On a one-parameter family of delay population equations

Abstract

studied e.g. by Seifert [lo], where b > 0 and a is a positive parameter. It is known that, if b > 1, then any positive solution x(t) of (1.1) converges to a/(b + 1) as t + +a. If b c 1, then a Hopf bifurcation occurs at a critical value a, of the parameter a. Searching the Hopf bifurcation phenomenon it is equivalent to the investigation into the problem of showing the existence of nonconstant periodic solutions. In particular a large part of this investigation has been devoted to problems arising from current models of population growth in certain biological contexts (see, for example, Cushing [3, 41, Simpson [l 11, and the references therein). One approach in this regard has been to demonstrate that, under certain conditions, nonconstant periodic solutions can bifurcate from a constant steady state. In (1.1) this steady state is the point a/(b + 1) if b 1, then a/(b + 1) is the carrying capacity of the system. The performance of Hopf bifurcation can be guaranteed if the conditions imposed to the system are appropriate to lead to the (sufficient) conditions of the so-called Hopf bifurcation theorem: namely the noncreasonance condition as well as the generic branching condition. What these conditions are and how they can be applied to several situations the reader may consult, for instance, Chafee [2], Hale [5, Chapter 111, Hale et al. [6, p. 1071, Kozjakin and Krasnoselskii [7] and Zeidler [12, Part I, Section 8.15 and Part IV, Section 79.91. The purpose of the present paper is to discuss the nature of the positive solutions of a delay differential equation of the general form