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Evolution equations with applications to population dynamics

Abstract : The main topic of this thesis is the analysis of evolution equations reflecting issues in ecology and population dynamics. In mathematical modelling, the impact of environmental elements and the interaction between species is read into the role of heterogeneity in equations and interactions in coupled systems. In this direction, we investigate three separate problems. The first problem addresses the evolution of a single population living in a periodic medium with a fast diffusion line; this corresponds to the study of a reaction-diffusion system with equations in different dimensions. Result on asymptotic behaviour are derived through the study of some generalised principal eigenvalues. We find that the presence of the road has no impact on the survival chances of the population, despite the deleterious effect expected from fragmentation. The second study regards a model describing the competition between two populations in a situation of asymmetrically aggressive interactions; this consists in a system of two ODEs. The evolution progresses through two possible scenarios, where only one population survives. The interpretation of one of the parameters as the aggressiveness of the attacker naturally raises questions of controllability. We characterise the set of initial conditions leading to the victory of the attacker through a suitable (possibly time-dependant) strategy. The third part of this thesis analyses the time decay of some evolution equations with classical and fractional time derivatives. Depending on the type of derivative and some degree of non-degeneracy of the spatial operator, quantitative polynomial or exponential estimates are entailed.
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Submitted on : Thursday, June 10, 2021 - 4:42:07 PM
Last modification on : Tuesday, July 13, 2021 - 3:28:21 AM


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  • HAL Id : tel-03257094, version 1


Elisa Affili. Evolution equations with applications to population dynamics. Analysis of PDEs [math.AP]. Sorbonne Université; Università degli studi (Milan, Italie), 2021. English. ⟨NNT : 2021SORUS015⟩. ⟨tel-03257094⟩



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