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Cooperative dynamic systems applied in biology

Abstract : This thesis work consists of new applications of the theory of cooperative dynamical systems to the study of models in Biology. A first model of compartmentalized dynamics coupling hemodynamics and cerebral energy metabolism. It has been proposed to study a natural extension of this model comprising two distinct intracellular compartments, one representing a neuron and the other an astrocyte in addition to the extracellular compartment (also called interstitial) and the capillary compartment. We began by observing that this system (even an extension of this system to N neurons and A astrocytes) is a cooperative system. It was then possible to apply the techniques developed by Hal L. Smith and demonstrate (in all dimensions) that the single stationary point is asymptotically stable. In the following, we have considered a variant of the reduced system of dimension 2 in which we consider a piecewise differentiable dynamic that has a jump when the variable x or the variable y exceeds a certain threshold. This piecewise system allows the introduction of an autoregulation induced by a feedback of the extracellular or capillary Lactate concentrations on the Capillary Blood Flow. New dynamical phenomena are uncovered and we discuss existence and nature of two equilibrium points, attractive segment, boundary equilibrium and periodic orbits depending of the Capillary Blood Flow. In the last chapter, we consider, in contrast with the preceding chapters, a forced dynamical system. This dynamical system models a population whose environment varies periodically over time. We apply our theorem to the example of a population dynamics of insects (for example mosquitoes) with a juvenile stage exposed to a quadratic competition and an adult stage. These dynamics are subject to a seasonal periodic forcing. In particular, in temperate countries, mosquitoes are very rare in winter and grow explosively after the first rainy episodes of the hot season.
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Hongjun Ji. Cooperative dynamic systems applied in biology. Dynamical Systems [math.DS]. Sorbonne Université, 2018. English. ⟨NNT : 2018SORUS514⟩. ⟨tel-03001246⟩

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