Abstract Sufficient conditions are obtained for the existence of a globally attracting positive periodic solution of the “food-limited” population system modelled by the equation N (t) = r(t)((K(t) − N(t − mω)) (K(t) + c(t)r(t) N(t − mω))) , where m is a nonnegative integer and K, r, c are continuous positive periodic functions of period ω.

AbstractLetP(t) denote the density of mature cells in blood circulation. Mackey and Glass (1977) have proposed the following equations: $$\dot P(t) = \frac{{\beta _0 \theta ^n }}{{\theta ^n + [P(t - \tau )]^n }} - \gamma P(t)$$ and $$\dot P(t) = \frac{{\beta _0 \theta ^n P(t - \tau )}}{{\theta ^n + [P(t - \tau )]^n }} - \gamma P(t)$$ as models of hematopoiesis. We obtain sufficient and also necessary and sufficient conditions for all positive solutions to oscillate about their respective positive steady states. We also obtain sufficient conditions for the positive equilibrium to be a global attractor.

We obtained sufficient conditions for the oscillation of all positive solutions of the system Ni(t)=Niai−∑j=1mbijNj(t−τ) , i = 1, 2, …, m about its steady state. We also obtained sufficient conditions for the existence of a nonoscillatory solution of this system.

We established sufficient conditions for the global attractivity of the positive equilibrium of the delay differential equation N(t)=−mu;N(t)+∑i=1m pi exp[−γN(t−τi)], t⩾0, m⩾1. For m = 1, equation (1) was used by Wazewska-Czyzewska and Lasota as a model for the survival of red-blood cells in an animal.

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.

Abstract In this paper we are dealing with first-order differential equations and inequalities with a deviating argument and we give some new necessary, sufficient, and necessary and sufficient conditions concerning: (a) the comparison of oscillatory and asymptotic behavior of solutions of: (1) first-order vector differential equations and, corresponding to them, operator and scalar differential equations with a deviating argument and (2) first-order differential equations and, corresponding to them, inequalities with a deviating argument; and (b) the existence of a certain type of nonoscillatory solutions of differential equations and inequalities with a deviating argument. What is important, in our opinion, is the fact that some of the obtained results concern not only the cases of constant sign coefficients, but also the cases where the coefficients of equations and inequalities under consideration are of non-constant sign.

Received July 9, 1986 Consider the neutral delay differential equation $x(r)-P(f)x(f-r)]+Q(t)x(r-0)-O, tat,, (1) where P, Q E C[ [ I~, co), (w + ] and T, D E Iw +. We obtain sutlicient conditions for all solutions of Eq. (1) to oscillate. Our conditions are “sharp” in the sense than when the coefficients P and Q are constants the conditions are also necessary. We also obtain sufficient conditions for Eq. (1) to have a nonoscillatory solution when P(r) is a constant in the interval