Stability of Delay Differential Equations with Applications in Biology and Medicine

Abstract

Delay differential or differential difference or functional differential equations arise in models of biological phenomena when the time delays occurring in these phenomena are taken into account. This will be illustrated below with some predator-prey models. If there are equilibrium states in these equations, it is important to determine sets of parameter values for which these states are stable (locally or globally), and critical parameter values at which stability may be lost. Often the loss of stability may be associated with Hopf bifurcation and the onset of oscillatory behavior. The local stability problem is usually analyzed by linearization around the equilibrium point. For an equation with a single discrete time lag T this could lead to a delay differential equation of retarded type of the form $$ \sum\limits_{k = 0}^n {{a_k}\frac{{{d^k}u(t)}}{{d{t^k}}} + \sum\limits_{k = 0}^m {{b_k}} } \frac{{{d^k}u(t - T)}}{{d{t^k}}} = 0 $$ ([1]) Associated with this equation is the characteristic equation $$ P(z) + Q(z){e^{{T_z}}} = 0 $$ ([2]) where $$ P(z) = \sum\limits_{k = 0}^n {{a_{{k^{{z^k}}}}}} ,\quad Q(z) = \sum\limits_{k = 0}^m {{b_{{k^{{z^k}}}}}} $$ It is well-known that the necessary and sufficient condition for uniform asymptotic stability is that all roots of the characteristic equation lie in the left half-plane, Re z < 0. However, it is a difficult mathematical problem to determine when this condition will be satisfied.