1.2 Interprétations
C.1.3 Bref état de l’art
C.2 Présentation des résultats
C.2.1 Population structurée en âge
C.2.2 Ajout d’un phénomène de mutation
C.2.3 Transmission Génétique Horizontale
D Introduction à la partie III
D.1 Introduction
D.2 Étude théorique
D.3 Modélisation de l’Agitation Sociale
D.3.1 Introduction
D.3.2 Analyse du modèle
Introduction0 (English)
A0 General Introduction
A0.1 A brief history
A0.1.1 First discrete models
A0.1.2 First continuous models
A0.2 Reaction-Diffusion Equations
A0.2.1 Ordinary Differential Equations
A0.2.2 Models with space
A0.2.3 Boundary conditions
A0.2.4 Reaction-Diffusion Equations
A0.2.5 Purpose of this manuscript
B0 Introduction to Part I
B0.1 Introduction
B0.1.1 Presentation of the problem
B0.1.2 Related topics and state of the art
B0.2 Presentation of the results
B0.2.1 Criterion for the non-existence of patterns
B0.2.2 Generalized principal eigenvalue
B0.2.3 Non-existence of patterns in unbounded domains
B0.2.4 Other extensions
C0 Introduction to part II
C0.1 Introduction
C0.1.1 The Hamilton-Jacobi approach on a s simple example
C0.1.2 Interpretation
C0.1.3 Brief state of the art
C0.2 Presentation of the results
C0.2.1 Age-structured populations
C0.2.2 Adding a mutation phenomenon
C0.2.3 Horizontal Gene Transfer
D0 Introduction to Part III
D0.1 Introduction
D0.2 Theoritical analysis
D0.3 Modelling Social Unrest
D0.3.1 Introduction
D0.3.2 Analysis on the model
I Qualitative Properties of Stable Solutions
1 Stable solutions of semilinear elliptic equations in unbounded domains
1.1 Introduction
1.1.1 General Framework
1.1.2 Definition of stability
1.2 The results
1.2.1 Convex domains
1.2.2 Symmetry properties
1.2.3 Counterexamples
1.3 Preliminaries
1.3.1 The classical case of bounded convex domains
1.3.2 Non-degenerate stable solutions – proof of Theorem 1.2
1.4 Properties of 1
1.4.1 Existence of a positive eigenfunction
1.4.2 Liouville result, or the simplicity of 1
1.5 Proof of the symmetry properties
1.5.1 Proof of Proposition 1.4
1.5.2 Proof of Theorem 1.6 and Corollary 1.7
1.6 Proof of Theorem 1.3
1.7 Appendix
1.7.1 Generalized principal eigenvalue
1.7.2 On the different definitions of stability
2 Variations on the Casten, Holland, and Matano Theorem
2.1 Introduction
2.1.1 Framework
2.1.2 The classical proof of Casten, Holland, and Matano
2.2 Extension to non-self adjoint operators
2.2.1 Quantitative criterion for the non-existence of patterns
2.2.2 Perturbation results
2.2.3 Extension to unbounded domains
2.3 Asymptotic results
2.3.1 Statement
2.3.2 Proofs
2.3.3 Other asymptotic symmetries
2.4 Flatness estimate
2.5 Appendix: isolation of stable solutions
3 The generalized Robin principal eigenvalue
3.1 Definition of the principal eigenvalue
3.1.1 Framework
3.1.2 The principal eigenvalue on bounded domains
3.1.3 Extension to unbounded domains
3.1.4 Existence of a positive eigenfunction
3.2 The self-adjoint case
3.2.1 Rayleigh-Ritz variational formula
3.2.2 The Maximum Principle
3.2.3 The Critical Maximum Principle
3.3 Non-self-adjoint operators
3.3.1 Case of a bounded drift
3.3.2 General case
II Dynamics of Concentration and Modeling of Natural Selection
4 Population structured by age and phenotype
4.1 Introduction
4.1.1 The model
4.1.2 Assumptions
4.2 Case without mutations
4.2.1 The eigenproblem
4.2.2 Concentration
4.2.3 Properties of concentration points
4.2.4 Numerical simulations
4.3 Case with mutations
4.3.1 Saturation and stationary problem
4.3.2 The Hamilton-Jacobi equation
4.3.3 Global existence and a priori estimate
4.3.4 The semi-relaxed limits
4.3.5 Uniqueness result
4.4 Appendix
4.4.1 Saturation of the population denstity
4.4.2 The eigenproblem
4.4.3 Convexity of the Hamiltonian
4.4.4 A priori bound on @tU »
5 Population structured by age and phenotype II: adding mutations
5.1 Introduction
5.1.1 Main results
5.1.2 The model
5.1.3 Formal Approach and Method
5.1.4 Outline of the paper
5.1.5 Assumptions
5.2 Definition of the ansatz and a priori estimates
5.2.1 Formal Limiting Eigenproblem
5.2.2 Construction of U » and a priori estimates
5.2.3 Further estimates
5.2.4 Asymptotics
5.3 Proof of the main theorem
5.3.1 Estimates on p »
5.3.2 Selection of the fittest phenotype
5.3.3 Adaptive dynamics
5.4 Construction, estimates, and asymptotics of U »
5.4.1 The eigenproblem
5.4.2 Construction of U »
5.4.3 A priori Lipschitz estimate
5.4.4 Semi convexity
5.4.5 Asymptotics of U »
5.4.6 A posteriori Lipschitz estimate
6 Horizontal Gene Transfer
6.1 Introduction
6.2 Model
6.2.1 Stochastic model
6.2.2 The PDE model
6.2.3 The Hamilton-Jacobi limit
6.2.4 Formal analysis on the Hamilton-Jacobi equation
6.3 Numerical tests
6.3.1 Stochastic model
6.3.2 Numerical scheme for the PDE model
6.3.3 The scheme for the Hamilton-Jacobi equation
6.3.4 Comparison with theoretical analysis
6.4 Conclusions
III Systems of Reaction-Diffusion Equations
7 A generalization of the SI epidemic model
7.1 Introduction
7.1.1 General Framework
7.1.2 Assumptions and notations
7.2 Main results
7.2.1 The inhibiting case
7.2.2 The general case
7.3 Stability analysis
7.3.1 Stability
7.3.2 Unstability
7.4 Asymptotic speed of propagation
7.4.1 Upper bound
7.4.2 Lower bound
7.5 Transition waves
7.5.1 Non-existence
7.5.2 Existence
8 Modelization of social unrest 301
8.1 Introduction
8.1.1 The dynamics of Social Unrest
8.1.2 Construction of the model
8.2 General framework
8.2.1 Assumptions and notations
8.2.2 Main results
8.2.3 Numerical simulations
8.2.4 Comparison with previous models
8.3 Analysis
8.3.1 Resumption of calm
8.3.2 Burst of Social Unrest
8.3.3 Geographical propagation
8.4 Tension Inhibiting – dyanmics of a riot
8.5 Tension Enhancing – dynamics of a revolution
8.5.1 Asymptotic behavior of solutions
8.5.2 Transition waves
Bibliography
