Rich dynamics have been demonstrated when a discrete time delay is introduced in a simple predator-prey system. For example, Hopf bifurcations and a sequence of period doubling bifurcations that appear to lead to chaotic dynamics have been observed. In this thesis we consider two different predator-prey models: the classical Gause-type predator-prey model and the chemostat predator-prey model. In both cases, we explore how different ways of modeling the time between the first contact of the predator with the prey and its eventual conversion to predator biomass affects the possible range of dynamics predicted by the models. The models we explore are systems of integro-differential equations with delay kernels from various distributions including the gamma distribution of different orders, the uniform distribution, and the Dirac delta distribution. We study the models using bifurcation theory taking the mean delay as the main bifurcation parameter. We use both an analytical approach and a computational approach using the numerical continuation software XPPAUT and DDE-BIFTOOL. First, general results common to all the models are established. Then, the differences due to the selection of particular delay kernels are considered. In particular, the differences in regions of stability of the coexistence equilibrium are investigated. Finally, the effects on the predicted range of dynamics between the classical Gause-type and the chemostat predator-prey models are compared. / Thesis / Doctor of Philosophy (PhD)

This thesis presents several nonlinear mathematical models applied to ecoepidemiology and evolution. A detailed study involving predator-prey type models considering an alternative resource for the predator was carried out, investigating the situation of infection in the prey and in the predator on separate models. Such study served as a theoretical contribution to the investigation of problems such as bovine tuberculosis in wild animal species presented in a specific model. We also developed models to explain the evolution of alarm calls in species of birds and mammals. The theoretical framework adopted for those evolution models is that of Population Ecology. The models were developed using Ordinary Diferential Equations (ODEs) to describe the population dynamics. The biological assumptions of the systems that we wanted to analyse were enumerated and explained / Esta tese apresenta vários modelos matemáticos não-lineares aplicados à ecopidemiologia e à evolução. Foi realizado um estudo detalhado envolvendo modelos do tipo predador-presa considerando um recurso alternativo para o predador, investigando situações de infecção na presa e no predador em modelos separados. Tal estudo, serviu de aporte teórico para a investigação de problemas como a tuberculose bovina em espécies de animais selvagens apresentado em um modelo específico. Também desenvolvemos modelos para explicar a evolução dos chamados de alarme em espécies de aves e mamíferos. O quadro teórico adotado para esses modelos de evolução é o da Ecologia de População. Nos modelos desenvolvidos usamos as Equações Diferenciais Ordinárias (EDOs) para descrever a dinâmica populacional. Consideramos pressupostos biológicos dos sistemas biológicos analisados

Philosophiae Doctor - PhD / In this thesis, we solve some time-dependent partial differential equations, and systems of such equations, that governs reaction-diffusion models in biology. we design and implement some novel exponential time differencing schemes to integrate stiff systems of ordinary differential equations which arise from semi-discretization of the associated partial differential equations. We split the semi-linear PDE(s) into a linear, which contains the highly stiff part of the problem, and a nonlinear part, that is expected to vary more slowly than the linear part. Then we introduce higher-order finite difference approximations for the spatial discretization. Resulting systems of stiff ODEs are then solved by using exponential time differencing methods. We present stability properties of these methods along with extensive numerical simulations for a number of different reaction-diffusion models, including single and multi-species models. When the diffusivity is small many of the models considered in this work are found to exhibit a form of localized spatiotemporal patterns. Such patterns are correctly captured by our proposed numerical schemes. Hence, the schemes that we have designed in this thesis are dynamically consistent. Finally, in many cases, we have compared our results with those obtained by other researchers.

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