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Table of contents
Introduction
I Reaction-diffusion systems and modelling
1 Calcium ions in dendritic spines Work in collaboration with Kamel Hamdache. Published in Nonlinear Analysis: Real World Applications, Volume 10, Issue 4, August 2009.
1.1 Introduction
1.1.1 Dendritic spines
1.1.2 The role of Ca2+ in spine twitching and synaptic plasticity
1.1.3 The original model
1.2 The modified model and main results
1.2.1 New variables
1.2.2 Modeling the twitching motion of the spine
1.2.3 The modified model
1.2.4 Main results
1.3 Proof of Theorem 1.1
1.3.1 A priori estimates
1.3.2 The Fixed Point operator
1.3.3 Conclusion of the proof
1.4 Proof of Theorem 1.2
1.4.1 A priori estimates
1.4.2 The Fixed Point operator
1.4.3 Conclusion of the proof
1.5 Proof of Theorems 1.3 and 1.4
1.5.1 Proof of Theorem 1.3
1.5.2 Proof of Theorem 1.4
1.6 Final remarks
1.6.1 On the cytoplasmic flux
1.6.2 On the diffusion coefficient
1.6.3 On the reactions between calcium and the proteins
1.7 Discussion
2 Viral infection and immune response Work in collaboration with Anna Marciniak-Czochra. Already submitted.
2.1 Introduction
2.1.1 Spatial effects of viral infection and immunity response
2.1.2 The original model
2.1.3 Biological hypotheses of the new model
2.1.4 The reaction-diffusion (RD) model
2.1.5 The hybrid model
2.2 Main results
2.2.1 Existence and uniqueness results
2.2.2 Asymptotic results for the RD system
2.2.3 Asymptotic results for the hybrid system
2.3 Numerical simulations
2.4 The fixed point operator and a priori estimates
2.4.1 Construction of the fixed point operator R
2.4.2 Positivity of solutions
2.4.3 A priori estimates
2.4.4 Continuity of the operator R
2.5 Proof of the theorems
2.5.1 Proof of Theorem 2.1
2.5.2 Proof of Theorem 2.2
2.5.3 Proof of Theorem 2.3
2.5.4 Proof of Theorem 2.4
2.5.5 Proof of Theorem 2.5
2.5.6 Proof of Theorem 2.6
2.6 Discussion
II Reaction-diffusion equations and systems on manifolds
3 The effect of growth on pattern formation
3.1 Introduction
3.2 Main results
3.2.1 Reaction-diffusion systems on growing manifolds
3.2.2 Properties of solutions: existence and uniqueness
3.2.3 The anti-blow-up effect of growth
3.2.4 The stabilising effect of growth
3.3 Proof of Theorem 3.1
3.3.1 Parametrisation and Riemannian metric
3.3.2 The general model with growth and curvature
3.3.3 The isotropic growth model
3.4 Proof of Theorem 3.2
3.5 Proof of Theorem 3.3
3.6 Proof of Theorem 3.4
3.7 Proof of Theorem 3.5
3.8 Discussion
4 Generalised travelling waves on manifolds
4.1 Definition of general travelling waves on manifolds
4.1.1 Complete Riemannian manifolds
4.1.2 Reaction-diffusion equations on manifolds
4.1.3 Fronts, waves and invasions
4.2 Properties of fronts on manifolds
4.3 Proofs
4.3.1 Proof of Theorem 4.1
4.3.2 Proof of Theorem 4.2
4.3.3 Proof of Theorem 4.3
5 Travelling waves on the real line
5.1 Introduction
5.1.1 Calcium waves and fertilized eggs
5.1.2 The reaction-diffusion model on the sphere
5.1.3 Murray’s approach
5.1.4 Modified equation
5.1.5 On intuition and the real dynamics of the travelling waves
5.2 Main results
5.3 Proofs
5.3.1 Supersolutions and subsolutions
5.3.2 Global solution for N
5.3.3 Global solution for S
5.3.4 Steady-state solutions and blocking of the waves
5.4 Discussion
6 Travelling waves on the sphere Work in collaboration with Henri Berestycki and Fran¸cois Hamel. To be submitted.
6.1 Elliptic equation on the truncated sphere
6.1.1 Trivial solutions
6.1.2 Non-trivial solution: variational approach
6.1.3 Pair of non-trivial solutions: topological approach
6.2 Reaction-diffusion equations on the truncated sphere
6.3 Elliptic equation on the N-sphere
6.3.1 Trivial solutions
6.3.2 Stability of solutions
6.3.3 Non-trivial solutions
6.4 Reaction-diffusion equation on the N-sphere
6.4.1 Bistable nonlinearity
6.4.2 Monostable nonlinearity
6.5 Discussion
III Elliptic equations and nonlinear eigenvalues on the sphere
7 Bifurcation and multiple periodic solutions on the sphere Work in collaboration with Henri Berestycki and Fran¸cois Hamel. To be submitted.
7.1 Bifurcation on S1
7.1.1 Properties of solutions
7.1.2 Proof of Conjecture 6.12 for S1
7.1.3 Bifurcation analysis
7.2 Bifurcation on SN
7.2.1 Eigenvalues and eigenvectors
7.2.2 Groups, actions and equivariance
7.2.3 Symmetries and reduction to an ODE
7.2.4 Existence and uniqueness of axis-symmetric solutions
7.2.5 Bifurcation analysis of axis-symmetric solutions
7.3 Discussion
IV Perspectives
8 Conclusions
8.1 Reaction-difusion systems and modelling
8.1.1 Calcium dynamics in neurons
8.1.2 Virus infection and immune response
8.2 Reaction-diffusion equations and systems on manifolds
8.2.1 The effect of growth on pattern formation
8.2.2 Generalised travelling waves on manifolds
8.2.3 Travelling waves on the real line
8.2.4 Travelling waves on the sphere
8.3 Elliptic equations and nonlinear eigenvalues on the sphere
8.3.1 Bifurcation and multiple periodic solutions on the sphere
References
