In this thesis we study the existence and stability of positive equilibrium of a general model for the dynamics of several interacting, age-structured population. We begin with the formulation and proof of a global existence theorem for the initial value problem. The proof of this theorem is used to develop an algorithm and a FORTRAN code for the numerical solution of initial value problems for the single species case. This computer program is used to study prototype models for the dynamics of a population whose fertility and mortality rates exhibit an "Allee effect". This is done from a bifurcation theoretic point of view, using the inherent net reproductive rate as a bifurcating parameter. An unstable "left" bifurcation is found. Multi-equilibria and various kinds of oscillations are studied as a function of r, the fertility window, and the nature of the density dependence.

Systems biology is a comprehensive quantitative analysis of the manner in which all the components of a biological system interact functionally along with time. Mathematical modeling and computational methods are indispensable in such kind of studies, especially for interpreting and predicting the complex interactions among all the components so as to obtain some desirable system properties. System dynamics, system robustness and control method are three crucial properties in systems biology. In this thesis, the above properties are studied in four different biological systems. The outbreak and spread of infectious diseases have been questioned and studied for years. The spread mechanism and prediction about the disease could enable scientists to evaluate isolation plans to have significant effects on a particular epidemic. A differential equation model is proposed to study the dynamics of HIV spread in a network of prisons. In prisons, screening and quarantining are both efficient control manners. An optimization model is proposed to study optimal strategies for the control of HIV spread in a prison system. A primordium (plural: primordia) is an organ or tissue in its earliest recognizable stage of development. Primordial development in plants is critical to the proper positioning and development of plant organs. An optimization model and two control mechanisms are proposed to study the dynamics and robustness of primordial systems. Probabilistic Boolean Networks (PBNs) are mathematical models for studying the switching behavior in genetic regulatory networks. An algorithm is proposed to identify singleton and small attractors in PBNs which correspond to cell types and cell states. The captured problem is NP-hard in general. Our algorithm is theoretically and computationally demonstrated to be much more efficient than the naive algorithm that examines all the possible states. The goal of studying the long-term behavior of a genetic regulatory network is to study the control strategies such that the system can obtain desired properties. A control method is proposed to study multiple external interventions meanwhile minimizing the control cost. Robustness is a paramount property for living organisms. The impact degree is a measure of robustness of a metabolic system against the deletion of single or multiple reaction(s). An algorithm is proposed to study the impact degree in Escherichia coli metabolic system. Moreover, approximation method based on Branching process is proposed for estimating the impact degree of metabolic networks. The effectiveness of our method is assured by testing with real-world Escherichia coli, Bacillus subtilis, Saccharomyces cerevisiae and Homo Sapiens metabolic systems. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

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D.Phil. (Electrical and Electronic Engineering) / The knowledge and understanding that a scientist has about the world is often embodied in the form of a model, which is a representation containing the essential structure of some object or event. The goal of the scientific method is to reduce the complexity of our observations on our surroundings (and ourselves) by creating, verifying and modifying simplified models. In turn, a technical scientist (commonly referred to as an “engineer”) uses appropriately simplified mathematical models to predict and control various processes. Yet the central question is as to how close such models are to reality in spite of considerable simplifying assumptions, and whether or not they are reliable and credible enough to be accepted as being valid. In the following, models applied in technical science (commonly referred to as “engineering”) are examined to find out whether or not such mathematical models are valid in biology as well. In fact, it is shown, that such models do fall short of a valid representation of biological phenomena. In turn, the concept of analogy, a method borrowed from cognitive science, is introduced as another way of knowledge representation and model construction.

We address the problem of the consistency between discrete and continuous models for density-dependent size-structured populations. Some earlier works have discussed the consistency of density independent age and size-structured models. Although the issue of consistency between these models has raised interest in recent years, it has not been discussed in depth, perhaps because of the non-linear nature of the equations involved. We construct a numerical scheme of the continuous model and show that the transition matrix of this scheme has the form of the standard discrete model. The construction is based on the theory of Upwind Numerical Schemes for non-Linear Hyperbolic Conservation Laws with one important difference, that we do have a non-linear source at the boundary; interestingly, this case has not been explored in depth from the purely mathematical point of view. We prove the consistency, non-linear stability and hence convergence of the numerical scheme which guarantee that both models yield results that are completely consistent with each other. Several examples are worked out: a simple linear age-structured problem, a density-independent size-structured problem and a non-linear size-structured problem. These examples confirm the convergence just proven theoretically. An ample revision of relevant biological and computational literature is also presented and used to establish realistic restrictions on the objects under consideration and to prepare significant examples to illustrate our points.

published_or_final_version / Statistics / Master / Master of Philosophy

In this thesis we examine mathematical models which have been suggested as possibile mechanisms for forming certain biological patterns. We analyse them in detail attempting to produce the requisite patterns both analytically and numerically. A reaction diffusion system in two spatial dimensions with anisotropic diffusion is examined in detail and the results compared with certain snakeskin patterns. We examine two other variants to the standard reaction diffusion system: a system where the reaction kinetics and the diffusion coefficients depend upon the cell density suggested as a possible model for the segmentation sequence in Drosophila and a system where the model parameters have one dimensional spatial gradients. We also analyse a model derived from known cellular processes used to model the branching behaviour in bryozoans and show that, in one dimension, such a model can, in theory, give all the required solution behaviour. A genetic switch model for pattern elements on butterfly wings is also briefly examined to obtain expressions for the solution behaviour under coldshock.

1 v. (various paging) : ill., tables, diags ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Computing Science, 1976

xi, 98 leaves ; 28 cm Notes Typescript Thesis (Ph. D.)--University of Oregon, 1983 Includes vita and abstract Bibliography: leaves 92-98 Another copy on microfilm is located in Archives

Orientador: Roberto André Kraenkel / Banca: Fernando Fagundes Ferreira / Banca: Gustavo Camelo Neto / Resumo: Este trabalho estuda equações de reação de difusão em domínios finitos com vista a aplicações no estudo do efeito de fragmentação de florestas sobre a dinâmica de populaçõs de espécies. Referimo-nos a habitats de tamanho finito como sendo habitats insulares. A imensa quantidade de dados observacionais relacionados às espécies biológicas presentes em ilhas ou fragmentos de florestas motiva este estudo. Mais especificamente, este trabalho tem como objetivo modelar a dinâmica de interaçãao espacial entre espécies invasoras e espécies nativas de palmeiras, em fragmentos de floresta amazônica, mostrando que os menores fragmentos são mais suscetíveis a espécies invasoras que os fragmentos maiores. O modelo apresentado é um sistema de equações de competição de Lotka-Volterra, com difusão / Abstract: The present work studies reaction-diffusion equations in finite domains, in view of application to the modeling of the effects of fragmentation on the dynamics of biological species. We refer to finite habitats as being insular habitats. The great amount of observational data related to biological species in islands or forest fragments motivates this work. More specifically, this work has as its objective to model the dynamics of the spatial interaction between invader and native species of palm trees in fragments of the Amazon Forest, showing that the smaller fragments are more vulnerable to the invader species than larger fragments. The mathematical model is a system of Lotka-Volterra equations with diffusion / Mestre