رسائل دكتوراة
English
2006
SOME STOCHASTIC MODELS IN MATHEMATICAL BIOLOGY
اسلام هـ . السانوسي, عبدالسلام بن علي الدايخ
كلية العلوم الرياضية والاحصاء-جامعة النيلين · السودان
Many recent scientific works have addressed the need for a better understandingof the underlying theory of modeling in biology. However, much more attention hasbeen paid to the area of deterministic modeling in biology than to stochastic modeling,although it is more realistic to consider biological processes as stochastic rather thandeterministic.The goal in writing this thesis is to~ introduce a contribution to fill the arising gap,~ provide some of deterministic and stochastic biological models, and to0 compare between such models.To achieve this, we present the thesis in the following chapters:Chapter One: Measure Theorv & Basic Modern ProbabilityThe underlying mathematical theory of stochastic modeling is stochasticprocesses, and the theory of stochastic processes is based on probability theory. Theaxiomatic development of probability theory was initiated by Kolmogorov in the earlyl930’s. Modem probability theory is technically a branch of measure theory, but it hasdeveloped characteristics and methods of its own, so any systematic exposition of thesubject must begin with some basic measure-theoretic facts. The fundamental conceptin this approach to probability theory is the probability space. In this Chapter, topicsfrom measure theory and probability theory are reviewed which are particularlyrelevant to stochastic processes.Chapter Two: Stochastic ProcessIn this chapter some basic concepts from the theory of stochastic processes arepresented, in particular those are needed for an exploration of stochastic differentialequations. ln the first section, after defining the stochastic process, the concept ofBrownian motion is introduced as one ofthe most important examples of such process.The Markov process is a stochastic process exhibits specific property, called Markovdependence, is presented in the second section. Another related concepts such as:Random Walks, Martingales, Ito Integration and Ito Formula are introduced in theremain sections.Chapter Three: Stochastic Differential EquationsOne ofthe main problems in the stochastic modeling is how to solve the analogstochastic differential equation explicitly or at least numerically, so this chapter isdevoted to many related concepts to the solutions of stochastic differential equations:the first part of this chapter is devoted to Analytical Solutions of SDEs, many relatedconcepts are introduced such as: Interpretation of Stochastic Differential Equations, anExistence and Uniqueness Solutions, Strong and Weak Solutions, MartingaleProblem,.... However, except in simple cases, it is generally not possible to obtainexplicit solutions to SDEs, so the second part is Numerical Methods for Solving SDEs,to reach the last part, which is the main stone in the remain chapters, that is theDiffusion Processes and SDEs.Chapter Four: Deterministic Models in BiologyNumerous d6I8tTl1lItlSIlC models from biology for single and two interactingspecies are presented, such as: Exponential Growth Model for Single population,Logistic Growth Model, Competition Model, Predator-Prey Model and HarvestingProblem. For these models, the behavior of deterministic model is discussed, and willbe developed to the corresponding stochastic model, in the next chapter.Chapter Five: Stochastic Models in BiologyBoth deterministic and stochastic models have important roles to play andshould therefore be considered together. We start with an introduction to show theimportance of considering stochastic models, and then introduce a stochastic analogueto each one of the previous deterministic models. The explicit solutions of some fewmodels are presented. For the majority, however, it is impossible to get such solutions.Ito SDEs for Interacting Populations offer much help to solve many population modelsnumerically, therefore, we devote a section to discuss these equations, and then usethem to solve many models in the remain sections. In biology we are often asked toinfer the nature of population development from a single data set, yet differentrealizations of the same process can vary enormously. Since even stochastic solutionsare only of limited help here, we shall construct simple computer simulation procedureswhich provide much needed insight into the underlying generating mechanisms.Indeed, such model-based simulations can highlight hitherto unforeseen features of aprocess and thereby suggest further profitable lines of biological investigation. AllMAT HEMATICA-4.1 and MATLAB-6.2 programs used to generate the graphs areprovided in the Appendix for easy referencing and developing to further studies.Chapter six: Conclusion and General RemarksAlthough full conditions for agreement between deterministic and expectedstochastic solutions are at present unknown, it could be, however, compared them. Inthis chapter, a comparison between the Cl6l6I'I1'lll'llSliC & stochastic models is given toshow the feature and the better use of each one.
