  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.
