Discrete delay, distributed delay and stability switches

Abstract

In modelling in the biological, physical and social sciences, it is sometimes necessary to take account of time delays inherent in the phenomena. The inclusion of delays explicitly in the equations is often a simplification or idealization that is introduced because a detailed description of the underlying processes is too complicated to be modelled mathematically, or because some of the details are unknown. In these cases, it may be necessary to choose between a model with discrete or sharp delays and a model with distributed or continuous delay. A question of great importance is whether two models with parallel structure, one with discrete delay and one with distributed delay, will exhibit the same qualitative modes of behaviour. More generally, how does the qualitative behaviour depend on the form and magnitude of the delays? In this paper we shall examine certain aspects of this question. The paper is divided into two parts. In the first part (Sections 2-6), we examine how the stability properties of certain models change when the delay is increased. It has frequently been observed that stability of an equilibrium may be lost when delays are increased. Less frequently. it has been seen that further increase in the delay may result in restabilization. In this paper, we examine the possibilities for several simple equations: (1) a first order linear differential-difference equation; (2) a second order delayed friction model; (3) a second order equation with delayed restoring force; and (4) a population growth model of J. Cushing. In (2) and (3) and in a general equation including both, we show that there may be arbitrarily many switches from stability to instability to stability as the delay is increased, but in (1) this is not possible. In (4), the equation has distributed delay. and