On a transcendental equation in the stability analysis of a population growth model

We consider the rate equation n = rn for the density n of a single species population in a constant environment. We assume only that there is a positive constant solution n*, that the rate of increase r depends on the history of n and that r decreases for great n. The stability properties of the solution n* depend on the location of the eigenvalues of the linearized functional differential equation. These eigenvalues are the complex solutions gamma of the equation gamma + alpha integral of 0-1exp[gamma a] ds (a) = 0 with alpha greater than 0 and s increasing, s(-1) = 0, s(0) = 1. We give conditions on alpha and s which ensure that all eigenvalues have negative real part, or that there are eigenvalues with positive real part. In the case of the simplest smooth function s (s = id + 1), we obtain a theorem which describes the distribution of all eigenvalues in the complex plane for every alpha greater than 0.