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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.

Studies in grassland management and ecology have always been challenging because of the large amount and great variation of the entities representing and affecting the system. Despite that, we were able to progress significantly in range experimentation in the Campos, in Southern Brazil. Along past thirty years, a large amount of data and information was generated, from vegetation production to components of intake. In an attempt to integrate the information available, seeking for a deeper understanding of the functioning of native grasslands, we propose adapting a mechanistic vegetation model, aggregated of a spatialized grazing component to create PampaGraze. This model was developed for temperate perennial grasslands, and was adapted and tested for subtropical, C4-dominated grasslands of the Campos of Southern Brazil (Chapter III). Despite the limited capacity of field data for validating, the model was able to relatively well simulate the trends in vegetation production along the year and seasons, while overpredicting herbage production during peak growing season. The structure of the model as it is did not allow for an accurate simulation slow-growing, tussock-forming species. Further, we developed and integrated a grazing model, based on a hybrid approach of the classical mechanistic equations of the prey model (STEPHENS & KREBS, 1986), and experimental data on foraging behaviour measured on native grasslands (Chapter IV). The model was very successful on predicting the components of intake, and responded well to variation of components in relation to changes in vegetation and to selectivity pressures, compared to available literature. Regardless of the limitations on the vegetation model, we were able to further explore the relationships of components of intake, identifying possible major limitations for herbage consumption, thus animal performance, in native grasslands. A significant progress was achieved with this thesis, but still long ways to go with this project. A list of suggestions for further developments can be found in Chapter V. We identified the emergent needs for field studies on parameters and morphogenesis, for improving predictions of the vegetation model, as well as structural points of the model that could be addressed for better representation of natural phenomena. This thesis is the first step towards a more detailed and reliable tool for studying and predicting the behaviour of vegetation dynamics and animal production in sub-tropical grasslands. This can allow us to explore relationships and scenarios beyond our experimental capacity, and investigate the connectivity of the system, as well as each mechanism separately. The stage has been set, awaiting further developments.

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.

Studies in grassland management and ecology have always been challenging because of the large amount and great variation of the entities representing and affecting the system. Despite that, we were able to progress significantly in range experimentation in the Campos, in Southern Brazil. Along past thirty years, a large amount of data and information was generated, from vegetation production to components of intake. In an attempt to integrate the information available, seeking for a deeper understanding of the functioning of native grasslands, we propose adapting a mechanistic vegetation model, aggregated of a spatialized grazing component to create PampaGraze. This model was developed for temperate perennial grasslands, and was adapted and tested for subtropical, C4-dominated grasslands of the Campos of Southern Brazil (Chapter III). Despite the limited capacity of field data for validating, the model was able to relatively well simulate the trends in vegetation production along the year and seasons, while overpredicting herbage production during peak growing season. The structure of the model as it is did not allow for an accurate simulation slow-growing, tussock-forming species. Further, we developed and integrated a grazing model, based on a hybrid approach of the classical mechanistic equations of the prey model (STEPHENS & KREBS, 1986), and experimental data on foraging behaviour measured on native grasslands (Chapter IV). The model was very successful on predicting the components of intake, and responded well to variation of components in relation to changes in vegetation and to selectivity pressures, compared to available literature. Regardless of the limitations on the vegetation model, we were able to further explore the relationships of components of intake, identifying possible major limitations for herbage consumption, thus animal performance, in native grasslands. A significant progress was achieved with this thesis, but still long ways to go with this project. A list of suggestions for further developments can be found in Chapter V. We identified the emergent needs for field studies on parameters and morphogenesis, for improving predictions of the vegetation model, as well as structural points of the model that could be addressed for better representation of natural phenomena. This thesis is the first step towards a more detailed and reliable tool for studying and predicting the behaviour of vegetation dynamics and animal production in sub-tropical grasslands. This can allow us to explore relationships and scenarios beyond our experimental capacity, and investigate the connectivity of the system, as well as each mechanism separately. The stage has been set, awaiting further developments.

Studies in grassland management and ecology have always been challenging because of the large amount and great variation of the entities representing and affecting the system. Despite that, we were able to progress significantly in range experimentation in the Campos, in Southern Brazil. Along past thirty years, a large amount of data and information was generated, from vegetation production to components of intake. In an attempt to integrate the information available, seeking for a deeper understanding of the functioning of native grasslands, we propose adapting a mechanistic vegetation model, aggregated of a spatialized grazing component to create PampaGraze. This model was developed for temperate perennial grasslands, and was adapted and tested for subtropical, C4-dominated grasslands of the Campos of Southern Brazil (Chapter III). Despite the limited capacity of field data for validating, the model was able to relatively well simulate the trends in vegetation production along the year and seasons, while overpredicting herbage production during peak growing season. The structure of the model as it is did not allow for an accurate simulation slow-growing, tussock-forming species. Further, we developed and integrated a grazing model, based on a hybrid approach of the classical mechanistic equations of the prey model (STEPHENS & KREBS, 1986), and experimental data on foraging behaviour measured on native grasslands (Chapter IV). The model was very successful on predicting the components of intake, and responded well to variation of components in relation to changes in vegetation and to selectivity pressures, compared to available literature. Regardless of the limitations on the vegetation model, we were able to further explore the relationships of components of intake, identifying possible major limitations for herbage consumption, thus animal performance, in native grasslands. A significant progress was achieved with this thesis, but still long ways to go with this project. A list of suggestions for further developments can be found in Chapter V. We identified the emergent needs for field studies on parameters and morphogenesis, for improving predictions of the vegetation model, as well as structural points of the model that could be addressed for better representation of natural phenomena. This thesis is the first step towards a more detailed and reliable tool for studying and predicting the behaviour of vegetation dynamics and animal production in sub-tropical grasslands. This can allow us to explore relationships and scenarios beyond our experimental capacity, and investigate the connectivity of the system, as well as each mechanism separately. The stage has been set, awaiting further developments.

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.