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The systems of interest in this study are the spread of epidemics and invasions from a small propagule introduced into an arena that was initially devoid of the given species or stage of illness. In reaction-diffusion models, populations are continuous. Populations at low densities have the same growth functions as populations at high densities. In nature, such low densities would signify extinction of a population or of a disease. This property can be removed from reaction-diffusion models by small changes in the formulation so that small populations become extinct. This can be achieved by the use of a threshold density or an Allee effect, so there is negative growth at low densities. Both these alterations were made to the Fisher model, a predator-prey model and a two stage and a three stage epidemic model. A semi-numerical method, termed the Shooting method, was developed to predict the shapes and velocities of these wave fronts. This was found to correctly predict the velocity, the peak density of the invading stage or species and the width of the wave front. It was found that in oscillatory cases of the multi species models, a high threshold can remove the wave train or wake which would normally follow the wave front, so the wave becomes a soliton. The next step is to investigate probable causes of persistence behind the initial wavefront. To do this, discrete time and space versions of the models were formulated so that experiments investigating persistence can be carried out in a two dimensional arena with less computational effort. The formulations were chosen so that at reasonable time and space steps the discrete models show no behaviour different to that of the reaction diffusion model, and so that the Shooting method could also be used to make predictions about these wavefronts. Three mechanisms of persistence are investigated; environmental heterogeneity, long range dispersal and self organised patterns.

Finite-difference numerical methods are developed for the solution of some systems in the biomedical sciences; namely, a predator-prey model and the SEIR (Susceptible/Exposed/ Infectious/Recovered) measles model. First-order methods are developed to solve the predator-prey model and one second-order method is developed to solve the SEIR measles model. The predator-prey model is extended to one-space dimension to incorporate diffusion. The SEIR measles model is extended to one-space dimension to incorporate (i) diffusion, (ii) convection and (iii) diffusion-convection. The SEIR measles model is extended further to model diffusion in two-space dimensions. The reaction terms in these systems of partial differntial equations contain nonlinear expressions. Nevetheless, it is seen that the numerical solutions are obtained by solving a linear algebraic system at each time step, as opposed to solving a nonlinear algebraic systems, which is often required when integrating non-linear partial differential equations. The development of each numerical method is made in the light of experience gained in solving the system of ordinary differential equations for each system. The numerical methods proposed for the solution of the initial-value problem for the predator-prey and measles models are characterized to be implicit. However, in each case it is seen that the numerical solutions are obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it is seen that the proposed methods have superior stability properties to those of the well-known, first-order, Euler method to which they are compared. Incorporating the proposed methods into the numerical solution of partial differential equations is seen to lead to economical and reliable methods for solving the systems.

Our goal in this thesis is to study the dynamics of the classical predator-prey model and the predator-prey model in the chemostat when a discrete delay is introduced to model the time between the capture of the prey and its conversion to biomass. In both models we use Holling type I response functions so that no oscillatory behavior is possible in the associated system when there is no delay. In both models, we prove that as the parameter modelling the delay is varied Hopf bifurcation can occur. However, we show that there seem to be differences in the possible sequences of bifurcations. Numerical simulations demonstrate that in the classical predator-prey model period doubling bifurcation can occur, possibly leading to chaos while that is not observed in the chemostat model for the parameters we use. For a delay differential equation, a prerequisite for Hopf bifurcation is the existence of a pair of pure imaginary eigenvalues for the characteristic equation associated with the linerization of the system. In this case, the characteristic equation is a transcendental equation with delay dependent coefficients. For our models, we develop two different methods to show how to find values of the bifurcation parameter at which pure imaginary eigenvalues occur. The method used for the classical predator-prey model was developed first. However, it was necessary to develop a more robust, less complicated method to analyze the predator-prey model in the chemostat with a discrete delay. The latter method was then generalized so that it could be applied to any second order transcendental equation with delay dependent coefficients. / Thesis / Doctor of Philosophy (PhD)

There is an inherent tension between a seller and a buyer as their relationship progresses towards closing the deal.The salesperson fears that the buyer wastes his time, which he could otherwise spend towards real potential buyers.The buyer questions the sincerity of the salesperson. Both evaluate the amount of trust and cooperation they should invest in the relationship. It is possible, after all, that the salesperson wants to take advantage of every weakness he detects in the buyer in order to guarantee the sale, and it is equally possible that the buyer tries to fool the salesperson, with false credit information for example. This thesis examines tensions that exist between salespeople and buyers by using an extended version of grounded theory, by which date is collected and analysed both qualitatively and quantitatively. It demonstrates that perceived predation reduces considerably the quality of the relationship.The thesis suggests that the phenomenon of predation exists in every sphere of human activity, including in the legal system.The PARDU Model ( pr edator-pr ey) initially developed to discuss the phenomenon of predation evolves, as the research progresses, towards the OPERA Model and the MESLYª grid, which offer pratical tools to better manage informational predation.

Résumé : II existe une tension inhérente au sein de la relation en cours de développement entre un vendeur et un acheteur. Le vendeur craint que l'acheteur lui fasse perdre son temps et l'acheteur, lui, doute de l'honnêteté du vendeur. Tous deux s'interrogent sur le niveau de confiance et de coopération à accorder à l'autre. Il est possible, après tout, que le vendeur soit à l'affût de la moindre faiblesse de l'acheteur pour profiter de la situation et s'enrichir à ses dépens, ou, vice-versa, que l'acheteur cherche à profiter du vendeur. La présente thèse examine les tensions entre vendeurs et acheteurs en ayant recours à la théorie enracinée élargie, qui comprend une série de boucles investigatrices formées de revues des écrits scientifiques et de cueillettes de données qualitatives et quantitatives. Elle cherche à démontrer que la prédation perçue (l'impression que l'autre abuse de nous de manière coordonnée) affecte négativement la bonne entente entre les parties prenantes de la transaction. La thèse suggère aussi que le phénomène de prédation existe dans toutes les sphères d'activités humaines, y compris dans le domaine juridique. Le modèle PARDU (prédateur-proie) initialement développé pour discuter du phénomène de prédation débouche, au fil de la recherche, sur le modèle OPERA et la grille MESLY®, qui offrent des applications pratiques pour mieux gérer la prédation informationnelle.||Abstract : There is an inherent tension between a seller and a buyer as their relationship progresses towards closing the deal. The salesperson fears that the buyer wastes his time, which he could otherwise spend towards real potential buyers. The buyer questions the sincerity of the salesperson. Both evaluate the amount of trust and coopération they should invest in the relationship. It is possible, after ail, that the salesperson wants to take advantage of every weakness he detects in the buyer in order to guarantee the sale, and it is equally possible that the buyer tries to fool the salesperson, with false credit information for example. This thesis examines tensions that exist between salespeople and buyers by using an extended version of grounded theory, by which date is collected and analysed both qualitatively and quantitatively. It demonstrates that perceived prédation reduces considerably the quality of the relationship. The thesis suggests that the phenomenon of prédation exists in every sphere of human activity, including in the légal system. The PARDU Model (predator-prey) initially developed to discuss the phenomenon of predation evolves, as the research progresses, towards the OPERA Model and the MESLY® grid, which offer pratical tools to better manage informational predation.

The thesis describes various modifications of the predator-prey model. The modifications are considering several harvesting methods. At the beginning a solution and a sensitivity analysis of the basic model are provided. The first modification is the percentage harvesting model, which could be easily converted to the basic model. Secondly a constant harvesting including a linearization is derived. A significant part is devoted to regulation models with special a focus on environmental applications and the stability of the system. Optimization algorithms for one and both species harvesting are derived and back-tested. One species harvesting is based on econometrical tools; the core of two species harvesting is the modified Newton's method. The economic applications of the model in macroeconomics and oligopoly theory are expanded using the methods derived in the thesis.

Research Doctorate - Doctor of Philosophy (PhD) / The original contribution of this thesis is to demonstrate the use of a generalised and parsimonious approach to building individual-based computer models of ecological systems with the objective of advancing our mechanistic understanding of these systems. Two models are presented; the first, a model of predator-prey interaction, produces the expected non-linear dynamics and illustrates the importance of the timing of variable updating and individual variation for the persistence of the populations. This model is applied to two near-exclusive, cycling predator-prey systems, those of the Canadian lynx and snowshoe hare and the Fennoscandian mustelid predators and their microtine prey. The reproduction of the patterns found in the empirical data of these systems by the model suggests that the underlying mechanism of these predator-prey systems may be more simple than is suggested by other more complex models reported in the literature. The second model describes a system similar to that of a grazing herbivore in a two-dimensional space. The emergence of complex behaviour resulting from the use of space in the model, including metapopulation-like local extinction and re-population and the effects of corridors and edge qualities on the species are demonstrated. The inclusion of a graphical display of the two-dimensional space in the computer interface to the model reveals important details of system behaviour not evident in the population means, including herding behaviour. The latter is dependent on herbivore mobility and the re-growth of resources in an heterogeneous environment, and emerges in the absence of social behaviour. The problem of detecting herding behaviour automatically is addressed, including the development of qualitative and quantitative definitions of herding in the model.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The aim of this work is to study the temporal and spatiotemporal evolution of a threedimensional system that describes a predator-prey dynamics, where the predator population can develop an infectious disease. Thus, the predators are split into two subpopulations: susceptible predators and infected predators. The rate at which susceptible become infected is described by a Holling type II functional response giving saturation when the number of susceptible predators increases. We assume that the disease develops only in the predators population and that all are born susceptible, ie, there is no vertical transmission. In the temporal evolution system, described by ordinary di�erential equations, we analyze the asymptotic behavior of the model, describing the necessary conditions for the occurrence of qualitative changes, relating them to the basic reproduction number of predators and the basic reproduction number of the disease. In numerical simulations these changes are graphically described, from the variation of the parameters that determine the predation efficiency of the infected predator and the mortality rate of susceptible and infected predators. Starting from the same local dynamics, we include spatial variation and consider movement by difusion to the population, obtaining a system described by partial diferential equations in which we can observe in addition to the temporal evolution of the spatial evolution of the system, or as populations are distributed spatially over time, when and how invasions occur in the domain. The temporal evolution of the system exhibits complex dynamics such as stable equilibrium, limit cycles, periodic oscillations and aperiodicity. The same dynamics are found in reaction-difusion system, considering that every point of the space represented by x displays a local dynamic . Spatially, invasions were observed in the form of wave fronts, making populations evenly distributed over time. / O objetivo central deste trabalho é estudar a evolução temporal e espaço-temporal do sistema tridimensional que descreve uma dinâmica presa-predador, onde a população de predadores pode desenvolver uma doença infecciosa. Desta forma, os predadores são divididos em duas subpopulações: predadores suscetí- veis e predadores infectados. A taxa com que os suscetíveis se tornam infectados é dada por uma resposta funcional tipo II, que exibe uma saturação conforme o número de predadores suscetíveis aumenta. Assumimos que a doença se desenvolve apenas na população de predadores e que todos nascem suscetíveis, ou seja, não há transmissão vertical. No sistema de evolução temporal, descrito por equações diferenciais ordinárias, analisamos o comportamento assintótico do modelo, descrevendo as condições necessárias para a ocorrência de mudanças qualitativas, relacionando-as ao número de reprodução básico dos predadores e ao número de reprodução básico da doença. Nas simulações numéricas essas mudanças são descritas gra�camente, a partir da variação dos parâmetros que determinam a e�ciência de predação do predador infectado e a taxa de mortalidade de predadores suscetíveis e infectados. Partindo da mesma dinâmica local, incluímos a variação espacial e consideramos movimenta ção por difusão para as populações, obtendo um sistema descrito por equações diferenciais parciais, com o qual podemos observar, além da evolução temporal, a evolução espacial do sistema, ou seja, como as populações se distribuem espacialmente com o passar do tempo, quando e como ocorrem as invasões do domínio. A evolução temporal do sistema exibe dinâmicas complexas, como equilíbrios estáveis, ciclos limites, oscilações periódicas e aperiodicidade. As mesmas dinâmicas são encontradas no sistema de reação-difusão, considerando-se que cada ponto do espaço, representado por x, exibe uma dinâmica local. Espacialmente, foram observadas invasões em forma de frentes de ondas, tornando as populações homogeneamente distribuídas com o passar do tempo.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2015-05-20T18:30:38Z No. of bitstreams: 2 Dissertação - Lucyjane de Almeida Silva - 2015.pdf: 1357945 bytes, checksum: c6e7c0e30627101c90a6eb4ae00a5c4d (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Erika Demachki (erikademachki@gmail.com) on 2015-05-20T18:32:39Z (GMT) No. of bitstreams: 2 Dissertação - Lucyjane de Almeida Silva - 2015.pdf: 1357945 bytes, checksum: c6e7c0e30627101c90a6eb4ae00a5c4d (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-05-20T18:32:39Z (GMT). No. of bitstreams: 2 Dissertação - Lucyjane de Almeida Silva - 2015.pdf: 1357945 bytes, checksum: c6e7c0e30627101c90a6eb4ae00a5c4d (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-03-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study the global qualitative behavior of three predator-prey models. We analyze the existence of limit cycle and canard cycle and we investigate the kinds of bifurcation that can occur. In the first model, Gause predator-prey with a refuge, we analyze the effects of a prey refuge on the ecosystem qualitative behavior. Employing the carrying capacity of the prey population in the Gause Model with a refuge we obtain the second model, for which we analyze the effects of the carrying capacity and we compare the results. In the third model we consider the continuous threshold harvesting strategies ocurring when the predator density is above a certain threshold. We note that the model has a complex dynamics with multiple internal equilibria and different types of bifurcation. / Neste trabalho estudamos o comportamento qualitativo global de três modelos predadorpresa. Analisamos a existência de ciclos limite e ciclos de canard e investigamos os tipos de bifurcações que podem ocorrer. No primeiro modelo, modelo predador-presa de Gause com refúgio, analisamos os efeitos do refúgio para as presas no comportamento dinâmico do ecossistema. Empregando a capacidade de suporte para a população de presas no modelo de Gause com refúgio obtemos o segundo modelo, para o qual analisamos os efeitos da capacidade de suporte e comparamos os resultados obtidos. No terceiro modelo consideramos as estratégias de colheita com limiar contínuo que é aplicada quando a densidade de predadores está acima de um certo limite e investigamos o comportamento dinâmico global. Observamos que o modelo possui uma dinâmica complexa com múltiplos pontos de equilíbrio e diferentes tipos de bifurcações.