We show how results by Diekmann et al. (2007) on the qualitative behaviour of solutionsof delay equations apply directly to a resource-consumer model with age-structured consumerpopulation.

We carried out a computational study of propagation speeds ofreaction-diffusion-advection fronts in three dimensional (3D) cellular andArnold-Beltrami-Childress (ABC) flows with Kolmogorov-Petrovsky-Piskunov(KPP)nonlinearity. The variational principle of front speeds reduces the problem to a principaleigenvalue calculation. An adaptive streamline diffusion finite element method is used inthe advection dominated regime. Numerical results showed that the front speeds areenhanced in cellular flows according to sublinear power lawO(δp),p ≈ 0.13, δ the flow intensity. In ABC flows however,the enhancement is O(δ) which can be attributed to thepresence of principal vortex tubes in the streamlines. Poincaré sections are used tovisualize and quantify the chaotic fractions of ABC flows in the phase space. The effectof chaotic streamlines of ABC flows on front speeds is studied by varying the threeparameters (a,b,c) of the ABC flows. Speed enhancement alongx direction is reduced as b (the parameter controlingthe flow variation along x) increases at fixed(a,c) > 0, more rapidly as the corresponding ABC streamlines becomemore chaotic.

The problem of transformation of quasimonochromatic wavetrains of surface gravity waveswith narrow spatial-temporal spectra on the bottom shelf is considered in the linearapproximation. By means of numerical modeling, the transmission and reflectioncoefficients are determined as functions of the depth ratio and wave number (frequency) ofan incident wave. The approximation formulae are proposed for the coefficients of wavetransformation. The characteristic features of these formulae are analyzed. It is shownthat the numerical results agree quite satisfactorily with the proposed approximationformulae.

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.

This contribution reviews the nonlinearstochastic properties of turbulent velocity and passive scalarintermittent fluctuations in Eulerian and Lagrangian turbulence.These properties are illustrated with original data sets of (i)velocity fluctuations collected in the field and in thelaboratory, and (ii) temperature, salinity and in vivofluorescence (a proxy of phytoplankton biomass, i.e. unicelledvegetals passively advected by turbulence) sampled from highlyturbulent coastal waters. The strength of three of the mostpopular models describing intermittent fluctuations (thelognormal, log-Lévy and log-Poisson models) to fit thedistribution of in vivo fluorescence has subsequently beencritically assessed. A theoretical formulation for the stochasticproperties of biologically active scalars is also provided andvalidated. Finally, the potential effect of the intermittentproperties of turbulent velocity fluctuations on processesrelevant to the life of plankton organisms are theoreticallyinvestigated. It is shown that the intermittent nature ofmicroscale turbulence may result in (i) a decrease in the rate ofnutrient fluxes towards non-motile phytoplankton cells (6-62 %),(ii) a decrease in the physical coagulation of phytoplankton cells(25-48 %) and in the subsequent phytoplankton aggregate volumes(22-41 %), and (iii) a decrease of the turbulence contribution topredator-prey encounter rates (25-50 %).

We are interested in optimizing the co-administration of two drugs for some acute myeloidleukemias (AML), and we are looking for in vitro protocols as a first step. This issue canbe formulated as an optimal control problem. The dynamics of leukemic cell populations inculture is given by age-structured partial differential equations, which can be reduced toa system of delay differential equations, and where the controls represent the action ofthe drugs. The objective function relies on eigenelements of the uncontrolled model and ongeneral relative entropy, with the idea to maximize the efficiency of the protocols. Theconstraints take into account the toxicity of the drugs. We present in this paper themodeling aspects, as well as theoretical and numerical results on the optimal controlproblem that we get.

In this paper, we study the numerical approximation of a size-structured population modelwhose dependency on the environment is managed by the evolution of a vital resource. Weshow that this is a difficult task: some numerical methods are not suitable for along-time integration. We analyze the reasons for the failure.

We review recent results on stability of traveling waves in partly parabolicreaction-diffusion systems with stable or marginally stable equilibria. We explain howattention to what are apparently mathematical technicalities has led to theorems thatallow one to convert spectral calculations, which are used in the sciences and engineeringto study stability of a wave, into detailed, theoretically-based information about thebehavior of perturbations of the wave.

We analyze the problem of shear-induced electrokinetic lift on a particle freely suspended near a solid wall, subject to a homogeneous (simple) shear. To this end, we apply the large-Péclet-number generic scheme recently developed by Yariv et al. (J. Fluid Mech., Vol. 685, 2011, p. 306). For a force- and torque-free particle, the driving flow comprises three components, respectively describing (i) a particle translating parallel to the wall; (ii) a particle rotating with an angular velocity vector normal to the plane of shear; and (iiii) a stationary particle in a shear flow. Symmetry arguments reveal that the electro-viscous lift, normal to the wall, is contributed by Maxwell stresses accompanying the induced electric field, while electro-viscous drag and torque corrections, parallel to the wall, are contributed by the Newtonian stresses accompanying the induced flow. We focus upon the near-contact limit, where all electro-viscous contributions are dominated by the intense electric field in the narrow gap between the particle and the wall. This field is determined by the gap-region pressure distributions associated with the translational and rotational components of the driving Stokes flow, with the shear-component contribution directly affecting only higher-order terms. Owing to the similarity of the corresponding pressure distributions, the induced electric field for equal particle–wall zeta potentials is proportional to the sum of translation and rotation speeds. The electro-viscous loads result in induced particle velocities, normal and tangential to the wall, inversely proportional to the second power of particle–wall separation.

Building on earlier work, we have given in [29] aproof of existence of Abrikosov vortex lattices in the Ginzburg-Landau model ofsuperconductivity and shown that the triangular lattice gives the lowest energy perlattice cell. After [29] was published, we realizedthat it proves a stronger result than was stated there. This result is recorded in thepresent paper. The proofs remain the same as in [29], apart from some streamlining.

In this paper, we consider a system of nonlinear delay-differential equations(DDEs) which models the dynamics of the interaction between chronic myelogenousleukemia (CML), imatinib, and the anti-leukemia immune response. Because of thechaotic nature of the dynamics and the sparse nature of experimental data, welook for ways to use computation to analyze the model without employing directnumerical simulation. In particular, we develop several tools usingLyapunov-Krasovskii analysis that allow us to test the robustness of the modelwith respect to uncertainty in patient parameters. The methods developed in thispaper are applied to understanding which model parameters primarily affect thedynamics of the anti-leukemia immune response during imatinib treatment. Thegoal of this research is to aid the development of more efficient modelingapproaches and more effective treatment strategies in cancer therapy.

A possible control strategy against the spread of an infectious disease is the treatmentwith antimicrobials that are given prophylactically to those that had contact with an infective person.The treatment continues until recovery or until it becomes obvious that there was no infectionin the first place. The model considers susceptible, treated uninfected exposed, treated infected,(untreated) infectious, and recovered individuals. The overly optimistic assumptions are made thattreated uninfected individuals are not susceptible and treated infected individuals are not infectious.Since treatment lengths are considered that have an arbitrary distribution, the model systemconsists of ordinary differential and integral equations. We study the impact of the treatment lengthdistribution on the large-time behavior of the model solutions, namely whether the solutions convergeto an equilibrium or whether they are driven into undamped oscillations.

We extrapolate from the exact master equations of epidemic dynamics on fully connectedgraphs to non-fully connected by keeping the size of the state space N + 1, whereN is thenumber of nodes in the graph. This gives rise to a system of approximate ODEs (ordinarydifferential equations) where the challenge is to compute/approximate analytically thetransmission rates. We show that this is possible for graphs with arbitrary degreedistributions built according to the configuration model. Numerical tests confirm that:(a) the agreement of the approximate ODEs system with simulation is excellent and (b) thatthe approach remains valid for clustered graphs with the analytical calculations of thetransmission rates still pending. The marked reduction in state space gives good results,and where the transmission rates can be analytically approximated, the model provides astrong alternative approximate model that agrees well with simulation. Given that thetransmission rates encompass information both about the dynamics and graph properties, thespecific shape of the curve, defined by the transmission rate versus the number ofinfected nodes, can provide a new and different measure of network structure, and themodel could serve as a link between inferring network structure from prevalence orincidence data.

We present sufficient conditions of local controllability for a class of models of treatment response to combined anticancer therapies which include delays in control strategies. The combined therapy is understood as combination of direct anticancer strategy e.g. chemotherapy and indirect modality (in this case antiangiogenic therapy). Controllability of the models in the form of semilinear second order dynamic systems with delays in control enables to answer the questions of realizability of different objectives of multimodal therapy in the presence of PK/PD effects. We compare results for the models without delays and conditions for relative local controllability of models with delays.

Stockwell transforms as hybrids of Gabor transforms and wavelet transforms have beenstudied extensively. We introduce in this paper multi-dimensional Stockwell transformsthat include multi-dimensional Gabor transforms as special cases. Continuous inversionformulas for multi-dimensional Stockwell transforms are proved.

The aim of this paper is to introduce and study multilinear pseudo-differential operatorson Zn and Tn =(Rn/2πZn) then-torus.More precisely, we give sufficient conditions and sometimes necessary conditions forLp-boundedness of theseclasses of operators. L2-boundedness results for multilinearpseudo-differential operators on Zn and Tn withL2-symbols are stated. The proofs of theseresults are based on elementary estimates on the multilinear Rihaczek transforms forfunctions in L2(Zn)respectively L2(Tn)which are also introduced.

We study the weak continuity of multilinear operators on the m-fold product of Lebesguespaces Lpj(Zn),j =1,...,m and thelink with the continuity of multilinear pseudo-differential operators on Zn.

Necessary and sufficient conditions for multilinear pseudo-differential operators onZn or Tn to be aHilbert-Schmidt operators are also given. We give a necessary condition for a multilinearpseudo-differential operators on Zn to be compact. A sufficientcondition for compactness is also given.

We prove that localization operators associated to ridgelet transforms withLp symbols are boundedlinear operators on L2(Rn).Operators closely related to these localization operators are shown to be in the traceclass and a trace formula for them is given.

Neutropenia is a significant dose-limiting toxicity of cancerchemotherapy, especially in dose-intensified regimens. It iswidely treated by injections of Granulocyte Colony-StimulatingFactor (G-CSF). However, optimal schedules of G-CSF administrationare still not determined. In order to aid in identifying moreefficacious and less neutropenic treatment protocols, we studied adetailed physiologically-based computer model of granulopoiesis,as affected by different treatment schedules of doxorubicin and/orGranulocyte Colony-Stimulating Factor (G-CSF). We validated themodel as evident from accurate prediction of clinical data onhuman granulopoiesis in healthy individuals and indoxorubicin-treated cancer patients, with or without G-CSFsupport. Based on our model, we suggest new G-CSF administrationregimens. These regimens include reduced G-CSF doses, optimallytimed post-chemotherapy. Application of these regimens can lead tominimization of G-CSF side effects, as well as more cost-effectiveand less myelotoxic protocols. Currently clinical trials are beingdesigned in order to test these new treatment regimens.

Wilkins' method has been successfully used since early 60s for numericalsimulation of high velocity contact elastic-plastic flows. The present workproposes some effective modifications of this method including moresophisticated material model including the Baushinger effect; modification ofthe algorithm based on correction of the initial configuration of a solid; acontact algorithm based on the idea of a mild contact.

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.