The study of the fluctuations in the steady state of a heated granular system isreviewed. A Boltzmann-Langevin description can be built requiring consistency with theequations for the one- and two-particle correlation functions. From the Boltzmann-Langevinequation, Langevin equations for the total energy and the transverse velocity field arederived. The existence of a fluctuation-dissipation relation for the transverse velocityfield is also studied.

One interesting example of a discrete mathematical model used in biology is a food web.The first biology courses in high school and in college present the fundamental nature ofa food web, one that is understandable by students at all levels. But food webs as part ofa larger system are often not addressed. This paper presents materials that can be used inundergraduate classes in biology (and mathematics) and provides students with theopportunity to explore mathematical models of predator-prey relationships, determinetrophic levels, dominant species, stability of the ecosystem, competition graphs, intervalgraphs, and even confront problems that would appear to have logical answers that are asyet unsolved.

Plant growth occurs due to cell proliferation in the meristem. We model the case ofapical meristem specific for branch growth and the case of basal meristem specific forbulbous plants and grass. In the case of apical growth, our model allows us to describethe variety of plant forms and lifetimes, endogenous rhythms and apical domination. In thecase of basal growth, the spatial structure, which corresponds to the appearance ofleaves, results from dissipative instability of the homogeneous in space solution. Westudy nonlinear dynamics and wave propagation of the corresponding reaction-diffusionsystems. Bifurcation of periodic at infinity waves is investigated numerically.

The interplay between dissipation and long-range repulsive/attractive forces inhomogeneous, dilute, mono-disperse particle systems is studied. Thepseudo-Liouville operator formalism, originallyintroduced for hard-sphere interactions, is modified such that it provides very goodpredictions for systems with weak long-range forces at low densities, with the ratio ofpotential to fluctuation kinetic energy as control parameter. By numerical simulations,the theoretical results are generalized with empirical, density dependentcorrection-functions up to moderate densities.

The main result of this study on dissipative cooling is an analytical prediction for the reduced cooling rate due torepulsive forces and for the increased rate due to attractive forces. In the latter case,surprisingly, for intermediate densities, similar cooling behavior is observed as insystems without long-range interactions. In the attractive case, in general, dissipationleads to inhomogeneities earlier and faster than in the repulsive case.

A number of exciting new laboratory techniques have been developed using the Watson-Crick complementarity properties of DNA strands to achieve the self-assembly of graphical complexes. For all of these methods, an essential step in building the self-assembling nanostructure is designing the component molecular building blocks. These design strategy problems fall naturally into the realm of graph theory. We describe graph theoretical formalism for various construction methods, and then suggest several graph theory exercises to introduce this application into a standard undergraduate graph theory class. This application provides a natural framework for motivating central concepts such as degree sequence, Eulerian graphs, Fleury’s algorithm, trees, graph genus, paths, cycles, etc. There are many open questions associated with these applications which are accessible to students and offer the possibility of exciting undergraduate research experiences in applied graph theory.

A numerical model to compute the dynamics of glaciers is presented. Ice damage due tocracks or crevasses can be taken into account whenever needed. This model allowssimulations of the past and future retreat of glaciers, the calving process or thebreak-off of hanging glaciers. All these phenomena are strongly affected by climatechange.

Ice is assumed to behave as an incompressible fluid with nonlinear viscosity, so that thevelocity and pressure in the ice domain satisfy a nonlinear Stokes problem. The shape ofthe ice domain is defined using the volume fraction of ice, that is one in the ice regionand zero elsewhere. The volume fraction of ice satisfies a transport equation with asource term on the upper ice-air free surface accounting for ice accumulation or melting.If local effects due to ice damage must be taken into account, the damage functionD is introduced, ranging between zero if no damage occurs and one.Then, the ice viscosity μ in the momentum equation must be replaced by(1 − D)μ. The damage function Dsatisfies a transport equation with nonlinear source terms to model cracks formation orhealing.

A splitting scheme allows transport and diffusion phenomena to be decoupled. Two fixedgrids are used. The transport equations are solved on an unstructured grid of small cubiccells, thus allowing numerical diffusion of the volume fraction of ice to be reduced asmuch as possible. The nonlinear Stokes problem is solved on an unstructured mesh oftetrahedrons, larger than the cells, using stabilized finite elements.

Two computations are presented at different time scales. First, the dynamics ofRhonegletscher, Swiss Alps, are investigated in 3D from 2007 to 2100 using severalclimatic scenarios and without considering ice damage. Second, ice damage is taken intoaccount in order to reproduce the calving process of a 2D glacier tongue submerged bywater.

We discuss the issues of implementation of a higher order discontinuous Galerkin (DG)scheme for aerodynamics computations. In recent years a DG method has intensively beenstudied at Central Aerohydrodynamic Institute (TsAGI) where a computational code has beendesigned for numerical solution of the 3-D Euler and Navier-Stokes equations. Ourdiscussion is mainly based on the results of the DG study conducted in TsAGI incollaboration with the NUMECA International. The capacity of a DG scheme to tacklechallenging computational problems is demonstrated and its potential advantages over FVschemes widely used in modern computational aerodynamics are highlighted.

We derive a biomembrane model consisting of a fluid enclosed by a lipid membrane. Themembrane is characterized by its Canham-Helfrich energy (Willmore energy with areaconstraint) and acts as a boundary force on the Navier-Stokes system modeling anincompressible fluid. We give a concise description of the model and of the associatednumerical scheme. We provide numerical simulations with emphasis on the comparisonsbetween different types of flow: the geometric model which does not take into account thebulk fluid and the biomembrane model for two different regimes of parameters.

This paper demonstrates the development of a simple model of carbon flow during plant growth. The model was developed by six undergraduate students and their instructor as a project in a plant ecophysiology course. The paper describes the structure of the model including the equations that were used to implement it in Excel®, the plant growth experiments that were conducted to obtain information for parameterizing and testing the model, model performance, student responses to the modeling project, and potential uses of the model by other students.

Blood clotting system (BCS) modelling is an important issue with a plenty of applications in medicine and biophysics. The BCS main function is to form a localized clot at the site of injury preventing blood loss. Mutual influence of fibrin clot consisting mainly of fibrin polymer gel and blood flow is an important factor for BCS to function properly. The process of fibrin polymer mesh formation has not adequately been described by current mathematical models. That is why it is not possible to define the borders of growing clot and model its interaction with a blood flow. This paper main goal is to propose physically well-founded mathematical model of fibrin polymerization and gelation. The proposed model defines the total length of fibrin polymer fibers in the unit volume, determines a position of the border between gel and liquid and allows to evaluate the permeability of growing gel. Without significant structural changes the proposed model could be modified to include the blood shear rate influence on the fibrin polymerization and gelation.

This special issue of Mathematical Modelling of Natural Phenomena on biomathematics education shares the work of fifteen groups at as many different institutions that have developed beautiful biological applications of mathematics that are different in three ways from much of what is currently available. First, many of these selections utilize current research in biomathematics rather than the well-known textbook examples that are at least a half-century old. Second, the selections focus on modules that are intended for instant classroom adoption, adaptation, and implementation. Instead of focusing on how to overcome the challenges of implementing mathematics into biology or biology into mathematics or on educational research on the effectiveness of some small implementation, the authors develop individual biological models sufficiently well such that they can be easily adopted and adapted for use in both mathematics and biology classrooms. A third difference in this collection is the substantive inclusion of discrete mathematics and innovative pedagogies. Because calculus-based models have received the majority of the biomathematics modeling attention until very recently, the focus on discrete models may seem surprising. The examples range from DNA nanostructures through viral capsids to neuronal processes and ecosystem problems. Furthermore, a taxonomy of quantitative reasoning and the role of modeling per se as a different practice are contextualized in contemporary biomathematics education.

We present a novel, cell-local shock detector for use with discontinuous Galerkin (DG)methods. The output of this detector is a reliably scaled, element-wise smoothnessestimate which is suited as a control input to a shock capture mechanism. Using anartificial viscosity in the latter role, we obtain a DG scheme for the numerical solutionof nonlinear systems of conservation laws. Building on work by Persson and Peraire, wethoroughly justify the detector’s design and analyze its performance on a number ofbenchmark problems. We further explain the scaling and smoothing steps necessary to turnthe output of the detector into a local, artificial viscosity. We close by providing anextensive array of numerical tests of the detector in use.

This paper is concerned with the numerical simulation of a thermodynamically compatibleviscoelastic shear-thinning fluid model, particularly well suited to describe therheological response of blood, under physiological conditions. Numerical simulations areperformed in two idealized three-dimensional geometries, a stenosis and a curved vessel,to investigate the combined effects of flow inertia, viscosity and viscoelasticity inthese geometries. The aim of this work is to provide new insights into the modeling andsimulation of homogeneous rheological models for blood and a basis for furtherdevelopments in modeling and prediction.

The space missions Voyager and Cassini together with earthbound observations revealed a wealth of structures in Saturn’s rings. There are, for example, waves being excited at ring positions which are in orbital resonance with Saturn’s moons. Other structures can be assigned to embedded moons like empty gaps, moon induced wakes or S-shaped propeller features. Furthermore, irregular radial structures are observed in the range from 10 meters until kilometers. Here some of these structures will be discussed in the frame of hydrodynamical modeling of Saturn’s dense rings. For this purpose we will characterize the physical properties of the ring particle ensemble by mean field quantities and point to the special behavior of the transport coefficients. We show that unperturbed rings can become unstable and how diffusion acts in the rings. Additionally, the alternative streamline formalism is introduced to describe perturbed regions of dense rings with applications to the wake damping and the dispersion relation of the density waves.

Pharmacokinetics is an excellent way to introduce biomathematical modeling at the sophomore level. Students have the opportunity to develop a mathematical model of a biological phenomenon to which they all can relate. Exploring pharmacokinetics takes students through the necessary stages of mathematical modeling: determining the goals of the model, deciphering between the biological aspects to include in the model, defining the assumptions of the model, and finally, building, analyzing, using, and refining the model to answer questions and test hypotheses. Readily accessible data allows students to use the model to test hypotheses that are meaningful to them on an individual level. Students make interdisciplinary connections between this model and their previous personal, mathematical, and other classroom experiences. By beginning with a simple model involving the half-life of a drug, students take advantage of their mathematical abilities to explore the biology. They can then use the new knowledge gained from analyzing the simple model to create more complicated models, thus gaining mathematical and modeling maturity through improving the biological accuracy of the model. Through this experiences, students actually get to do applied mathematics, and they take ownership of the model.

In this article a variational reduction method, how to handle the case of heterogenousdomains for the Transport equation, is presented. This method allows to get rid of therestrictions on the size of time steps due to the thin parts of the domain. In the thinpart of the domain, only a differential problem, with respect to the space variable, is tobe approximated numerically. Numerical results are presented with a simple example. Thevariational reduction method can be extended to thin domains multi-branching in 3dimensions, which is a work in progress.

The paper is devoted to the method of computer simulation of protein interactions takingpart in photosynthetic electron transport reactions. Using this method we have studiedkinetic characteristics of protein-protein complex formation for four pairs of proteinsinvolved in photosynthesis at a variety of ionic strength values. Computer simulationsdescribe non-monotonic dependences of complex formation rates on the ionic strength as theresult of long-range electrostatic interactions. Calculations confirm that the decrease inthe association second order rate constant at low values of the ionic strength is causedby the protein pairs spending more time in “wrong” orientations which do not satisfy thedocking conditions and so do not form the final complex capable of the electrontransfer.

A hybrid model of red blood cell production, where cells are considered as discreteobjects while intra-cellular proteins and extra-cellular biochemical substances aredescribed with continuous models, is proposed. Spatial organization and regulation of redblood cell production (erythropoiesis) are investigated. Normal erythropoiesis issimulated in two dimensions, and the influence on the output of the model of someparameters involved in cell fate (differentiation, self-renewal, and death by apoptosis)is studied.

Linear oscillators are used for modeling a diverse array of natural systems, for instance acoustics, materials science, and chemical spectroscopy. In this paper I describe simple models of structural interactions in biological molecules, known as elastic network models, as a useful topic for undergraduate biology instruction in mathematical modeling. These models use coupled linear oscillators to model the fluctuations of molecular structures around the equilibrium state. I present many learning activities associated with building and understanding these models, ranging from analytical to computational. I provide a number of web resources where students can obtain structural data, perform calculations, and suggest research directions for independent projects.