This paper introduces a new consideration in the well known chemostat model of a one-species with a periodic input of single nutrient with period ω , which is described by a system of differential delay equations. The delay represents the interval time between the consumption of nutrient and its metabolization by the microbial species. We obtain a necessary and sufficient condition ensuring the existence of a positive periodic solution with period ω . Our proof is based firstly on the construction of a Poincaré type map associated to an ω -periodic integro-differential equation and secondly on the existence of zeroes of an appropriate function involving the fixed points of the above mentioned map, which is proved by using Whyburn’s Lemma combined with the Leray–Schauder degree. In addition, we obtain a uniqueness result for sufficiently small delays.

Two-point boundary value problems of Dirichlet type are investigated for a Ermakov-Painlevé II equation which arises out of a reduction of a three-ion electrodiffusion Nernst-Planck model system. In addition, it is shown how Ermakov invariants may be employed to solve a hybrid Ermakov-Painlevé II triad in terms of a solution of the single component integrable Ermakov-Painlevé II reduction. The latter is related to the classical Painlevé II equation.

Let 𝕋 = {a _n } _n ∪{0} be a time scale with zero Minkowski (or box) dimension, where {a _n } _n is a monotonically decreasing sequence converging to zero, and a ₁ = 1. In this paper, we find an upper bound for the eigenvalue counting function of the linear problem − u ^ΔΔ = λu ^σ, with Dirichlet boundary conditions. We obtain that the nth-eigenvalue is bounded below by . We show that the bound is optimal for the q-difference equations arising in quantum calculus.