LAUSR

In this paper we prove that the following delay differential equation $$\begin{aligned} \frac{d}{dt}x(t)=rx(t)\left( 1-\int _{0}^{1}x(t-s)ds\right) , \end{aligned}$$ d dt x ( t ) = r x ( t ) 1 - ∫ 0 1 x ( t - s ) d s , has a periodic solution of period two for $$r>\frac{\pi ^{2}}{2}$$ r > π 2 2 (when the steady state, $$x=1$$ x = 1 , is unstable). In order to find the periodic solution, we study an integrable system of ordinary differential equations, following the idea by Kaplan and Yorke (J Math Anal Appl 48:317–324, 1974 ). The periodic solution is expressed in terms of the Jacobi elliptic functions.