Fractal dimension of a random invariant set

AbstractIn recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a parametrization of a ‘finite-dimensional’ set by a finite number of coordinates a bound on the fractal (upper box-counting) dimension is required. There are non-trivial problems in extending Debussche's techniques to this case, which can be overcome by careful use of the Poincaré recurrence theorem. We prove that under the same conditions as in Debussche's paper and an additional concavity assumption, the fractal dimension enjoys the same bound as the Hausdorff dimension. We apply our theorem to the 2d Navier–Stokes equations with additive noise, and give two results that allow different long-time states to be distinguished by a finite number of observations

Structural stability of invasion graphs for generalized Lotka--Volterra systems

In this paper we study in detail the structure of the global attractor for a generalized Lotka-Volterra system with Volterra--Lyapunov stable structural matrix. We provide the full characterization of this structure and we show that it coincides with the invasion graph as recently introduced in [15]. We also study the stability of the structure with respect to the perturbation of the problem parameters. This allows us to introduce a definition of structural stability in Ecology in coherence with the classical mathematical concept where there exists a detailed geometrical structure, governing the transient and asymptotic dynamics, which is robust under perturbation.Comment: Declaration on the lack of competing interest has been adde

Lower semicontinuity of attractors for non-autonomous dynamical systems

This paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions

Forwards attractors for non-autonomous Lotka-Volterra cooperative systems: a detailed geometrical description

Non-autonomous differential equations exhibit a highly intricate dynamics, and various concepts have been introduced to describe their qualitative behavior. In general, it is rare to obtain time dependent invariant compact attracting sets when time goes to plus infinity. Moreover, there are only a few papers in the literature that explore the geometric structure of such sets. In this paper we investigate the long time behaviour of cooperative $n$-dimensional non-autonomous Lotka-Volterra systems is population dynamics. We provide sufficient conditions for the existence of a globally stable (forward in time) entire solution in which one species becomes extinct, or where all species except one become extinct. Furthermore, we obtain the precise geometrical structure of the non-autonomous forward attractor in one, two, and three dimensions by establishing heteroclinic connections between the globally stable solution and the semi-stable solutions in cases of species permanence and extinction. We believe that understanding time-dependent forward attractors paves the way for a comprehensive analysis of both transient and long-term behavior in non-autonomous phenomena

The effect of noise on the Chaffee-Infante Equation: a nonlinear case study

We investigate the effect of perturbing the Chafee-Infante scalar reaction diffusion equation, ut - [Delta]u = [beta]u - u3, by noise. While a single multiplicative Itô noise of sufficient intensity will stabilise the origin, its Stratonovich counterpart leaves the dimension of the attractor essentially unchanged. We then show that a collection of multiplicative Stratonovich terms can make the origin exponentially stable, while an additive noise of sufficient richness reduces the random attractor to a single point

Endoglin overexpression modulates cellular morphology, migration, and adhesion of mouse fibroblasts

10 p.-9 fig.-1 tab.Endoglin is the gene mutated in hereditary hemorrhagic telangiectasia type 1 (HHT1), a dominantly inherited vascular disorder. Endoglin glycoprotein is a component of the transforming growth factor type ß (TGF-ß) receptor system which is highly expressed by endothelial cells, and at lower levels on fibroblasts and smooth muscle cells, suggesting the involvement of these lineages in the HHT1 vascular dysplasia. Overexpression of endoglin in mouse NCTC929 fibroblasts led to decreased migration in chemotactic and wound healing assays, as well as changes in the cellular morphology. When plated on uncoated surfaces, endoglin transfectants formed intercellular clusters, endoglin being not specifically localized to the cell-cell junctions, but homogenously distributed on the cellular surface. Although the expression of α5ß1 integrin and of an activation epitope of ß1 integrin were unchanged, a polyclonal antibody to α5ß1 integrin was able to inhibit cluster formation, suggesting the involvement of integrin ligand/s. In fact, coating with fibronectin, laminin, or an RGD-containing 80 kDa fragment of fibronectin were able to prevent the cellular clustering. Furthermore, synthesis of plasminogen activator inhibitor 1 (PAI-1), and to a weak extent that of fibronectin, were inhibited in endoglin transfectants. Thus, the presence of endoglin in mouse NCTC929 fibroblasts is associated with reduced production of certain extracellular matrix (ECM) components, which might explain their altered morphology, migration and intercellular cluster formation.This work has been supported by grants from Camisión Interministerial de Ciencia yTecnología (CICYT-SAF97-0034 to C. Bernabéu, and CICYT-SAF97-0064-C03-02 to A. García-Pardo), Comunidad Autónoma de Madrid (CAM) and Biomed Program of the European Community (BMH4-CT95-0995 to C. Bernabéu).Peer reviewe

Effect of fermented soy beverage in aged female mice model

11 Pág.Soy beverage is a rich source of phytoestrogens isoflavones, with potential benefits on health. The effect of those compounds depends greatly on their bacterial metabolization into their aglycone forms. This study evaluated the health effects of two soy beverages, non-fermented (SB) and fermented with Bifidobacterium pseudocatenulatum INIA P815 (FSB), in acyclic and cyclic C57BL/6J aged female mice as a model of menopause and premenopause, respectively. SB and FSB treatments were administrated for 36 days and, subsequently, body weight, lipid and inflammatory profile and fertility were analyzed and compared. In addition, hepatic gene expression and faecal microbiota composition were also assessed. After fermentation, FSB presented a high content in the aglycones daidzein and genistein and a higher antioxidant activity. FSB treated cyclic mice showed a significant increase in the number of retrieved oocytes and zigotes. Differences in serum lipids were observed in triglycerides, which were lower in FSB than in SB groups. None of the treatments influenced the inflammatory profile or caused a dramatic change in the intestinal microbiota profile or hepatic gene expression in any of the groups. Our data showed that FSB provided greater health benefits than SB in lipid profile and fertility in cyclic mice. These beneficial effects could be attributed to the fermentation process, which produces more bioavailable and bioactive compounds, achieving a greater impact on health.This work was supported by projects RTA2017-00002-00-00, PID2020-11960RB-I00, and RTI2018-093548-B-I00 from the Spanish Ministry of Science and Innovation. We appreciate the help of Doctors Javier Ortego and Eva Calvo from CISA-CSIC in the analysis of inflammatory profile. We are grateful to the ICTAN (Institute of Food Science, Technology and Nutrition, Madrid, Spain) Analysis Services Unit for providing chromatography and mass spectrometry facilities.Peer reviewe