Asymptotics of relaxation cycles in the generalized logistic delay equation (Q6112929)

For a generalized logistic delay differential equation of the form \[\dot u(t)=\lambda [1-F(u(t- 1))]u(t),\] where \(\lambda\in\mathbb{R}\) and \(F\) is an increasing continuous function with \(A:=\lim_{u\to\infty}F(u)\in (1,\infty]\), the author studies the asymptotic behaviour of solutions. For a large parameter \(\lambda>0\) and \(A\in (1,\infty)\), the main result establishes the existence of an asymptotically orbitally stable slowly oscillating periodic solution, and estimates for the periods are provided. This result is achieved via a fixed point theorem and a biological interpretation is given.