Dynamics of the logistic equation with delay
This paper is about the dynamics of the equation \[ u'(t) = \lambda[1-\alpha u(t) -(1-\alpha)u(t-1)]u(t) \] for \(1 < \alpha <1\). It is the generalisation of the author's work on Hutchinson equation. It is known from the roots of the characteristic equation \(\mu = -\lambda[\alpha +(1-\alpha)e^{-\mu}]\) that all positive solutions tend to \(u_0 = 1\) as \(t\to\infty\) if \(\alpha > 1/2\). But if \(0< \alpha <1/2\), there is a bifurcation point \(\lambda = \lambda_0(\alpha)\). Under some conditions, it is shown that the equation has an asymptotically orbitally stable periodic solution close to \(u_0\). Further, global dynamics is investigated and asymptotics of the relaxation cycle for large \(\lambda\) is discussed.
scientific article; zbMATH DE number 6498243
scientific article