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Stability for the models of neuronal network and chemotaxis

Abstract : This thesis is aimed to study some biological models in neuronal network and chemotaxis with the spectral analysis method. In order to deal with the main concerning problems, such as the existence and uniqueness of the solutions and steady states as well as the asymptotic behaviors, the associated linear or linearized model is considered from the aspect of spectrum and semigroups in appropriate spaces then the nonlinear stability follows. More precisely, we start with a linear runs-and-tumbles equation in dimension d≥1 to establish the existence of a unique positive and normalized steady state and the exponential asymptotic stability in weighted L¹ space based on the Krein-Rutman theory together with some moment estimates from kinetic theory. Then, we consider time elapsed model under general assumptions on the firing rate and prove the uniqueness of the steady state and its nonlinear exponential stability in case without or with delay in the weak connectivity regime from the spectral analysis theory for semigroups. Finally, we study the model under weaker regularity assumption on the firing rate and the existence of the solution as well as the same exponential stability are established generally no matter taking delay into account or not and no matter in weak or strong connectivity regime.
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Submitted on : Monday, December 11, 2017 - 11:59:37 PM
Last modification on : Wednesday, October 14, 2020 - 4:01:18 AM


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Qilong Weng. Stability for the models of neuronal network and chemotaxis. General Mathematics [math.GM]. Université Paris sciences et lettres, 2017. English. ⟨NNT : 2017PSLED026⟩. ⟨tel-01661428⟩



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