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The case of juvenile versus adult competitive effects on adult fertility in an agedependent population dynamics model introduced by Cushing and Li [1] is considered. It is shown that stronger juvenile competition stabilises the model, while low sensitivity of adult fertility to juvenile cohort sizes leads to destabilisation and oscillatory solutions. As a special case, there is included a form of Gurtin-MacCamy's equation for which no Hopf bifurcations are possible. We have tried to maintain a certain balance between analytical and numerical methods of investigation. The paper contains, therefore, a considerable amount of numerically computed illustrations.

We offer sufficient conditions for the oscillation of all solutions of the partial difference equations y(m+1,n)+β(m,n)y(m,n+1)−δ(m,n)y(m,n)+P(m,n,y(m−k,n−l)) = Q (m,n,y(m − k,n −l)) , and y(m+1,n)+β(m,n)y(m,n+1)−δ(m,n)y(m,n)+ ∑ i=1 τ P i (m,n,y(m − k i , n − l i )) = ∑ i=1 τ Q i (m,n,y(m − k i , n − l i )) . Several examples, which dwell upon the importance of our results, are also included.

The aim of this paper is twofold. First, we deal with the extension to the random framework of the piecewise Fröbenius method to solve Airy differential equations. This extension is based on mean square stochastic calculus. Second, we want to explore the capability to provide not only reliable approximations for both the average and the standard deviation functions associated to the solution stochastic process, but also to save computational time as it happens in dealing with the analogous problem in the deterministic scenario. This includes a comparison of the numerical results with respect to those obtained by other commonly used operational methods such as polynomial chaos and Monte Carlo simulations. To conduct this comparative study, we have chosen the Airy random differential equation because it has highly oscillatory solutions. This feature allows us to emphasize differences between all the considered approaches. © 2011 Elsevier Ltd. All rights reserved., This work has been partially supported by the Spanish M.C.Y.T. and FEDER grants MTM2009-08587, DPI2010-20891-C02-01 as well as the Universitat Politecnica de Valencia grant PAID-06-09 (Ref. 2588).

For the half-linear difference equation Δ[|Δy(k)|α−1Δy(k)]=∑i=1npi(k)|y(gi(k))|α−1y(gi(k)),k≥a where α > 0, we shall offer sufficient conditions for the oscillation of all solutions, as well as necessary and sufficient conditions for the existence of both bounded and unbounded nonoscillatory solutions. Several examples which dwell upon the importance of our results are also included.

Consider the higher-order linear difference equation where m ≥ 1 is an odd integer, and { p n } is a sequence of nonnegative real numbers. We obtain several new sufficient conditions for the oscillation of all solutions of equation (*). Examples which dwell upon the importance of our results are also included.