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 (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

Reduction for Michaelis-Menten-Henri kinetics in the presence of diffusion

The Michaelis-Menten-Henri (MMH) mechanism is one of the paradigm reaction mechanisms in biology and chemistry. In its simplest form, it involves a substrate that reacts (reversibly) with an enzyme, forming a complex which is transformed (irreversibly) into a product and the enzyme. Given these basic kinetics, a dimension reduction has traditionally been achieved in two steps, by using conservation relations to reduce the number of species and by exploiting the inherent fast-slow structure of the resulting equations. In the present article, we investigate how the dynamics change if the species are additionally allowed to diffuse. We study the two extreme regimes of large diffusivities and of small diffusivities, as well as an intermediate regime in which the time scale of diffusion is comparable to that of the fast reaction kinetics. We show that reduction is possible in each of these regimes, with the nature of the reduction being regime dependent. Our analysis relies on the classical method of matched asymptotic expansions to derive approximations for the solutions that are uniformly valid in space and time

Macroscopic and large scale phenomena coarse graining, mean field limits and ergodicity

This book is the offspring of a summer school school “Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity”, which was held in 2012 at the University of Twente, the Netherlands. The focus lies on mathematically rigorous methods for multiscale problems of physical origins. Each of the four book chapters is based on a set of lectures delivered at the school, yet all authors have expanded and refined their contributions. Francois Golse delivers a chapter on the dynamics of large particle systems in the mean field limit and surveys the most significant tools and methods to establish such limits with mathematical rigor. Golse discusses in depth a variety of examples, including Vlasov--Poisson and Vlasov--Maxwell systems. Lucia Scardia focuses on the rigorous derivation of macroscopic models using $\Gamma$-convergence, a more recent variational method, which has proved very powerful for problems in material science. Scardia illustrates this by various basic examples and a more advanced case study from dislocation theory. Alexander Mielke's contribution focuses on the multiscale modeling and rigorous analysis of generalized gradient systems through the new concept of evolutionary $\Gamma$-convergence. Numerous evocative examples are given, e.g., relating to periodic homogenization and the passage from viscous to dry friction. Martin Göll and Evgeny Verbitskiy conclude this volume, taking a dynamical systems and ergodic theory viewpoint. They review recent developments in the study of homoclinic points for certain discrete dynamical systems, relating to particle systems via ergodic properties of lattices configurations