Reaction-diffusion equations with degenerate nonlinear diffusion are in widespread use asmodels of biological phenomena. This paper begins with a survey of applications toecology, cell biology and bacterial colony patterns. The author then reviews mathematicalresults on the existence of travelling wave front solutions of these equations, and theirgeneration from given initial data. A detailed study is then presented of the form ofsmooth-front waves with speeds close to that of the (unique) sharp-front solution, for theparticular equation ut =(uux)x + u(1− u). Using singular perturbation theory, the author derives anasymptotic approximation to the wave, which gives valuable information about the structureof smooth-front solutions. The approximation compares well with numerical results.

Bacillus subtilis swarms rapidly over the surface of a synthetic mediumcreating remarkable hyperbranched dendritic communities. Models to reproduce such effectshave been proposed under the form of parabolic Partial Differential Equations representingthe dynamics of the active cells (both motile and multiplying), the passive cells(non-motile and non-growing) and nutrient concentration. We test the numerical behavior ofsuch models and compare them to relevant experimental data together with a criticalanalysis of the validity of the models based on recent observations of the swarmingbacteria which show that nutrients are not limitating but distinct subpopulations growingat different rates are likely present.

This paper is related to the spectral stability of traveling wave solutions of partialdifferential equations. In the first part of the paper we use the Gohberg-Rouche Theoremto prove equality of the algebraic multiplicity of an isolated eigenvalue of an abstractoperator on a Hilbert space, and the algebraic multiplicity of the eigenvalue of thecorresponding Birman-Schwinger type operator pencil. In the second part of the paper weapply this result to discuss three particular classes of problems: the Schrödingeroperator, the operator obtained by linearizing a degenerate system of reaction diffusionequations about a pulse, and a general high order differential operator. We studyrelations between the algebraic multiplicity of an isolated eigenvalue for the respectiveoperators, and the order of the eigenvalue as the zero of the Evans function for thecorresponding first order system.

Cell motility is an integral part of a diverse set of biological processes. The quest formathematical models of cell motility has prompted the development of automated approachesfor gathering quantitative data on cell morphology, and the distribution of molecularplayers involved in cell motility. Here we review recent approaches for quantifying cellmotility, including automated cell segmentation and tracking. Secondly, we present our ownnovel method for tracking cell boundaries of moving cells, the Electrostatic ContourMigration Method (ECMM), as an alternative to the generally accepted level set method(LSM). ECMM smoothly tracks regions of the cell boundary over time to compute localmembrane displacements using the simple underlying concept of electrostatics. It offerssubstantial speed increases and reduced computational overheads in comparison to the LSM.We conclude with general considerations regarding boundary tracking in the context ofmathematical modelling.

The oriented movement of biological cells or organisms in response to a chemical gradientis called chemotaxis. The most interesting situation related to self-organizationphenomenon takes place when the cells detect and response to a chemical which is secretedby themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) manyparticularized models have been proposed to describe the aggregation phase of thisprocess. Most of efforts were concentrated, so far, on mathematical models in which theformation of aggregate is interpreted as finite time blow-up of cell density. In recentlyproposed models cells are no more treated as point masses and their finite volume isaccounted for. Thus, arbitrary high cell densities are precluded in such description and athreshold value for cells density is a priori assumed. Different modelingapproaches based on this assumption lead to a class of quasilinear parabolic systems withstrong nonlinearities including degenerate or singular diffusion. We give a survey ofanalytical results on the existence and uniqueness of global-in-time solutions, theirconvergence to stationary states and on a possibility of reaching the density threshold bya solution. Unsolved problems are pointed as well.

We study discrete spectrum in spectral gaps of an elliptic periodic second orderdifferential operator in L 2(ℝd )perturbed by a decaying potential. It is assumed that a perturbation is nonnegative andhas a power-like behavior at infinity. We find asymptotics in the large coupling constantlimit for the number of eigenvalues of the perturbed operator that have crossed a givenpoint inside the gap or the edge of the gap. The corresponding asymptotics is power-likeand depends on the observation point.

Anisotropic adaptive methods based on a metric related to the Hessian of the solution areconsidered. We propose a metric targeted to the minimization of interpolation errorgradient for a nonconforming linear finite element approximation of a given piecewiseregular function on a polyhedral domain Ω ofℝ d , d ≥ 2. We alsopresent an algorithm generating a sequence of asymptotically quasi-optimal meshes relativeto such a nonconforming discretization and give numerical asymptotic behavior of the errorreduction produced by the generated mesh

This paper is devoted to the study of global existence of periodic solutions of a delayedtumor-immune competition model. Also some numerical simulations are given to illustratethe theoretical results

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 present a new method for generating a d-dimensional simplicial meshthat minimizes the L p -norm,p > 0, of the interpolation error or its gradient. The methoduses edge-based error estimates to build a tensor metric. We describe and analyze thebasic steps of our method

Epidermal wound healing is a complex process that repairs injured tissue. The complexityof this process increases when bacteria are present in a wound; the bacteria interactiondetermines whether infection sets in. Because of underlying physiological problemsinfected wounds do not follow the normal healing pattern. In this paper we present amathematical model of the healing of both infected and uninfected wounds. At the core ofour model is an account of the initiation of angiogenesis by macrophage-derived growthfactors. We express the model as a system of reaction-diffusion equations, and we presentresults of computations for a version of the model with one spatial dimension.

This paper extends the volume filling chemotaxis model [18, 26] by taking into account the cell population interactions. Theextended chemotaxis models have nonlinear diffusion and chemotactic sensitivity dependingon cell population density, which is a modification of the classical Keller-Segel model inwhich the diffusion and chemotactic sensitivity are constants (linear). The existence andboundedness of global solutions of these models are discussed and the numerical patternformations are shown. The further improvement is proposed in the end.

Collective cell motility and its guidance via cell-cell contacts is instrumental in several morphogenetic and pathological processes such as vasculogenesis or tumor growth. Multicellular sprout elongation, one of the simplest cases of collective motility, depends on a continuous supply of cells streaming along the sprout towards its tip. The phenomenon is often explained as leader cells pulling the rest of the sprout forward via cell-cell adhesion. Building on an empirically demonstrated analogy between surface tension and cell-cell adhesion, we demonstrate that such a mechanism is unable to recruit cells to the sprout. Moreover, the expansion of such hypothetical sprouts is limited by a form of the Plateau-Taylor instability. In contrast, actively moving cells – guided by cell-cell contacts – can readily populate and expand linear sprouts. We argue that preferential attraction to the surfaces of elongated cells can provide a generic mechanism, shared by several cell types, for multicellular sprout formation.

We present a user-friendly version of a double operator integration theory which stillretains a capacity for many useful applications. Using recent results from the lattertheory applied in noncommutative geometry, we derive applications to analogues of theclassical Heinz inequality, a simplified proof of a famous inequality ofBirman-Koplienko-Solomyak and also to the Connes-Moscovici inequality. Our methods aresufficiently strong to treat these inequalities in the setting of symmetric operator normsin general semifinite von Neumann algebras.

Solid tumors and hematological cancers contain small population of tumor cells that arebelieved to play a critical role in the development and progression of the disease. Thesecells, named Cancer Stem Cells (CSCs), have been found in leukemia, myeloma, breast,prostate, pancreas, colon, brain and lung cancers. It is also thought that CSCs drive themetastatic spread of cancer. The CSC compartment features a specific and phenotypicallydefined cell population characterized with self-renewal (through mutations), quiescence orslow cycling, overexpression of anti-apoptotic proteins, multidrug resistance and impaireddifferentiation. CSCs show resistance to a number of conventional therapies, and it isbelieved that this explains why it is difficult to completely eradicate the disease andwhy recurrence is an ever-present threat. A hierarchical phenomenological model isproposed based on eight compartments following the stem cell lineage at the normal andcancer cell levels. As an empirical test, the tumor grading and progression, typicallycollected in the pathologic lab, is used to correlate the outcome of this model with thetumor development stages. In addition, the model is able to quantitatively account for thetemporal development of the population of observed cell types. Two types of therapeutictreatment models are considered, with dose-density chemotherapy (a pulsatile scenario) aswell as continuous, metronomic delivery. The drug hit is considered at the stem cellprogenitor and early differentiated specialized cell levels for both normal and cancercells, while the quiescent stem cell and fully differentiated compartments are consideredfavorable outcome for cancer treatment. Circulating progenitors are neglected in thisanalysis. The model provides a number of experimentally testable predictions. The relativeimportance of the cell kill and survival is demonstrated through a deterministicparametric study. The significance of the stem cell compartment is underlined based onthis simulation study. This predictive mathematical model for cancer stem cell hypothesisis used to understand tumor responses to chemotherapeutic agents and judge theefficacy.

We present in this work a numerical study of a problem governed by Navier-Stokesequations and heat equation. The mathematical problem under consideration is obtained bymodelling the natural convection of an incompressible fluid, in laminar flow between twohorizontal concentric coaxial cylinders, the temperature of the inner cylinder is supposedto be greater than the outer one. The numerical simulation of the flow is carried out bycollocation-Legendre method. The influence of Prandtl and Rayleigh numbers isinvestigated

We consider the zigzag half-nanotubes (tight-binding approximation) in a uniform magneticfield which is described by the magnetic Schrödinger operator with a periodic potentialplus a finitely supported perturbation. We describe all eigenvalues and resonances of thisoperator, and theirs dependence on the magnetic field. The proof is reduced to theanalysis of the periodic Jacobi operators on the half-line with finitely supportedperturbations.

We prove the instability of threshold resonances and eigenvalues of the linearized NLSoperator. We compute the asymptotic approximations of the eigenvalues appearing from theendpoint singularities in terms of the perturbations applied to the original NLS equation.Our method involves such techniques as the Birman-Schwinger principle and the Feshbachmap.