A Note on the Global Attractivity of a Discrete Model of Nicholson ’ s Blowflies *

In this paper, we further study the global attractivity of the positive equilibrium of the discrete Nicholson's blowflies model Nn

Corresponding author.
By a solution of (1) and (2) we mean a sequence {Nn} which satisfies (1) for n--0, 1,2,... as well as the initial condition (2). Clearly, the unique solution {Nn} of the above initial value problem is positive for all large n [ [3] and So and Yu [1] respectively.
The recent result is the following [1].
THEOREM A Assume that p > (5 and that Then any nontrival solution Nn of (1) and (2) satisfies lira Nn N*.
In this note, our purpose is to improve condition (4). Exactly speaking, we will show some conditions for the global attractivity of N* when (4) does not hold. Our results are discrete analogues of the results in [2]. To prove our main results, we need some known results.

II. MAIN RESULTS
The following theorem provides a new sufficient condition for the equilibrium N*-(I/a)ln(p/6) to be a global attractor. To prove this theorem, it is sufficient to prove limn_ox =0. Lemma implies that {xn} is bounded above. Let # lim sup x and A lim inf x. (9) Then -aN* < A _< # < oo. We claim that A-#-0. For the case {x} is eventually nonnegative or eventually nonpositive, this has been proved in the proof of Theorem 2 in [3]. Therefore it is sufficient to consider the case that {x} is an oscillatory solution of (8).
Rewriting Eq. (8) into the following form: Now summing (16) up from n-Qi-k-1 (assuming Qi-k-_> n*) to n-Qi-1. we have Substituting (15) into the above inequality, we get Remark 2 Theorem 4.1 in [1] only applies to the case M <_ 1, while Theorem in this paper not only applies to M< but also to M> 1. So the results in this paper improve those in [1]. The conditions in Theorem are satisfied. Thus is a global attractor or (22). But Theorem 4.1 in [1] cannot apply to this case.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points. Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable: