Delay Induced Bifurcation to Periodicity

Abstract

This chapter is intended to provide a reasonably self-contained demonstration of the various computations associated with Hopf-bifurcation in delay differential equations in which delay becomes a bifurcation parameter. We begin with a brief motivation. In most biological populations, the accumulation of metabolic products may seriously inconvenience a population and one of the consequences can be a fall in the birth rate and an increase in the mortality rate. If we assume (see Volterra [1931]) that the total toxic action on birth and death rates is expressed by an integral term in the logistic equation, one can then, consider the following integrodifferential equation

$$\frac{dN(t)}{dt}=rN(t)-cN(t)\left ( \int_{0}^{\infty } K(s)N(t-s))ds\right )^n$$

(2.1.1)

where K denotes the residual intensity of pollution and n ∈ (0, ∞). A model related to (2.1.1) in theme has been numerically studied by Borsellino and Torre [1974]. In order to derive the qualitative findings of Borsellino and Torre by means of an analytically manageable model, Cushing [1977] has proposed a model of the form

$$\frac{dN(t))}{dt}=N(t)\left \{ \alpha -\beta N(t)-\gamma \left ( \int_{0}^{\infty } K(s)N(t-s)ds\right )^2 \right \}$$

(2.1.2)

where α, β, γ are positive constants and

$$K(s)=\frac{1}{\tau }s\:\textup{exp}(-s/\tau ), \tau >0.$$

(2.1.3)