Stability of cluster solutions in a cooperative consumer chain model

This is the author's accepted manuscript. The final published article is available from the link below. Copyright @ Springer-Verlag Berlin Heidelberg 2012.We study a cooperative consumer chain model which consists of one producer and two consumers. It is an extension of the Schnakenberg model suggested in Gierer and Meinhardt [Kybernetik (Berlin), 12:30-39, 1972] and Schnakenberg (J Theor Biol, 81:389-400, 1979) for which there is only one producer and one consumer. In this consumer chain model there is a middle component which plays a hybrid role: it acts both as consumer and as producer. It is assumed that the producer diffuses much faster than the first consumer and the first consumer much faster than the second consumer. The system also serves as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir. In the small diffusion limit we construct cluster solutions in an interval which have the following properties: The spatial profile of the third component is a spike. The profile for the middle component is that of two partial spikes connected by a thin transition layer. The first component in leading order is given by a Green's function. In this profile multiple scales are involved: The spikes for the middle component are on the small scale, the spike for the third on the very small scale, the width of the transition layer for the middle component is between the small and the very small scale. The first component acts on the large scale. To the best of our knowledge, this type of spiky pattern has never before been studied rigorously. It is shown that, if the feedrates are small enough, there exist two such patterns which differ by their amplitudes.We also study the stability properties of these cluster solutions. We use a rigorous analysis to investigate the linearized operator around cluster solutions which is based on nonlocal eigenvalue problems and rigorous asymptotic analysis. The following result is established: If the time-relaxation constants are small enough, one cluster solution is stable and the other one is unstable. The instability arises through large eigenvalues of order O(1). Further, there are small eigenvalues of order o(1) which do not cause any instabilities. Our approach requires some new ideas: (i) The analysis of the large eigenvalues of order O(1) leads to a novel system of nonlocal eigenvalue problems with inhomogeneous Robin boundary conditions whose stability properties have been investigated rigorously. (ii) The analysis of the small eigenvalues of order o(1) needs a careful study of the interaction of two small length scales and is based on a suitable inner/outer expansion with rigorous error analysis. It is found that the order of these small eigenvalues is given by the smallest diffusion constant ε22.RGC of Hong Kon

Existence and Stability of a Spike in the Central Component for a Consumer Chain Model

We study a three-component consumer chain model which is based on Schnakenberg type kinetics. In this model there is one consumer feeding on the producer and a second consumer feeding on the first consumer. This means that the first consumer (central component) plays a hybrid role: it acts both as consumer and producer. The model is an extension of the Schnakenberg model suggested in \cite{gm,schn1} for which there is only one producer and one consumer. It is assumed that both the producer and second consumer diffuse much faster than the central component. We construct single spike solutions on an interval for which the profile of the first consumer is that of a spike. The profiles of the producer and the second consumer only vary on a much larger spatial scale due to faster diffusion of these components. It is shown that there exist two different single spike solutions if the feed rates are small enough: a large-amplitude and a small-amplitude spike. We study the stability properties of these solutions in terms of the system parameters. We use a rigorous analysis for the linearized operator around single spike solutions based on nonlocal eigenvalue problems. The following result is established: If the time-relaxation constants for both producer and second consumer vanish, the large-amplitude spike solution is stable and the small-amplitude spike solution is unstable. We also derive results on the stability of solutions when these two time-relaxation constants are small. We show a new effect: if the time-relaxation constant of the second consumer is very small, the large-amplitude spike solution becomes unstable. To the best of our knowledge this phenomenon has not been observed before for the stability of spike patterns. It seems that this behavior is not possible for two-component reaction-diffusion systems but that at least three components are required. Our main motivation to study this system is mathematical since the novel interaction of a spike in the central component with two other components results in new types of conditions for the existence and stability of a spike. This model is realistic if several assumptions are made: (i) cooperation of consumers is prevalent in the system, (ii) the producer and the second consumer diffuse much faster than the first consumer, and (iii) there is practically an unlimited pool of producer. The first assumption has been proven to be correct in many types of consumer groups or populations, the second assumption occurs if the central component has a much smaller mobility than the other two, the third assumption is realistic if the consumers do not feel the impact of the limited amount of producer due to its large quantity. This chain model plays a role in population biology, where consumer and producer are often called predator and prey. This system can also be used as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir

Railway bridge structural health monitoring and fault detection: state-of-the-art methods and future challenges

Railway importance in the transportation industry is increasing continuously, due to the growing demand of both passenger travel and transportation of goods. However, more than 35% of the 300,000 railway bridges across Europe are over 100-years old, and their reliability directly impacts the reliability of the railway network. This increased demand may lead to higher risk associated with their unexpected failures, resulting safety hazards to passengers and increased whole life cycle cost of the asset. Consequently, one of the most important aspects of evaluation of the reliability of the overall railway transport system is bridge structural health monitoring, which can monitor the health state of the bridge by allowing an early detection of failures. Therefore, a fast, safe and cost-effective recovery of the optimal health state of the bridge, where the levels of element degradation or failure are maintained efficiently, can be achieved. In this article, after an introduction to the desired features of structural health monitoring, a review of the most commonly adopted bridge fault detection methods is presented. Mainly, the analysis focuses on model-based finite element updating strategies, non-model-based (data-driven) fault detection methods, such as artificial neural network, and Bayesian belief network–based structural health monitoring methods. A comparative study, which aims to discuss and compare the performance of the reviewed types of structural health monitoring methods, is then presented by analysing a short-span steel structure of a railway bridge. Opportunities and future challenges of the fault detection methods of railway bridges are highlighted

Anemia and iron homeostasis in a cohort of HIV-infected patients in Indonesia

Contains fulltext : 97632.pdf (publisher's version ) (Open Access)BACKGROUND: Anemia is a common clinical finding in HIV-infected patients and iron deficiency or redistribution may contribute to the development of low hemoglobin levels. Iron overload is associated with a poor prognosis in HIV and Hepatitis C virus infections. Iron redistribution may be caused by inflammation but possibly also by hepatitis C co-infection. We examined the prevalence of anemia and its relation to mortality in a cohort of HIV patients in a setting where injecting drug use (IDU) is a main mode of HIV transmission, and measured serum ferritin and sTfR, in relation to anemia, inflammation, stage of HIV disease, ART and HCV infection. METHODS: Patient characteristics, ART history and iron parameters were recorded from adult HIV patients presenting between September 2007 and August 2009 in the referral hospital for West Java, Indonesia. Kaplan-Meier estimates and Cox's regression were used to assess factors affecting survival. Logistic regression was used to identity parameters associated with high ferritin concentrations. RESULTS: Anemia was found in 49.6% of 611 ART-naive patients, with mild (Hb 10.5 -12.99 g/dL for men; and 10.5-11.99 g/dL for women) anemia in 62.0%, and moderate to severe anemia (Hb < 10.5 g/dL) in 38.0%. Anemia remained an independent factor associated with death, also after adjustment for CD4 count and ART (p = 0.008). Seroprevalence of HCV did not differ in patients with (56.9%) or without anemia (59.6%). Serum ferritin concentrations were elevated, especially in patients with anemia (p = 0.07) and/or low CD4 counts (p < 0.001), and were not related to hsCRP or HCV infection. Soluble TfR concentrations were low and not related to Hb, CD4, hsCRP or ART. CONCLUSION: HIV-associated anemia is common among HIV-infected patients in Indonesia and strongly related to mortality. High ferritin with low sTfR levels suggest that iron redistribution and low erythropoietic activity, rather than iron deficiency, contribute to anemia. Serum ferritin and sTfR should be used cautiously to assess iron status in patients with advanced HIV infection

Edge bifurcations in singularly perturbed reaction-diffusion equations: a case study

In this note we study the appearance and the relevace of edge bifurcations in the stability problem associated to a front pattern in a certain class of (singularly perturbed)bi-stable reaction-diffusion equations