Blooming in a non-local, coupled phytoplankton-nutrient model

Recently, it has been discovered that the dynamics of phytoplankton concentrations in an ocean exhibit a rich variety of patterns, ranging from trivial states to oscillating and even chaotic behavior [J. Huisman, N. N. Pham Thi, D. M. Karl, and B. P. Sommeijer, Nature, 439 (2006), pp. 322–325]. This paper is a first step towards understanding the bifurcational structure associated with nonlocal coupled phytoplankton-nutrient models as studied in that paper. Its main subject is the linear stability analysis that governs the occurrence of the first nontrivial stationary patterns, the deep chlorophyll maxima (DCMs) and the benthic layers (BLs). Since the model can be scaled into a system with a natural singularly perturbed nature, and since the associated eigenvalue problem decouples into a problem of Sturm–Liouville type, it is possible to obtain explicit (and rigorous) bounds on, and accurate approximations of, the eigenvalues. The analysis yields bifurcation-manifolds in parameter space, of which the existence, position, and nature are confirmed by numerical simulations. Moreover, it follows from the simulations and the results on the eigenvalue problem that the asymptotic linear analysis may also serve as a foundation for the secondary bifurcations, such as the oscillating DCMs, exhibited by the model

Thailand\u27s State Enterprise Labor Relations Act: Denying Public Employees the Right of Association and the Right to Organize and Bargain Collectively

On April 15, 1991, Thailand\u27s new legislative body enacted the State Enterprise Labor Relations Act, removing public employees from the dominion of the Labor Relations Act and dissolving the existing public labor unions. This Act has had a crippling effect on the entire Thai labor movement, which historically relied on the leadership and influence of public unions to promote private industry worker interests. This Comment argues that the State Enterprise Labor Relations Act contains many provisions which violate internationally accepted labor standards, specifically the right of association and the right to organize and bargain collectively. This Comment further asserts that the Act should be amended to conform with these standards so that it meets the needs of both the Thai government and public enterprise workers

Singularly perturbed and non-local modulation equations for systems with interacting instability mechanisms

Two related systems of coupled modulation equations are studied and compared in this paper. The modulation equations are derived for a certain class of basic systems which are subject to two distinct, interacting, destabilizing mechanisms. We assume that, near criticality, the ratio of the widths of the unstable wavenumber-intervals of the two (weakly) unstable modes is small { as for instance can be the case in double-layer convection. Based on these assumptions we rst derive a singularly perturbed modulation equation and then a modulation equation with a non-local term. The reduction of the singularly perturbed system to the non-local system can be interpreted as a limit in which the width of the smallest unstable interval vanishes. We study and compare the behaviour of the stationary solutions of both systems. It is found that spatially periodic stationary solutions of the non-local system exist under the same conditions as spatially periodic stationary solutions of the singularly perturbed system. Moreover, these solutions can be interpreted as representing the same quasi-periodic patterns in the underlying basic system. Thus, the `Landau-reduction' to the non-local system has no signicant in uence on the stationary quasi-periodic patterns. However, a large variety of intricate heteroclinic and homoclinic connections is found for the singularly perturbed system. These orbits all correspond to so-called `localised structures' in the underlying system: they connect simple periodic patterns at x!1. None of these patterns can be described by the non-local system. So, one may conclude that the reduction to the non-local system destroys a rich and important set of patterns

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

Pattern formation in the 1-D Gray-Scott model

In this work, we analyze a pair of one-dimensional coupled reaction-diusion equations known as the Gray{Scott model, in which self-replicating patterns have been observed. We focus on stationary and traveling patterns, and begin by deriving the asymptotic scaling of the parameters and variables necessary for the analysis of these patterns. Single{pulse and multiple{pulse stationary waves are shown to exist in the appropriately{scaled equations on the innite line. A (single) pulse is a narrow interval in which the concentration U of one chemical is small, while that of the second, V , is large, and outside of which the concentration U tends (slowly) to the homogeneous steady state U 1, while V is everywhere close to V 0. In addition, we establish the existence of a plethora of periodic steady states consisting of periodic arrays of pulses interspersed by intervals in which the concentration V is exponentially small and U varies slowly. These periodic states are spatially inhomogeneous steady patterns whose length scales are determined exclusively by the reactions of the chemicals and their diusions, and not by other mechanisms such as boundary conditions. A complete bifurcation study of these solutions is presented. We also establish the non-existence of traveling solitary pulses in this system. This non-existence result re ects the system's degeneracy and indicates that some event, for example pulse-splitting, `must' occur when a pair of pulses moving apart from each other (as has been observed in simulations): these pulses evolve towards the non-existent traveling solitary pulses. The main mathematical techniques employed in this analysis of the stationary and traveling patterns are geometric singular perturbation theory and adiabatic Melnikov theory. Finally, the theoretical results are compared to those obtained from direct numerical simulation of the coupled partial dierential equations on a `very large' domain, using a moving grid code. It has been checked that the boundaries do not in uence the dynamics. A subset of the family of stationary single pulses appears to be stable. This subset determines the boundary of a region in parameter space in which the self-replicating process takes place. In that region, we observe that the core of a time-dependent self-replicating pattern turns out to be precisely a stationary periodic pulse-pattern of the type that we construct. Moreover, the simulations reveal some other essential components of the pulse-splitting process and provide an important guide to further analysis

Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations

Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian rate of spatially periodic traveling-waves of systems of reaction diffusion equations. In the case that wave-speed is identically zero for all periodic solutions, we recover and slightly sharpen a well-known result of Schneider obtained by renormalization/Bloch transform techniques; by the same arguments, we are able to treat the open case of nonzero wave-speeds to which Schneider's renormalization techniques do not appear to appl