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 Nicholson's blowflies     

Oscillation criteria for delay equations

This paper is concerned with the oscillatory behavior of first-order delay differential equations of the form

(1)

where is non-decreasing, for and . Let the numbers and be defined by

It is proved here that when and all solutions of Eq. (1) oscillate in several cases in which the condition

2k+\frac{2}{{\lambda}_{1}}-1 \end{displaymath}">

holds, where is the smaller root of the equation .

     

Oscillatory properties of strongly superlinear differential equations with deviating arguments

We regard a graph G as a set {1,…, v} together with a nonempty set E of two-element subsets of {1,…, v}. Let p = (p₁,…, p_v) be an element of $\text{R}$^nv representing v points in $\text{R}$ⁿ and consider the realization G(p) of G in $\text{R}$ⁿ consisting of the line segments [p_i, p_j] in $\text{R}$ⁿ for {i, j} ?E. The figure G(p) is said to be rigid in $\text{R}$ⁿ if every continuous path in $\text{R}$^nv, beginning at p and preserving the edge lengths of G(p), terminates at a point q ? $\text{R}$^nv which is the image (Tp₁,…, Tp_v) of p under an isometry T of $\text{R}$ⁿ. We here study the rigidity and infinitesimal rigidity of graphs, surfaces, and more general structures. A graph theoretic method for determining the rigidity of graphs in $\text{R}$² is discussed, followed by an examination of the rigidity of convex polyhedral surfaces in $\text{R}$³.     

Stable steady state of some population models

Applying an analytical method and several limiting equations arguments, some sufficient conditions are provided for the existence of a unique positive equilibriumK for the delay differential equationx=–x+D(x _t ), which is the general form of many population models. The results are concerned with the global attractivity, uniform stability, and uniform asymptotic stability ofK. Application of the results to some known population models, which shows the effectiveness of the methods applied here, is also presented.