G. Ladas

Environmental periodicity and time delays in a “food-limited” population model

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.

Explicit conditions for the oscillation of difference equations

We obtain sufficient conditions for the oscillation of all solutions of the difference equation nAn+1−An+∑i=1mpiAn−ki=0, n=0, 1, 2, … n, where the pis are real numbers and the kis are integers. The conditions are given explicitly in terms of the pis and the kis.

Global attractivity in nicholson's blowflies

We established sufficient conditions for the global attractivity of the positive equilibrium of the delay differential equation [Ndot](t) ≡ −δN(t) + PN(t–τ)e−aN(t–τ) which was used by Gurney, Blythe and Nisbet [1] in describing the dynamics of Nicholsons blowflies

Oscillations of first-order neutral delay differential equations

Abstract Consider the neutral delay differential equation (∗) ( d dt )[y(t) + py(t − τ)] + qy(t − σ) = 0, t ⩾ t 0 , where τ, q, and σ are positive constants, while p ϵ (−∞, −1) ∪ (0, + ∞). (For the case p ϵ [−1, 0] see Ladas and Sficas, Oscillations of neutral delay differential equations (to appear)). The following results are then proved. Theorem 1. Assume p σ, and q(σ − τ) (1 + p) > ( 1 e ) . Then every solution of Eq. (∗) oscillates. Theorems 3. Assume p > 0. Then every nonoscillatory solution y(t) of Eq. (∗) tends to zero as t → ∞. Theorem 4. Assume p > 0. Then a necessary condition for all solutions of Eq. (∗) to oscillate is that σ > τ. Theorem 5. Assume p > 0, σ > τ, and q(σ − τ) (1 + p) > ( 1 e ) . Then every solution of Eq. (∗) oscillates. Extensions of these results to equations with variable coefficients are also obtained.

Oscillation and asymptotic behavior of neutral differential equations with deviating arguments

Consider the neutral differential equation where q≠0, p, τ, and σ are real numbers. Let y(t) be a nonoscillatory solution of Eq. (1). Then limtt→∞y(t) is determined for all cases, except: . Two conjectures (as well as evidence indicating their possible validity) are given to cover the missing cases i), ii), and iii). It is also shown that if qτ≧0, or if qτ<0 and p≧0, then each of the following conditions implies that every solution of Eq. (1) is oscillatory: .

Global attractivity in nonlinear delay difference equations

We obtain a set of sufficient conditions under which all positive solutions of the nonlinear delay difference equation x n+1 =x n f(x n-k ), n=0, 1, 2, ..., are attracted to the positive equilibrium of the equation. Our result applies, for example, to the delay logistic model N t+1 =αN t /(1+βN t-k ) and to the delay difference equation x n+1 =x n exp(r(1-x n-k ))

Oscillations and global attractivity in models of hematopoiesis

AbstractLetP(t) denote the density of mature cells in blood circulation. Mackey and Glass (1977) have proposed the following equations:n

Stability of solutions of linear nonautonomous difference equations

dot P(t) = frac{{beta _0 theta ^n }}{{theta ^n + [P(t - tau )]^n }} - gamma P(t)