Existence of a non-constant periodic solution of a non-linear autonomous functional differential equation representing the growth of a single species population.

We consider an integro-differential equation for the densityn of a single species population where the birth rate is constant and the death rate depends on the values ofn in an interval of length τ - 1 > 0. We prove the existence of a non-constant periodic solution under the conditions birth rate b > π/2 and τ- 1 small enough. The basic idea of proof (due to R. D. Nussbaum) is to employ a theorem about non-ejective fixed points for a translation operator associated with the solutions of the equation.