Nonautonomous difference equations with applications

This work is divided in two parts. In the first part we develop the theory of discrete nonautonomous dynamical systems. In particular, we investigate skew-product dynamical system, periodicity, stability, center manifold, and bifurcation. In the second part we present some concrete models that are used in ecology/biology and economics. In addition to developing the mathematical theory of these models, we use simulations to construct graphs that illustrate and describe the dynamics of the models. One of the main contributions of this dissertation is the study of the stability of some concrete nonlinear maps using the center manifold theory. Moreover, the second contribution is the study of bifurcation, and in particular the construction of bifurcation diagrams in the parameter space of the autonomous Ricker competition model. Since the dynamics of the Ricker competition model is similar to the logistic competition model, we believe that there exists a certain class of two-dimensional maps with which we can generalize our results. Finally, using the Brouwer’s fixed point theorem and the construction of a compact invariant and convex subset of the space, we present a proof of the existence of a positive periodic solution of the nonautonomous Ricker competition model.

Ecology and conservation of bottlenose dolphins in Madeira Archipelago, Portugal

This study aimed to provide an insight on the ecology of the bottlenose dolphin population in Madeira archipelago. To achieve this, population structure; group dynamics, site-fidelity, residency and movement patterns within and out of the study area; survival and abundance estimates and spatial and temporal distribution and habitat preferences related to physiographic parameters using data collected between 2001-2011, were investigated. Photo-identification data analysis revealed strong evidences that bottlenose dolphins seen in the archipelago of Madeira belong to an open population with regular recruitment of new animals to the area. This population exhibited a typical fission-fusion society, in which short-term acquaintances prevail, with only a few long-lasting associations. Photo-identification methods demonstrated that there is a large variability in residency pattern, with resident, transient and migrant individuals. Only a small number of dolphins were found to be resident (4.3%). Social network diagram as well as SLAR analysis supported the existence of a mixed population of residents, migrants and transients. Mark-recapture methods estimated a high survival rate, within the range of other long-lived cetacean species. The resident community is composed of app. 180 individuals. In addition, around 400 dolphins of different residency patterns were found to use the south area of Madeira Island. Spatial distribution indicated that bottlenose dolphins were regularly found in shallow and closer to shore areas, suggesting the existence of biological processes influenced by bathymetry. Moreover, temporal patterns revealed no strong seasonal fluctuation in the presence of bottlenose dolphins in Madeira archipelago waters. Bottlenose dolphins are listed under the Annex II of the EU habitats Directive that requires the designation of Special Areas of Conservation (SACs) for this species; as such, the knowledge gained through this work can be used by governmental authorities to the establishment and management of areas for the conservation of bottlenose dolphin in Madeira archipelago.