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Spatial dynamic of interfaces in ecology : deterministic and stochastic models

Abstract : Traveling fronts arising from reaction diffusion equations model various phenomena observed in physics and biology. From a biological standpoint, a traveling front can be seen as the invasion of an uninhabited environment by a species. Since biological systems are finite and thus undergo demographic fluctuations, these deterministic wavefronts only represent an approximation of the population dynamics, in which we presuppose that the local density of individuals is infiniteso that the fluctuations self-average. In this sense, reaction diffusion equations can be seen as hydrodynamic limits of some individual based models. In this thesis, we investigate the long time behaviour of some finite microscopic systems modeling such front propagations and compare them to the one of their large population asymptotics.The first part of this thesis is dedicated to the study of the dynamics of a population colonising a slowly varying environment. This question has been widely studied from the PDE point of view. However, the results given by the viscosity solutions theory turn out to be biologically unsatisfactory in some situations. We thus suggest to study an individual based model for front propagation in the limit, when the scale of heterogeneity of the environment tends to infinity. In this framework, we show that the spreading speed of the population may be much slower than the speed of the front in the PDE describing the large population asymptotics of the system. This qualitative disagreement between the two behaviours is related to the so-called tail problem observed in PDE theory, due to the absence of local extinction in FKPP-type equations.In a second part, we study the impact of the type of the deterministic limit waves on the related stochastic models to explain this drastic slow-down in the particle system. Indeed, wavefronts arising from monostable reaction diffusion PDEs are classified into two types: pulled and pushed waves. It is well-known that small perturbations have a huge impact on pulled waves. In sharp contrast, pushed waves are expected to be less sensitive. Nevertheless, some recent numerical experiments have suggested the existence of a third class of waves in stochasticfronts. It is a subclass of pushed fronts very sensitive to fluctuations. In this thesis, we investigate the internal mechanisms of such fronts to explain the transition between these three regimes.
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Submitted on : Wednesday, January 19, 2022 - 2:03:28 PM
Last modification on : Thursday, January 20, 2022 - 3:39:44 AM
Long-term archiving on: : Wednesday, April 20, 2022 - 6:42:40 PM


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Julie Tourniaire. Spatial dynamic of interfaces in ecology : deterministic and stochastic models. General Mathematics [math.GM]. Institut Polytechnique de Paris, 2021. English. ⟨NNT : 2021IPPAX093⟩. ⟨tel-03534375⟩



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