Oscillations, feedback and bifurcations in mathematical models of angiogenesis and haematopoiesis

Abstract

Angiogenesis and haematopoiesis are physiological processes which include the dynamics of tumour growth from a dormant to a malignant state. These dynamics involve many interacting oscillatory processes that operate on several scales of space and time. It is important to understand such processes as anti-angiogenic drugs can have both inhibitory and stimulatory effects on tumour growth. In this chapter, we consider ordinary and delay differential equations which model angiogenesis and haematopoiesis. By introducing a feedback mechanism, bifurcations involving critical points and limit cycles are investigated as parameter values are altered in the system. It is shown that the dynamics are history dependent and hysteresis is possible. Mathematica TM program files have also been listed so that the reader can reproduce the results listed in this chapter. The reader can also download working Mathematica programs from the web.