Population growth and persistence in a heterogeneous environment: the role of diffusion and advection

The spatio-temporal dynamics of a population present one of the most fascinating aspects and challenges for ecological modelling. In this article we review some simple mathematical models, based on one dimensional reaction-diffusion-advection equations, for the growth of a population on a heterogeneous habitat. Considering a number of models of increasing complexity we investigate the often contrary roles of advection and diffusion for the persistence of the population. When it is possible we demonstrate basic mathematical techniques and give the critical conditions providing the survival of a population, in simple systems and in more complex resource-consumer models which describe the dynamics of phytoplankton in a water column.Comment: Introductory review of simple conceptual models. 45 pages, 15 figures v2: minor change

A high-resolution Petrov-Galerkin method for the convection-diffusion-reaction problem. Part II-A multidimensional extension

A multidimensional extension of the HRPG method using the lowest order block finite elements is presented. First, we design a nondimensional element number that quantifies the characteristic layers which are found only in higher dimensions. This is done by matching the width of the characteristic layers to the width of the parabolic layers found for a fictitious 1D reaction–diffusion problem. The nondimensional element number is then defined using this fictitious reaction coefficient, the diffusion coefficient and an appropriate element size. Next, we introduce anisotropic element length vectors li and the stabilization parameters αi, βi are calculated along these li. Except for the modification to include the new dimensionless number that quantifies the characteristic layers, the definitions of αi, βi are a direct extension of their counterparts in 1D. Using αi, βiand li, objective characteristic tensors associated with the HRPG method are defined. The numerical artifacts across the characteristic layers are manifested as the Gibbs phenomenon. Hence, we treat them just like the artifacts formed across the parabolic layers in the reaction-dominant case. Several 2D examples are presented that support the design objective—stabilization with high-resolutio

CFD modeling of a fixed-bed biofilm reactor coupling hydrodynamics and biokinetics

Peer ReviewedPostprint (author's final draft

Modeling and Simulation of Thermo-Fluid-Electrochemical Ion Flow in Biological Channels

In this article we address the study of ion charge transport in the biological channels separating the intra and extracellular regions of a cell. The focus of the investigation is devoted to including thermal driving forces in the well-known velocity-extended Poisson-Nernst-Planck (vPNP) electrodiffusion model. Two extensions of the vPNP system are proposed: the velocity-extended Thermo-Hydrodynamic model (vTHD) and the velocity-extended Electro-Thermal model (vET). Both formulations are based on the principles of conservation of mass, momentum and energy, and collapse into the vPNP model under thermodynamical equilibrium conditions. Upon introducing a suitable one-dimensional geometrical representation of the channel, we discuss appropriate boundary conditions that depend only on effectively accessible measurable quantities. Then, we describe the novel models, the solution map used to iteratively solve them, and the mixed-hybrid flux-conservative stabilized finite element scheme used to discretize the linearized equations. Finally, we successfully apply our computational algorithms to the simulation of two different realistic biological channels: 1) the Gramicidin-A channel considered in~\cite{JeromeBPJ}; and 2) the bipolar nanofluidic diode considered in~\cite{Siwy7}

Two-dimensional hydrodynamic core-collapse supernova simulations with spectral neutrino transport. I. Numerical method and results for a 15 M_sun star

Supernova models with a full spectral treatment of the neutrino transport are presented, employing the Prometheus/Vertex neutrino-hydrodynamics code with a ``ray-by-ray plus'' approximation for treating two- (or three-) dimensional problems. The method is described in detail and critically assessed with respect to its capabilities, limitations, and inaccuracies in the context of supernova simulations. In this first paper of a series, 1D and 2D core-collapse calculations for a (nonrotating) 15 M_sun star are discussed, uncertainties in the treatment of the equation of state -- numerical and physical -- are tested, Newtonian results are compared with simulations using a general relativistic potential, bremsstrahlung and interactions of neutrinos of different flavors are investigated, and the standard approximation in neutrino-nucleon interactions with zero energy transfer is replaced by rates that include corrections due to nucleon recoil, thermal motions, weak magnetism, and nucleon correlations. Models with the full implementation of the ``ray-by-ray plus'' spectral transport were found not to explode, neither in spherical symmetry nor in 2D with a 90 degree lateral wedge. The success of previous 2D simulations with grey, flux-limited neutrino diffusion can therefore not be confirmed. Omitting the radial velocity terms in the neutrino momentum equation leads to ``artificial'' explosions by increasing the neutrino energy density in the convective gain layer by about 20--30% and thus the integral neutrino energy deposition in this region by about a factor of two. (abbreviated)Comment: 46 pages plus 13 pages online material; 49 figures; referee's comments included, version accepted by Astronomy & Astrophysic