We propose and analyze a nonlinear mathematical model of hematopoiesis,describing the dynamics of stem cell population subject to impulsiveperturbations. This is a system of two age-structured partial differentialequations with impulses. By integrating these equations over theage, we obtain a system of two nonlinear impulsive differential equations withseveral discrete delays. This system describes the evolution of the totalhematopoietic stem cell populations with impulses. We first examine theasymptotic behavior of the model in the absence of impulsions.Secondly, we add the impulsive perturbations and we investigate the qualitativebehavior of the model including the global asymptotic stability of the trivialsolution and the existence of periodic solution in the case of periodicimpulsive perturbations. Finally, numerical simulations are carriedout to illustrate the behavior of the model. This study maybe helpful tounderstand the reactions observed in the hematopoietic system after differentforms of stress as direct destruction by some drugs or irradiation.

The model of geographical space of the Earth is represented as a multileveldiscrete gridstructure. This provides a scientific basis for the construction of a naturalgeoinformation systemof the globe with invariant organization of data in the form of a universalgrid, i.e. a different-levelset of elementary cells with discrete topology. A grid structure is a multilevelsystem of referencepoints, with respect to which a spatial analysis of the territory is carriedout. Reference points haveattributes reflecting the territorys hierarchic level and locationcharacteristics. It is found that thereis a connection between the topology of the cells and specific structures of theEarth, and betweenthe lineament network and the hydrological network in particular.

In this paper we investigate the role of spatial effects in determining thedynamics of a subclass of signalling pathways characterised by their ability todemonstrate oscillatory behaviour. To this end, we formulate a simple spatial model of thep53 network that accounts for both a negative feedback and a transcriptional delay. We show that the formation of protein density patterns can depend on the shape of the cell, position of the nucleus, and the protein diffusion rates. Thetemporal changes in the total amounts of protein are also subject to spatial influences. The level of DNA damage required to induce sustained oscillations, forinstance, depends on the morphology of the cell. The model also provides a newinterpretation of experimentally observed undamped oscillations in p53 levels in single cells.Our simulations reveal that alternate sequences of high- and low-amplitudeoscillations can occur. We propose that the digital pulses may correspond to snap-shots ofour high-amplitude sequences. Shorter waiting-times between subsequent time-lapse fluorescence microscopy images in combination with lower detection thresholds may reveal the irregular high-frequency oscillations suggested by our spatialmodel.

This paper is devoted to the introduction of a new variant of the extendedfinite element method (Xfem) for the approximation of elastostatic fractureproblems. This variant consists in a reduced basis strategy for the definitionof the crack tip enrichment. It is particularly adapted when the asymptoticcrack-tip displacement is complex or even unknown. We give a mathematical resultof quasi-optimal a priori error estimate and some computational tests includinga comparison with some other strategies.

Tumor growth and progression is a complex phenomenon dependent on theinteraction of multiple intrinsic and extrinsic factors. Necessary for tumordevelopment is a small subpopulation of potent cells, so-called cancer stemcells, that can undergo an unlimited number of cell divisions and which areproposed to divide symmetrically with a small probability to produce more cancerstem cells. We show that the majority of cells in a tumor must indeed benon-stem cancer cells with limited life span and limited replicative potential. Tumor development is dependent as well on the proliferative potential and deathof these cells, and on the migratory ability of all cancer cells. Withincreasing number of cells in the tumor, competition for space limits tumorprogression, and in agreement with in vitro observation, the majority of cancercells become quiescent, with proliferation primarily occurring on the outer rimwhere space is available. We present an agent-based model of early tumordevelopment that captures the spatial heterogeneity of stemness andproliferation status. We apply the model to simulations of radiotherapy topredict treatment outcomes for tumors with different stem cell pool sizes anddifferent quiescence radiosensitivities. We show by first presuming homogeneousradiosensitivity throughout the tumor, and then considering the greaterresistance of quiescent cells, that stem cell pool size and stem cellrepopulation during treatment determine treatment success. The results fortumor cure probabilities comprise upper bounds, as there is evidence that cancerstem cells are also more radioresistant than other tumor cells. Beyond justdemonstrating the influence of mass effects of stem to non-stem cell ratios andproliferating to quiescent cell ratios, we show that the spatiotemporalevolution of the developing heterogeneous population plays a pivotal role indetermining radioresponse and treatment optimization.

A spatial and temporal diversity of processes and phenomena is observed inmountaintaigalandscapes as the object of management. A knowledge of the mechanisms of theirinteractionsmakes it possible to effectively manage these processes and phenomena. Themethodology of systemanalysis and mathematical models of interaction mechanisms are used in theanalysis of thesephenomena. A visualization of modelling results is carried out usinggeoinformation mappingmethods.

The circulatory system is one of thefirst to function during development. The earliest event in thesystem's development is vasculogenesis, whereby vascularprogeniter cells form clusters called blood islands, which laterfuse to form capillary networks. There exists a very goodin vitro system that mimics this process. When HUVECs(Human Umbilical Vein Endothelial Cells) are cultured on Matrigel,they spontaneously form a capillary network structure. Twotheoretical models have been proposed to explain the patternformation of this in vitro system. Both models utilizechemotaxis to generate spatial instability, and one modelspecifies VEGF as the chemoattractant. However, there are severalunknown factors concerning the experimental model. First, thepattern formation process occurs at the interface between theliquid medium and Matrigel, and it is unclear whether diffusion inthe liquid or gel is critical. Second, the diffusion coefficientof VEGF, which determines the spatial scale of the capillarystructure, has not been properly measured. In the present study,we modified the experimental system to clarify the effect ofdiffusion in Matrigel, and experimentally measured the diffusioncoefficient of VEGF in this system. The relationship with thespatial scale of the pattern generated is discussed.

In this paper, with the assumptions that an infectious disease has a fixedlatent period in a population and the latent individuals of the population maydisperse, we reformulate an SIR model for the population living in two patches(cities, towns, or countries etc.), which is a generalization of the classicKermack-McKendrick SIR model. The model is given by a system of delaydifferential equations with a fixed delay accounting for the latency andnon-local terms caused by the mobility of the individuals during the latentperiod. We analytically show that the model preserves some properties that theclassic Kermack-McKendrick SIR model possesses: the disease always dies out,leaving a certain portion of the susceptible population untouched (calledfinal sizes). Although we can not determine the two final sizes, we are able toshow that the ratio of the final sizes in the two patches is totally determinedby the ratio of the dispersion rates of the susceptible individuals between thetwo patches. We also explore numerically the patterns by which the disease diesout, and find that the new model may have very rich patterns for the diseaseto die out. In particular, it allows multiple outbreaks of the disease before itgoes to extinction, strongly contrasting to the classic Kermack-McKendrick SIRmodel.

We present new a posteriori error estimates for the finite volume approximationsof elliptic problems. They are obtained by applying functional a posteriorierror estimates to natural extensions of the approximate solution and its fluxcomputed by the finite volume method. The estimates give guaranteed upper boundsfor the errors in terms of the primal (energy) norm, dual norm (for fluxes), andalso in terms of the combined primal-dual norms. It is shown that the estimatesprovide sharp upper and lower bounds of the error and their practicalcomputation requires solving only finite-dimensional problems.

The seasonality hypothesis states that climates characterized by large annual cycles select for large body sizes. In order to study the effects of seasonality on the evolution of body size, we use a model that is based on physiological rules and first principles. At the ecological time scale, our model results show that both larger productivity and seasonality may lead to larger body sizes. Our model is the first dynamic and process-based model to support the seasonality hypothesis and hence demonstrates the importance of basing models on physiological processes. We focus not only on variability at the ecological time scale, but also on the temporal variations in seasonality existing at geological time scales. A particularly strong forcing of seasonality exists on the scale of 20,000-400,000 years, the scale of Milankovitch cycles. Therefore, we simulated the evolutionary response of body size to a Milankovitch-type of forcing of climate and food density. Results illustrate that for a given level of investment in reserves body size may track climatic cycles, and that below a certain seasonality threshold the body size will decrease rapidly, leading to extinction.

Skeletal patterning in the vertebrate limb,i.e., the spatiotemporal regulation of cartilage differentiation(chondrogenesis) during embryogenesis and regeneration, is oneof the best studied examples of a multicellular developmental process.Recently [Alber et al., The morphostatic limit for a model ofskeletal pattern formation in the vertebrate limb, Bulletin ofMathematical Biology, 2008, v70, pp. 460-483], a simplified two-equationreaction-diffusion system was developed to describe the interaction of two ofthe key morphogens: the activator and an activator-dependent inhibitor ofprecartilage condensation formation. A discontinuous Galerkin (DG)finite element method was applied to solve this nonlinear system on complexdomains to study the effects of domain geometry on the pattern generated [Zhu etal., Application of Discontinuous Galerkin Methods for reaction-diffusionsystems in developmental biology, Journal of Scientific Computing, 2009, v40,pp. 391-418]. In this paper, we extend these previous results and develop a DGfinite element model in a moving and deforming domain for skeletal patternformation in the vertebrate limb. Simulations reflect the actual dynamics oflimb development and indicate the important role played by the geometryof the undifferentiated apical zone.

We examine the theoretical and applications-specific issues relating to modelingthetemporal and spatial dynamics of forest ecosystems, based on the principles ofinvestigating dynamicalmodels. When developing the predictive dynamical models of forest resources,there is apossibility of achieving uniqueness of the solutions to equations by taking intoaccount the initialand boundary conditions of the solution, and the conditions of the geographicalenvironment. Wepresent the results of a computer modeling and predictive mapping for theregional and subregionalmodel of the forest resource dynamics.

We develop a stage-structured model that describes the dynamics of two competing species each of which have sexual and clonal reproduction. This is typical of many plants including irises. We first analyze the dynamical behavior of a single species model. We show that when the inherent net reproductive number is smaller than one then the population will go to extinction and if it is larger than one then an interior equilibrium exists and it is globally asymptotically stable. Then we analyze the two-species model and establish conditions on the reproduction and survivorship rates that lead to competitive exclusion. We show that the winner species is the one that attains higher density at which its net reproductive number equals unity. Numerical results corroborating the theoretical ones are also presented.

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.

The present paper introduces a tumormodel with two time scales, the time t during which the tumorgrows and the cycle time of individual cells. The model alsoincludes the effects of gene mutations on the population densityof the tumor cells. The model is formulated as a free boundaryproblem for a coupled system of elliptic, parabolic and hyperbolicequations within the tumor region, with nonlinear and nonlocalterms. Existence and uniqueness theorems are proved, andproperties of the free boundary are established.

The paper deals with an iterative method for numerical solving frictionlesscontact problems for two elastic bodies. Each iterative step consists of aDirichlet problem for the one body, a contact problem for the other one and twoNeumann problems to coordinate contact stresses. Convergence is proved by theBanach fixed point theorem in both continuous and discrete case. Numericalexperiments indicate scalability of the algorithm for some choices of therelaxation parameter.

Cell-based, mathematical models helpmake sense of morphogenesis—i.e. cells organizing intoshape and pattern—by capturing cell behavior in simple, purelydescriptive models. Cell-based models then predict thetissue-level patterns the cells produce collectively. The firststep in a cell-based modeling approach is to isolatesub-processes, e.g. the patterning capabilities of one or afew cell types in cell cultures. Cell-based models can thenidentify the mechanisms responsible for patterning in vitro.This review discusses two cell culture models of morphogenesisthat have been studied using this combinedexperimental-mathematical approach: chondrogenesis (cartilagepatterning) and vasculogenesis (de novo blood vessel growth). Inboth these systems, radically different models can equallyplausibly explain the in vitro patterns. Quantitativedescriptions of cell behavior would help choose betweenalternative models. We will briefly review the experimentalmethodology (microfluidics technology and traction forcemicroscopy) used to measure responses of individual cells to theirmicro-environment, including chemical gradients, physical forcesand neighboring cells. We conclude by discussing how to includequantitative cell descriptions into a cell-based model: theCellular Potts model.

Mathematical modeling provides a particularly important tool for studying thestreamrunoff formation processes, and its role is enhanced in the case of a sparse,obsolete monitoringnetwork characteristic of most regions of Siberia. When analyzingspatio-temporal regularities ofthe water and sediment runoff in river systems, serious problems are caused bylack of the basichydrological model capable of handling real-time data of hydrologicalmeasurements.
Calculations of unsteady flows in stream channels draw heavily onone-dimensional numericalmodels which are relatively easy to use and yield reliable results. Numericalinvestigations into thehydraulic regime of natural streams involve specific difficulties caused by thepresence of nonlinearfrictional forces in a turbulent flow, a variability in the channel geometry,the braiding of flows, thepresence of floodplain depressions, riffles, etc. For especially complexstretches of rivers, a onedimensionalapproximation no longer fits the reality sufficiently adequately, so that theplanar flowstructure must be taken into account. For this purpose Saint Vennant's planesystem of equationswas used as the basis in order to develop further the numerical model due tothis author whichis intended for calculating the flow field, flow rates, levels, and impurityconcentrations in naturalwater bodies of an arbitrary configuration or in a part of them. Fundamentallaws of fluid mechanicsare used as the basis for the model.
Spatial modeling of flows in complex regions necessitates reliable, consistentmethods providingacceptable accuracy. As far as hydrological problems are concerned, the controlvolumemethod that allows the use of curvilinear grids was found to be the mostpowerful tool for obtainingthe initial finite-difference relations.
This paper offers a number of examples illustrating the capabilities of theplanar model forstreams which is intended to resolve real problems arising at the design,construction and operation stages of engineering structures in river channelsand floodplains.

In this survey, we briefly review some of our recent studies on predator-preymodels with discrete delay. We first study the distribution of zeros of a seconddegree transcendental polynomial. Then we apply the general results on thedistribution of zeros of the second degree transcendental polynomial to variouspredator-prey models with discrete delay, including Kolmogorov-typepredator-prey models, generalized Gause-type predator-prey models withharvesting, etc. Bogdanov-Takens bifurcations in delayed predator-prey modelswith nonmonotone functional response and in delayed predator-prey model withpredator harvesting are also introduced.

Most mammalian tissues are organized into a hierarchical structure of stem,progenitor, and differentiated cells. Tumors exhibit similar hierarchy, even ifit is abnormal in comparison with healthy tissue. In particular, it is believedthat a small population of cancer stem cells drives tumorigenesis in certainmalignancies. These cancer stem cells are derived from transformed stem cells ormutated progenitors that have acquired stem-cell qualities, specifically theability to self-renew. Similar to their normal counterparts, cancer stem cellsare long-lived, can self-renew and differentiate, albeit aberrantly, and arecapable of generating tissue, resulting in tumor formation. Although identifiedand characterized in several forms of malignancy, the specific multi-stepprocess that causes the formation of cancer stem cells is uncertain. Here, amaturity-structured mathematical model is developed to investigate thesequential order of mutations that causes the fastest emergence of cancer stemcells. Using model predictions, we discuss conditions for which geneticinstability significantly speeds cancer onset and suggest that unbalancedstem-cell self-renewal and inhibition of progenitor differentiation contributeto aggressive forms of cancer. To our knowledge, this is the first continuousmaturity-structured mathematical model used to investigate mutation acquisitionwithin hierarchical tissue in order to address implications of cancer stem cellsin tumorigenesis.