Geometric Stability Switches for Delay Equations with Delay-Dependent Parameters

In most applications of delay differential equations in population dynamics, the need of incorporation of time delays is often the result of the existence of some stage structure. Since the through-stage survival rate is often a function of time delays, it is thus easy to conceive that these models may involve some delay dependent parameters. The presence of such parameters often greatly complicates the task of an analytical study of such models. The objective of this talk is to provide practical guidelines that combine graphical information with analytical work to effectively study the local stability of some models involving delay-dependent parameters. Specifically, we shall show that the stability of a given steady state is simply determined by the graphs of some functions of the delay which can be expressed explicitly and thus can be easily depicted by Maple and other popular software. In fact, for most application problems, we need only look at some of such functions, locate their zeros, and determine the sign of their derivatives at the zeros. This function often has only two zeros, providing thresholds for stability switches (Geometric Stability Switch Criterion). The common scenario is that as time delay increases, stability changes from stable to unstable to stable, implying that a large delay can be stabilizing. This scenario often contradicts the one provided by similar models with only delay independent parameters. A simple application is discussed.

We consider some applications from population dynamics. The first model discussed is a model by Bence & Nisbet (1989) for a population of sessile invertebrates. The model in its simpler presentation leads to a first order characteristic equation with delay dependent parameters. We shall provide a negative criterion for the occurrence of stability switches and, for a fixed set of parameters, we discuss the conditions for the occurrrence of stability switches. A second model by Bence & Nisbet (1989) even on a population of sessile invertebrates leads to a second order characteristic equation to which we apply the geometric stability switch criterion.

As a further example of a second order characteristic equation we consider a model of bacteria-bacteriophage interaction by Beretta & Kuang (2001) discussing the local asymptotic stability of the positive equilibrium as function of the dealay in the meaningful parameter space. At last we will consider the limit case in which the model parameters are independent from the delay and the geometric stability switch criterion recovers some known results in literature (Freedman & Kuang (1991)).