Evolution equations with applications to population dynamics

TL;DR

This thesis analyzes evolution equations in ecology, focusing on population survival in heterogeneous environments, competitive interactions, and decay rates of fractional and classical derivatives, revealing insights into population dynamics and controllability.

Contribution

It introduces new results on population survival in heterogeneous media, characterizes control strategies in competitive models, and provides decay estimates for fractional evolution equations.

Findings

Roads do not affect population survival in heterogeneous media.

Controlled strategies can determine the winner in asymmetric competition.

Decay rates depend on the type of derivatives and spatial operator properties.

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, each corresponding to a chapter of this thesis. 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. We derive results on asymptotic behaviour through the study of some generalised principal eigenvalues. We find that the road has no impact on the survival chances of the population, despite the deleterious effect expected from fragmentation. The second investigation regards a model describing the competition between two populations in a situation of asymmetrically aggressive interactions; this consists of a system of two ODEs. The evolution progresses through two possible scenarios, where only one population survives. Then, the interpretation of one of the parameters as the aggressiveness of the attacker population 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 and last 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.