Juan Perán

Dynamics of a Rational System of Difference Equations in the Plane

We consider a rational system of first-order difference equations in the plane with four parameters such that all fractions have a common denominator. We study, for the different values of the parameters, the global and local properties of the system. In particular, we discuss the boundedness and the asymptotic behavior of the solutions, the existence of periodic solutions, and the stability of equilibria.

Global stability of an age-structured population model

We consider a nonlinear discrete-time population model for the dynamics of an age-structured species. This model has the form of a Lure feedback system (well-known in control theory) and is a particular case of the system studied by Townley et al. in Townley et al. (2012). The main objective is to show that, in this case, the range of nonlinearities for which the existence of globally asymptotically stable non-zero equilibrium can be guaranteed is considerably larger than that in the main result in Townley et al. (2012). We illustrate our results with several biologically meaningful examples.

The antipodal mapping theorem and difference equations in Banach spaces

We employ the Borsuk–Krasnoselskii antipodal theorem to prove a new fixed point theorem in ordered Banach spaces. Then, the applicability of the result is shown by presenting sufficient conditions for the existence of solutions to initial value problems for first-order difference equations in Banach spaces. To prove that result we shall employ set valued analysis techniques.

Unconditional convergence of difference equations

We put forward the notion of unconditional convergence to an equilibrium of a difference equation. Roughly speaking, it means that can be constructed a wide family of higher order difference equations, which inherit the asymptotic behavior of the original difference equation. We present a sufficient condition for guaranteeing that a second-order difference equation possesses an unconditional stable attractor. Finally, we show how our results can be applied to two families of difference equations recently considered in the literature.MSC:39A11.

Development of Virtual Illumination Functions for Thermographic NDT

This study develops and analyses a novel mathematical approach for thermographic NDT data processing. This approach is based on the definition of a simplified 2D thermal diffusion model to which usual processing algorithms are applied to produce analysis functions, which considers information related to temperature values, material properties and heating conditions. The analysis of the developed functions has been conducted with thermal data computationally generated in order to provide broad variety of case studies regarding different types of defects as well as stimulations. The results obtained have demonstrated that these functions may provide information useful in defect characterisation.

Effect of harvest timing on the dynamics of the Ricker–Seno model

The moment of intervention is a key question in harvest programmes and is currently generating an increasing interest. However, little is known about its effect on the population stability. This lack of knowledge is greater in the case of global stability, which is always desirable as it allows to predict the fate of populations regardless of the initial size. Here we use a discrete-time equation to model the dynamics of populations harvested at any time during the reproductive season. We study the effect of the time of intervention on the global stability of populations governed by the Ricker model, which is one of the most relevant models in discrete-time population dynamics. We prove that harvest timing never has a negative effect on the global stability of these populations, extending recent results in the literature. We also study the effect of delayed harvesting on the constancy stability of the controlled populations.

Locally compact multivector extensions

Abstract The aim of this paper is to develop a locally compact extension of an arbitrary normed space in such a way that the initial algebraic structure is prolonged in some sense. To obtain such an extension, we weaken vector space axioms allowing a set-valued addition and introduce in this scheme a topological structure, by means of a hypertopology, and a compatible proximity. Finally, the locally compact multivector extension appears as an ultrafilter space. We also provide a Young measure related interpretation of these extensions when the normed space is an Lp space.