Let x(e:t) denote the arterial concentration of CO2. Mackey and Glass (1977) have proposed that x(t) is governed by the autonomous delay differential equation (*) where γ, β, z.Θ are positive parameters, τ is a non-negative delay and Vm denotes the maximum ‘ventilation’ rate of CO2. We obtain sufficient and also necessary and sufficient conditions for all positive solutions of (*) to oscillate about the positive equilibrium x * of (*). We also obtain sufficient conditions for x * to be a global attractor.

Consider the delay differential equation y(0 + ay(t) + Pf{y(t r)) = 0, (*) where a, /?, and r are positive constants and / is a continuous function such that uf(u) > 0 for u e [-A, B], 0, and lim = 1, v ; w—»o u where A and B are positive numbers. When /(«) = sinw, (*) is the so-called "sunflower" equation, which describes the motion of the tip of the sunflower plant. We obtain necessary and sufficient conditions for the oscillation of all solutions of (*), whose graph lies eventually in the strip R+ x [—A, B], in terms of the characteristic equation of the linearized equation z(t) + az(t) + f}z(t r) = 0.

We obtained sufficient and necessary and sufficient conditions for the oscillation of all positive solutions of where r, k, τ ∊ (0, ∞) and c ∊ (0,∞). We also obtained sufficient conditions for the global attractivity of the positive equilibrium K.

Abstract Consider the neutral delay differential equation where p ∈ R, τ ≥ 0, q1 > 0, σ1 ≥ 0, for i = 1, 2, …, k. We prove the following result. Theorem. A necessary and sufficient condition for the oscillation of all solutions of Eq. (1) is that the characteristic equation has no real roots.