Reaction-diffusion equations and some applications to Biology

Abstract : In Chapter 1 we model calcium ions and some proteins inside a moving dendritic spine (a microstructure the neuron). We propose two models, one with diffusing proteins and another with proteins fixed in the cytoplasm. We prove that the first model is well-posed, that the second model is almost well-posed and that there is a continuous link between both models. In Chapter 2 we applied the techniques of Chapter 1 for a model of viral infection of cells and immune response in cultivated cells. We propose as well two models, one with diffusing cells and another with fixed cells. We prove that both models are well-posed and that there is a continuous link between them. We also study the asymptotic behaviour and stability of solutions for large times and perform numerical simulations in Matlab. In Chapter 3 we study the effect of growth on pattern formation. We show that growth has two positive effects on pattern formation. First, an \emph{anti-blow up} effect because it allows the solution on a growing domain to blow-up later than on a fixed domain, and if growth is fast enough then it can even prevent the blow-up. Second, a \emph{stabilising} effect because the eigenvalues on a growing domain have smaller real part than those on a fixed domain. In Chapter 4 we extend the definition of travelling waves to manifolds and study some of their properties. In Chapter 5 we study travelling waves on the real line. We prove that there are two waves moving in opposite directions and that they eventually block each other, giving rise to a non-trivial steady-state solution. This example shows that for non-homogeneous models the travelling waves are not necessarily invasions. In Chapter 6 we study travelling waves on the sphere. We prove that for sub-domains of the sphere with Dirichlet boundary conditions the travelling wave is always blocked, but for the whole sphere the wave can either invade or be blocked, depending on the initial conditions. In Chapter 7 we study an elliptic nonlinear eigenvalue problem on the sphere. On the 1-sphere we prove the existence of multiple non-trivial solutions using bifurcation techniques, and on the n-sphere we use the same arguments to prove the existence of multiple axis-symmetric solutions, i.e. solutions depending only on the vertical angle.


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Contributor : Mauricio Labadie <>
Submitted on : Monday, February 6, 2012 - 11:24:20 AM
Last modification on : Monday, February 6, 2012 - 11:25:55 AM

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Mauricio Labadie. Reaction-diffusion equations and some applications to Biology. Adaptation and Self-Organizing Systems. Université Pierre et Marie Curie - Paris VI, 2011. English. <tel-00666581>

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