Global convergence in systems of differential equations arising from chemical reaction networks

It is shown that certain classes of differential equations arising from the modelling of chemical reaction networks have the following property: the state space is foliated by invariant subspaces each of which contains a unique equilibrium which, in turn, attracts all initial conditions on the associated subspace.Comment: Some typos and minor errors from the previous version have been correcte

Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations

Producción CientíficaLinear skew-product semidynamical systems generated by random systems of delay differential equations are considered, both on a space of continuous functions as well as on a space of p-summable functions. The main result states that in both cases, the Lyapunov exponents are identical, and that the Oseledets decompositions are related by natural embeddings.NCN grant Maestro 2013/08/A/ST1/00275MICIIN/FEDER Grant RTI2018-096523-B-100H2020-MSCA-ITN-2014 643073 CRITICS

Two dynamical approaches to the notion of exponential separation for random systems of delay differential equations

This paper deals with the exponential separation of type II, an important concept for random systems of differential equations with delay, introduced in \JM\ et al.~\cite{MiNoOb1}. Two different approaches to its existence are presented. The state space $X$ will be a separable ordered Banach space with $\dim X\geq 2$, dual space $X^{*}$ and positive cone $X^+$ normal and reproducing. In both cases, appropriate cooperativity and irreducibility conditions are assumed to provide a family of generalized Floquet subspaces. If in addition $X^*$ is also separable, one obtains a exponential separation of type II. When this is not the case, but there is an Oseledets decomposition for the continuous semiflow, the same result holds. Detailed examples are given for all the situations, including also a case where the cone is not normal.Comment: arXiv admin note: text overlap with arXiv:1705.0131

Principal Floquet subspaces and exponential separations of type II with applications to random delay differential equations

Producción CientíficaThis paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type II, important for its application to random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type II in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear random delay differential equations with finite delay is given.2020-01-012020-01-01Ministerio de Economía, Industria y Competitividad - FEDER (Project MTM2015-66330-P

Linearization and invariant manifolds on the carrying simplex for competitive maps

A result due to M.W. Hirsch states that most competitive maps admit a carrying simplex, i.e., an invariant hypersurface of codimension one which attracts all nontrivial orbits. The common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex. However, this manifold is normally non-smooth. Therefore, not every tool coming from Differential Geometry can be applied. In this paper we prove that the restriction of the map to the carrying simplex in a neighborhood of an interior fixed point is topologically conjugate to the restriction of the map to its pseudo-unstable manifold by an invariant foliation. This implies that the linearization techniques are applicable for studying the local dynamics of the interior fixed points on the carrying simplex. We further construct the stable and unstable manifolds on the carrying simplex. Our results give partial responses to Hirsch's problem regarding the smoothness of the carrying simplex. We discuss some applications in classical models of population dynamics. (C) 2019 Elsevier Inc. All rights reserved.Peer reviewe