Emergence of steady and oscillatory localized structures in a phytoplankton-nutrient model

Co-limitation of marine phytoplankton growth by light and nutrient, both of which are essential for phytoplankton, leads to complex dynamic behavior and a wide array of coherent patterns. The building blocks of this array can be considered to be deep chlorophyll maxima, or DCMs, which are structures localized in a finite depth interior to the water column. From an ecological point of view, DCMs are evocative of a balance between the inflow of light from the water surface and of nutrients from the sediment. From a (linear) bifurcational point of view, they appear through a transcritical bifurcation in which the trivial, no-plankton steady state is destabilized. This article is devoted to the analytic investigation of the weakly nonlinear dynamics of these DCM patterns, and it has two overarching themes. The first of these concerns the fate of the destabilizing stationary DCM mode beyond the center manifold regime. Exploiting the natural singularly perturbed nature of the model, we derive an explicit reduced model of asymptotically high dimension which fully captures these dynamics. Our subsequent and fully detailed study of this model - which involves a subtle asymptotic analysis necessarily transgressing the boundaries of a local center manifold reduction - establishes that a stable DCM pattern indeed appears from a transcritical bifurcation. However, we also deduce that asymptotically close to the original destabilization, the DCM looses its stability in a secondary bifurcation of Hopf type. This is in agreement with indications from numerical simulations available in the literature. Employing the same methods, we also identify a much larger DCM pattern. The development of the method underpinning this work - which, we expect, shall prove useful for a larger class of models - forms the second theme of this article

Optimal and Robust Feedback Controller Estimation for a Vibrating Plate using Subspace Model Identification

This paper presents a method to estimate the H2 optimal and a robust feedback controller by means of Subspace Model Identification using the internal model control (IMC) approach. Using IMC an equivalent feed forward control problem is obtained, which is solved by the Causal Wiener filter for the H2 optimal controller. The robust variant, called the Cautious Wiener filter, optimizes the average performance w.r.t. probabilistic model errors. The identification of the Causal and Cautious Wiener filters are control-relevant. The method is illustrated by experiments on a 4-inputs 4-outputs vibrating plate with additional mass variation

Fully broadband vAPP coronagraphs enabling polarimetric high contrast imaging

We present designs for fully achromatic vector Apodizing Phase Plate (vAPP) coronagraphs, that implement low polarization leakage solutions and achromatic beam-splitting, enabling observations in broadband filters. The vAPP is a pupil plane optic, inducing the phase through the inherently achromatic geometric phase. We discuss various implementations of the broadband vAPP and set requirements on all the components of the broadband vAPP coronagraph to ensure that the leakage terms do not limit a raw contrast of 1E-5. Furthermore, we discuss superachromatic QWPs based of liquid crystals or quartz/MgF2 combinations, and several polarizer choices. As the implementation of the (broadband) vAPP coronagraph is fully based on polarization techniques, it can easily be extended to furnish polarimetry by adding another QWP before the coronagraph optic, which further enhances the contrast between the star and a polarized companion in reflected light. We outline several polarimetric vAPP system designs that could be easily implemented in existing instruments, e.g. SPHERE and SCExAO.Comment: 11 pages, 5 figures, presented at SPIE Astronomical Telescopes and Instrumentation 201

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

Child Maltreatment in Families Receiving Mandatory Versus Voluntary Child Protection Support:A Matched Cohort Study

Child safety is an important outcome of child protection services (CPSs); however, this is often assessed in terms of official registries (e.g., rereports). Little empirical evidence is available about how the frequency of child maltreatment changes during CPS intervention by using self-report measures. The present study evaluates the frequency of child maltreatment experienced by children receiving mandatory child protection support compared to carefully matched children receiving voluntary child protection support. The current study is part of an ongoing Dutch longitudinal study on family violence consisting of several cohorts with similar designs. Both parents and children reported on the frequency of child maltreatment using validated questionnaires at two timepoints, 12 months apart. To facilitate careful comparison, both groups were matched using propensity scores based on background variables, resulting in two groups of N = 178 children. GLMM analyses showed a significant decrease in the mean number of child maltreatment incidents over time in the total group. However, this decrease did not differ for children receiving mandatory and voluntary child protection support. The findings indicate that, despite possible motivational challenges in the mandatory group, mandatory child protection support elicits comparable results as voluntary support. Implications for further research and practice are discussed.</p

Phase Slips and the Eckhaus Instability

We consider the Ginzburg-Landau equation, $\partial_t u= \partial_x^2 u + u - u|u|^2$, with complex amplitude $u(x,t)$. We first analyze the phenomenon of phase slips as a consequence of the {\it local} shape of $u$. We next prove a {\it global} theorem about evolution from an Eckhaus unstable state, all the way to the limiting stable finite state, for periodic perturbations of Eckhaus unstable periodic initial data. Equipped with these results, we proceed to prove the corresponding phenomena for the fourth order Swift-Hohenberg equation, of which the Ginzburg-Landau equation is the amplitude approximation. This sheds light on how one should deal with local and global aspects of phase slips for this and many other similar systems.Comment: 22 pages, Postscript, A

Child Maltreatment in Families Receiving Mandatory Versus Voluntary Child Protection Support:A Matched Cohort Study

Child safety is an important outcome of child protection services (CPSs); however, this is often assessed in terms of official registries (e.g., rereports). Little empirical evidence is available about how the frequency of child maltreatment changes during CPS intervention by using self-report measures. The present study evaluates the frequency of child maltreatment experienced by children receiving mandatory child protection support compared to carefully matched children receiving voluntary child protection support. The current study is part of an ongoing Dutch longitudinal study on family violence consisting of several cohorts with similar designs. Both parents and children reported on the frequency of child maltreatment using validated questionnaires at two timepoints, 12 months apart. To facilitate careful comparison, both groups were matched using propensity scores based on background variables, resulting in two groups of N = 178 children. GLMM analyses showed a significant decrease in the mean number of child maltreatment incidents over time in the total group. However, this decrease did not differ for children receiving mandatory and voluntary child protection support. The findings indicate that, despite possible motivational challenges in the mandatory group, mandatory child protection support elicits comparable results as voluntary support. Implications for further research and practice are discussed.</p