I completed my PhD in Applied Mathematics under guidance of Alona Ben-Tal at School of Natural and Computational Sciences, Massey University, New Zealand. My thesis entitled: Mathematical modelling of the cardiovascular system to study the effects of respiratory sinus arrhythmia and heart failure. This scholarship was funded by Health research council of New Zealand. I developed a lumped parameter model of the cardiovascular system with a specific aim of simulating the system dynamics over a range of heart rates. The models contained several new modelling features that have been introduced progressively throughout the thesis starting with isolated models and continuing with closed loop models of the circulation. Specifically, the contraction of the cardiac chambers was modelled using a time-dependent muscle force with constant elasticity instead of time dependent elasticity. A new hypothesis was proposed about the mechanical contraction of the atria to generate realistic pressure volume loops. The inter-ventricular interaction was modelled as well. Additionally, hysteresis was incorporated in the aortic valve to produce an end-systolic reverse (negative) flow.