Deep chlorophyll maxima (DCMs) are widespread in large parts of the world's oceans. These deep layers of high chlorophyll concentration reflect a compromise of phytoplankton growth exposed to two opposing resource gradients: light supplied from above and nutrients supplied from below. It is often argued that DCMs are stable features. Here we show, however, that reduced vertical mixing can generate oscillations and chaos in phytoplankton biomass and species composition of DCMs. These fluctuations are caused by a difference in the timescales of two processes: (1) rapid export of sinking plankton, withdrawing nutrients from the euphotic zone and (2) a slow upward flux of nutrients fuelling new phytoplankton production. Climate models predict that global warming will reduce vertical mixing in the oceans. Our model indicates that reduced mixing will generate more variability in DCMs, thereby enhancing variability in oceanic primary production and in carbon export into the ocean interior.

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a Modelling, Analysis and Simulation Modelling, Analysis and SimulationBlooming in a non-local, coupled phytoplankton-nutrient model A. Zagaris, A. Doelman, N.N. Pham Thi, B.P. Sommeijer REPORT MAS-E0708 JUNE 2007Centrum voor Wiskunde en Informatica (CWI) is the national research institute for Mathematics and Computer Science. It is sponsored by the Netherlands Organisation for Scientific Research (NWO). CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics.CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms. Blooming in a non-local, coupled phytoplankton-nutrient model ABSTRACT 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 (2006), Reduced mixing generates oscillations and chaos in the oceanic deep chlorophyll maximum, Nature 439 322-325]. This paper is a first step towards understanding the bifurcational structure associated to non-local, 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. . This paper is a first step towards understanding the bifurcational structure associated to non-local, 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 t...

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C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and Simulation AbstractPositivity results are derived for explicit two-step methods formulated in linear multistep form and in one-leg form. It turns out that the latter formulation allows a slightly larger step size with respect to positivity.

On the basis of an application from aquatic ecology, we discuss the behaviour of the widely used time integration package VODE by Brown et al. [2]. When used in a default setting this code smoothly produces a negative steady state solution, which is not realistic in this application.