In this note, we consider a nonlinear diffusion equation with a bistable reaction termarising in population dynamics. Given a rather general initial data, we investigate itsbehavior for small times as the reaction coefficient tends to infinity: we prove ageneration of interface property.

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.

Two main approaches have been considered for modelling the dynamics of the SIS model oncomplex metapopulations, i.e, networks of populations connected by migratory flows whoseconfigurations are described in terms of the connectivity distribution of nodes (patches)and the conditional probabilities of connections among classes of nodes sharing the samedegree. In the first approach migration and transmission/recovery process alternatesequentially, and, in the second one, both processes occur simultaneously. Here we followthe second approach and give a necessary and sufficient condition for the instability ofthe disease-free equilibrium in generic networks under the assumption of limited (orfrequency-dependent) transmission. Moreover, for uncorrelated networks and under theassumption of non-limited (or density-dependent) transmission, we give a bounding intervalfor the dominant eigenvalue of the Jacobian matrix of the model equations around thedisease-free equilibrium. Finally, for this latter case, we study numerically theprevalence of the infection across the metapopulation as a function of the patchconnectivity.

In this paper we deal with a model describing the evolution in time of the density of aneural population in a state space, where the state is given by Izhikevich’s two -dimensional single neuron model. The main goal is to mathematically describe theoccurrence of a significant phenomenon observed in neurons populations, thesynchronization. To this end, we are making the transition to phasedensity population, and use Malkin theorem to calculate the phase deviations of a weaklycoupled population model.

2D shallow water equations with depth-averaged k−εmodel is considered. A meshless method based on multiquadric radial basis functions isdescribed. This methods is based on the collocation formulation and does not require thegeneration of a grid and any integral evaluation. The application of this method to a flowin horizontal channel, taken as an experimental device, is presented. The results ofcomputations are compared with experimental data and are found to be satisfactory

We study controllability for a nonhomogeneous string and ring under an axial stretchingtension that varies with time. We consider the boundary control for a string anddistributed control for a ring. For a string, we are looking for a controlf(t) ∈ L 2(0,T) that drives the state solution to rest. We show that for a ring, two forcesare required to achieve controllability. The controllability problem is reduced to amoment problem for the control. We describe the set of initial data which may be driven torest by the control. The proof is based on an auxiliary basis property result.

The aim of this study was to describe and analyze the regulation and spatio-temporaldynamics of melanocyte migration in vitro and its coupling to celldivision and interaction with the matrix. The melanocyte lineage is particularlyinteresting because it is involved in both embryonic development andoncogenesis/metastasis (melanoma). Biological experiments were performed on two melanocytecell lines established from wild-type and β-catenin-transgenic mice(bcat*). The multi-functional β-catenin molecule is known to be able toregulate the transcription of various genes involved in cell proliferation and migration,particularly in the melanocyte lineage. We also investigated fibronectin, anextra-cellular matrix protein that binds integrins, thereby providing adhesion points forcells and encouraging migration. As the migration of individual cells were followed,automated methods were required for processing the large amount of data generated by thetime-lapse video-microscopy. A model-based approach for automated cell tracking wasevaluated on a sample by comparison with manual tracking. This method was found reliablein studying overall cell behaviour. Its application to all the observed sequences providedinsight into the factors affecting melanocyte migration in vitro:melanocytes of mutated form of β-catenin showed higher division rates andno contact inhibition of growth was induced by the resulting increase in cell density.However, cell density limited the amplitude of cell displacements, although their motilitywas less affected. The high fibronectin concentration bound to substratum promoted cellmigration and motility, the effect being more intense for wild-type cells than for cellswith β-catenin over-expression. During the division process, cellmigration speed increased rapidly then decreased slowly. Analyses of such data is expectedto lead both to biological answers and to a framework for a better modeling processes inthe future.

We study a class of bistable reaction-diffusion systems used to model two competingspecies. Systems in this class possess two uniform stable steady states representingsemi-trivial solutions. Principally, we are interested in the case where the ratio of thediffusion coefficients is small, i.e. in thenear-degenerate case. First, limiting arguments are presented to relatesolutions to such systems to those of the degenerate case where one species is assumed notto diffuse. We then consider travelling wave solutions that connect the two stablesemi-trivial states of the non-degenerate system. Next, a general energy function for thefull system is introduced. Using this and the limiting arguments, we are able to determinethe wave direction for small diffusion coefficient ratios. The results obtained onlyrequire knowledge of the system kinetics.

Epidemiologic data suggest that schools and daycare facilities likely play a major rolein the dissemination of influenza. Pathogen transmission within such small, inhomogenouslymixed populations is difficult to model using traditional approaches. We developedsimulation based mathematical tool to investigate the effects of social contact networkson pathogen dissemination in a setting analogous to a daycare center or grade school. Herewe show that interventions that decrease mixing within child care facilities, includinglimiting the size of social clusters, reducing the contact frequency between socialclusters, and eliminating large gatherings, could diminish pathogen dissemination.Moreover, these measures may amplify the effectiveness of vaccination or antiviralprophylaxis, even if the vaccine is not uniformly effective or antiviral compliance isincomplete. Similar considerations should apply to other small, imperfectly mixedpopulations, such as offices and schools.

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.

Experimental evidence points to a rich variety of physical scenarios that arise when alaminar flame propagates through a pre-mixture of evaporating liquid fuel and a gaseousoxidant. In this paper new results of time-dependent numerical simulations of richoff-stoichiometric spray flame propagation in a two-dimensional channel are presented. Aconstant density model is adopted, thereby eliminating the Darrieus-Landau instability. Itis demonstrated that there exists a narrow band of vaporization Damkohler numbers (theratio of a characteristic flow time to a characteristic evaporation time) for which theflame propagation is oscillatory. For values outside this range steady state propagationis attained but with a curved (cellular) flame front. The critical range for thenon-steady propagation is also found to be a function of the Lewis number of the deficientreactant.

Plant growth depends essentially on nutrients coming from the roots and metabolitesproduced by the plant. Appearance of new branches is determined by concentrations ofcertain plant hormones. The most important of them are Auxin and Cytokinin. Auxin isproduced in the growing, Cytokinin in either roots or in growing parts. Many dynamicalmodels of this phenomena have been studied in [1]. In [5], the authors deal with onebranch model. In this work, we focus our interest on a multiple branch model. We deal withthe transport equation in domains of different sizes. A variational reduction type method[3] based on asymptotic partial decomposition introduced in [2] (see also [4]) is used. Inthis work we consider the transport equation in decomposed domain with a general righthand side

Approximate aggregation techniques consist of introducing certain approximations thatallow one to reduce a complex system involving many coupled variables obtaining a simplerʽʽaggregated systemʼʼ governed by a few variables. Moreover, they give results that allowone to extract information about the complex original system in terms of the behavior ofthe reduced one. Often, the feature that allows one to carry out such a reduction is thepresence of different time scales in the system under consideration. In this work we dealwith aggregation techniques in stochastic discrete time models and their application tothe study of multiregional models, i.e., of models for an age structured populationdistributed amongst different spatial patches and in which migration between the patchesis usually fast with respect to the demography (reproduction-survival) in each patch.Stochasticity in population models can be of two kinds: environmental and demographic. Wereview the formulation and the main properties of the dynamics of the different models forpopulations evolving in discrete time and subjected to the effects of environmental anddemographic stochasticity. Then we present different stochastic multiregional models withtwo time scales in which migration is fast with respect to demography and we review themain relationships between the dynamics of the original complex system and the aggregatedsimpler one. Finally, and within the context of models with environmental stochasticity inwhich the environmental variation is Markovian, we make use these techniques to analyzequalitatively the behavior of two multiregional models in which the original complexsystem is intractable. In particular we study conditions under which the population goesextinct or grows exponentially.

We analyze a stochastic neuronal network model which corresponds to an all-to-all networkof discretized integrate-and-fire neurons where the synapses are failure-prone. Thisnetwork exhibits different phases of behavior corresponding to synchrony and asynchrony,and we show that this is due to the limiting mean-field system possessing multipleattractors. We also show that this mean-field limit exhibits a first-order phasetransition as a function of the connection strength — as the synapses are made morereliable, there is a sudden onset of synchronous behavior. A detailed understanding of thedynamics involves both a characterization of the size of the giant component in a certainrandom graph process, and control of the pathwise dynamics of the system by obtainingexponential bounds for the probabilities of events far from the mean.

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.