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Reaction-diffusion Equations with Nonlinear and Nonlocal Advection Applied to Cell Co-culture

Abstract : This thesis is devoted to the study for a class of reaction-diffusion equations with nonlocal advection. The motivation comes from the cell movement with segregation phenomenon observed in cell co-culture experiments. The first part of the thesis mainly develops the theoretical framework of our model, namely the well-posedness and asymptotic behavior of solutions in both single-species and multi-species cases.In Chapter 1, we show a single scalar equation with a step function kernel may display Turing and Turing-Hopf bifurcations with the dominant wavenumber as large as we want. We find the bifurcation properties of the homogeneous steady state is closed related to the Fourier coefficients of the nonlocal kernel.In Chapter 2, we study a two-species nonlocal advection model with contact inhibition when the viscosity equals zero. By employing the notion of the solution integrated along the characteristics, we rigorously prove the well-posedness and segregation property of such a hyperbolic nonlocal advection system. Besides, under the framework of Young measure theory, we investigate the asymptotic behavior of solutions. From a numerical perspective, we find that under the effect of segregation, the nonlocal advection model admits a competitive exclusion principle.In the last Chapter, we are interested in applying our models to a cell co-culturing experiment. To that aim, we choose a hyperbolic Keller-Segel model on a bounded domain. By utilizing the experimental data, we simulate a 6-day process of cell growth in a circular petri dish and discuss the impact of both the segregation property and initial distributions on the finial population proportions.
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Xiaoming Fu. Reaction-diffusion Equations with Nonlinear and Nonlocal Advection Applied to Cell Co-culture. Numerical Analysis [math.NA]. Université de Bordeaux, 2019. English. ⟨NNT : 2019BORD0216⟩. ⟨tel-02396865⟩

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