Analysis of differential-delay equations for biology

In this thesis, we investigate the role of time delay in several differential-delay equation focusing on the negative autogenous regulation. We study these models for little or no delay to when the model has a very large delay parameter. We start with the logistic differential-delay equation applying techniques that would be used in subsequent chapters for other models being studied. A key goal of this research is to identify where the structure of the system does change. First, we investigate these models for critical point and study their behaviour close to these points. Of keen interest is the Hopf bifurcation points where we analyse the parameter associated with the Hopf point. The weakly nonlinear analysis carried out using the method of multiple time scale is used to give more insight to these model. The centre manifold method is shown to support the result derived using the multiple time scale. Then the second study carried out is the study of the transition from a sinelike wave to a square wave. This is analysed and a scale deduced at which this transition gradually takes place. One of the key areas we focused on in the large delay is to solve for a certain constant a' associated with the period of oscillation. The effect of the delayed parameter is shown throughout this thesis as a major contributor to the properties of both the logistic delay and the negative autogenous regulation

Analysis of differential-delay equations for biology

In this thesis, we investigate the role of time delay in several differential-delay equation focusing on the negative autogenous regulation. We study these models for little or no delay to when the model has a very large delay parameter. We start with the logistic differential-delay equation applying techniques that would be used in subsequent chapters for other models being studied. A key goal of this research is to identify where the structure of the system does change. First, we investigate these models for critical point and study their behaviour close to these points. Of keen interest is the Hopf bifurcation points where we analyse the parameter associated with the Hopf point. The weakly nonlinear analysis carried out using the method of multiple time scale is used to give more insight to these model. The centre manifold method is shown to support the result derived using the multiple time scale. Then the second study carried out is the study of the transition from a sinelike wave to a square wave. This is analysed and a scale deduced at which this transition gradually takes place. One of the key areas we focused on in the large delay is to solve for a certain constant a' associated with the period of oscillation. The effect of the delayed parameter is shown throughout this thesis as a major contributor to the properties of both the logistic delay and the negative autogenous regulation