Theses & DissertationsDepartment of Mathematics and Statisticshttp://hdl.handle.net/10214/21892019-08-25T20:10:33Z2019-08-25T20:10:33ZPoint Process Modelling of Presence-Only Species Data: Methodological AdvancesDaniel, Jeffreyhttp://hdl.handle.net/10214/169272019-08-22T05:06:59Z2019-08-21T00:00:00ZPoint Process Modelling of Presence-Only Species Data: Methodological Advances
Daniel, Jeffrey
Presence-only datasets commonly arise in ecological and environmental applications. Such data are often modelled as a realization of a spatial point process, and recent work has unified several seemingly disparate presence-only methods under a point process framework. These unifying results have spurred particular interest in the use of regularized point process models in order to improve predictive performance and aid interpretation. This thesis presents several new methods for modelling presence-only species data with spatial point processes in both the regularized and unregularized settings. In Chapter 2 we present a unified framework for fitting regularized Gibbs point process models in order to accommodate spatial dependence among presence records. Our approach encapsulates both penalized pseudolikelihood and a new approach based on penalized logistic composite likelihood, which tends to perform better in simulations. We also investigate model selection using composite information criteria and propose a new criterion, cERIC, which outperforms other criteria in a simulation study. We apply our methods in an analysis of the distribution of a rainforest tree species within the Barro Colorado Island census plot. In Chapter 3 we propose a multivariate Poisson point process model for presence-only multispecies data. We regularize our model with an adaptive sparse group lasso penalty in order to exploit structure in the model coefficients when the species intensities being modelled depend upon overlapping subsets of covariates. We compare our regularized multivariate model with separate regularized univariate Poisson models in a simulation study and in an application modelling the distributions of 18 bumble bee species within Ontario, Canada. In both settings, the multivariate model tends to outperform the separate univariate models. Finally, in Chapter 4 we return to the unregularized setting and derive local background sampling, an algorithm for fitting Poisson point process models via logistic regression in which background points are sampled with probability proportional to an initial pilot estimate of intensity. In simulations and in another analysis of the Ontario bumble bee data, local background sampling yields more efficient estimates of the model coefficients than the standard uniform background sampling technique.
2019-08-21T00:00:00ZTopological Climate Change and Anthropogenic ForcingKypke, Koljahttp://hdl.handle.net/10214/162482019-07-04T13:40:26Z2019-06-18T00:00:00ZTopological Climate Change and Anthropogenic Forcing
Kypke, Kolja
The mathematical theory of bifurcation is applied to an energy balance model of the Earthâ€™s climate. Bifurcation theory explains how nonlinear systems can exhibit drastically different solutions when certain parameters, though varied gradually, cause catastrophic changes. Such a model is better able to forecast major changes in the climate than traditional methods that capture gradual variations. This thesis presents the first mathematical proof of the existence of a cusp bifurcation in a paleoclimate energy balance model. This result leads to rational explanations for three outstanding problems of paleoclimate science: the Pliocene paradox, the abruptness of the Eocene-Oligocene transition, and the warm, equable Cretaceous-Eocene climate problem. The refinement of this paleoclimate energy balance model using modern climate data adapts it to modern day and near-future parameters up to the year 2300, focusing on climate forcing caused by human activity. Results suggest an even greater future warming effect in the Arctic and Antarctic than currently projected, as a result of a bifurcation that causes a jump to a much warmer climate state in these regions.
2019-06-18T00:00:00ZControlling Games, Replicator Dynamics and Predator-Prey Models with Transmission DynamicsJaber, Ahmed Shawkihttp://hdl.handle.net/10214/161272019-08-14T15:05:47Z2019-05-16T00:00:00ZControlling Games, Replicator Dynamics and Predator-Prey Models with Transmission Dynamics
Jaber, Ahmed Shawki
The concept of optimal control is one of the significant techniques to observe the evolution of various dynamical systems that can be modeled in a mathematical framework. In optimal control problem, the aim is to minimize a performance measure function by defining a control and state trajectories for a dynamical system over a specified period. The context of optimal control is widely used in various disciplines such as engineering, economics, biomathematics, and ecology.
In this thesis, we use control theory methods to undertake the study of specific models of dynamical systems. For instance, we apply the structure of optimal control on the replicator dynamic systems associated with certain classes of games, to further study the game equilibria (or Nash equilibrium points). In essence, we aim to control the game model of population groups who use some pure and mixed strategies and to move their Nash choices to a newer Nash strategy choice with a different outcome. In our first game, we control the Nash strategies to minimize defectors from a social norm, in the second game, we minimize the nonvaccinators in a population contemplating vaccination against infectious disease. In a related way, we utilize classical control theory to analyze an epidemiological model with two different biological populations (species). The aim is to examine endemic equilibrium points of a susceptible-infections-susceptible (SIS) or a susceptible-infections-recovered (SIR) models and control them with a vaccine uptake rate in order to decrease the overall level of infection in both species. All the optimal control problems we introduce are treated with a numerical approach to get the required solutions and to comments on the effect of model parameters on the optimal system states.
2019-05-16T00:00:00ZBiocontrol of Competing Species with Allee effectHodgins, Valeriehttp://hdl.handle.net/10214/160722019-07-04T13:35:23Z2019-05-13T00:00:00ZBiocontrol of Competing Species with Allee effect
Hodgins, Valerie
Biocontrol of a system relies on the addition of a predator, competitor or parasite to control another species. We develop a model of four autonomous ordinary differential equa- tions with biocontrol of two species of fungus in competition. Both populations have motile and sessile spores with the sessile spores subject to an Allee effect. The one-dimensional sin- gle species model, the two-dimensional competition model, and the two-dimensional single species model, are studied using phase portrait analysis and linear stability analysis to gain insight into how the four dimensional model behaves. The effect of stocking duration, the start time of stocking, and varying initial data on the amount of control agent required to ensure the controlled population survives is explored. We find that by incorporating motile and sessile spores we eliminate any possible trivial equilibriums. Furthermore, applying the control agent sooner results in less being required to ensure the survival of the stocked population.
2019-05-13T00:00:00Z