In a simple FitzHugh-Nagumo neuronal model with one fast and two slow variables, asequence of period-doubling bifurcations for small-scale oscillations precedes thetransition into the spiking regime. For a wide range of values of the timescale separationparameter, this scenario is recovered numerically. Its relation to the singularlyperturbed integrable system is discussed.

In this paper, we review some of our recent results in the study of the dynamics ofinteracting Bose gases in the Gross-Pitaevskii (GP) limit. Our investigations focus on thewell-posedness of the associated Cauchy problem for the infinite particle system describedby the GP hierarchy.

We develop a framework for analysing the outcome of resource competition based onbifurcation theory. We elaborate our methodology by readdressing the problem ofcompetition of two species for two resources in a chemostat environment. In the case ofperfect-essential resources it has been extensively discussed using Tilman’srepresentation of resource quarter plane plots. Our mathematically rigorous analysisyields bifurcation diagrams with a striking similarity to Tilman’s method including theinterpretation of the consumption vector and the resource supply vector. However, ourapproach is not restricted to a particular class of models but also works with othertrophic interaction formulations. This is illustrated by the analysis of a modelconsidering interactively-essential or complementary resources instead ofprefect-essential resources. Additionally, our approach can also be used for otherecosystem compositions: multiple resources–multiple species communities with equilibriumor oscillatory dynamics. Hence, it gives not only a new interpretation of Tilman’sgraphical approach, but it constitutes an extension of competition analyses to communitieswith many species as well as non-equilibrium dynamics.

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.

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.

Physics based simulation is widely seen as a way of increasing the information aboutaircraft designs earlier in their definition, thus helping with the avoidance ofunanticipated problems as the design is refined. This paper reports on an effort to assessthe automated use of computational fluid dynamics level aerodynamics for the developmentof tables for flight dynamics analysis at the conceptual stage. These tables are then usedto calculate handling qualities measures. The methodological questions addressed area)geometry and mesh treatment for automated analysis from a high level conceptual aircraftdescription and b) sampling and data fusion to allow the timely calculation of large datatables. The test case used to illustrate the approaches is based on a refined designpassenger jet wind tunnel model. This model is reduced to a conceptual description, andthe ability of this geometry to allow calculations relevant to the final design to bedrawn is then examined. Data tables are then generated and handling qualitiescalculated.

Consider the dynamics of a thin film flowing down an inclined plane under the action of gravity and in the presence of a first-order exothermic chemical reaction. The heat released by the reaction induces a thermocapillary Marangoni instability on the film surface while the film evolution affects the reaction by influencing heat/mass transport through convection. The main parameter characterizing the reaction-diffusion process is the Damköhler number. We investigate the complete range of Damköhler numbers. We analyze the steady state, its linear stability and nonlinear regime. In the latter case, long-wave models are compared with integral-boundary-layer ones and bifurcation diagrams for permanent solitary wave solutions of the different models are constructed. Time-dependent computations with the integral-boundary-layer models show that the system approaches a train of coherent structures that resemble the solitary pulses obtained in the bifurcation diagrams.

This paper is devoted to the study of spreading speeds and traveling waves for a class ofreaction-diffusion equations with distributed delay. Such an equation describes growth anddiffusion in a population where the individuals enter a quiescent phase exponentially andstay quiescent for some arbitrary time that is given by a probability density function.The existence of the spreading speed and its coincidence with the minimum wave speed ofmonostable traveling waves are established via the finite-delay approximation approach. Wealso prove the existence of bistable traveling waves in the case where the associatedreaction system admits a bistable structure. Moreover, the global stability and uniquenessof the bistable waves are obtained in the case where the density function has zerotail

Patients with acute myeloblastic leukemia (AML) are divided according to the FrenchAmerican British (FAB) classification into eight subgroups (M0 to M7) on the basis of their degreeof maturation/differentiation. However, even if immunophenotypical characterization by flow cytometryis routinely used to distinguish between AML and acute lymphoblastic leukemia (ALL), itis not yet well established for the identification within the AML subgroups. Here we show that certainsubgroups of AML can be identified with a good accuracy (80% of the patients we analyzed)by means of the expression of specific markers.

We consider the spread of an infectious disease on a heterogeneous metapopulation definedby any (correlated or uncorrelated) network. The infection evolves under transmission,recovery and migration mechanisms. We study some spectral properties of a connectivitymatrix arising from the continuous-time equations of the model. In particular we show thatthe classical sufficient condition of instability for the disease-free equilibrium, wellknown for the particular case of uncorrelated networks, works also for the general case.We give also an alternative condition that yields a more accurate estimation of theepidemic threshold for correlated (either assortative or dissortative) networks.

A model of chemotaxis is analyzed that prevents blow-up of solutions. The modelconsists of a system of nonlinear partial differential equations for the spatial populationdensity of a species and the spatial concentration of a chemoattractant in n-dimensionalspace. We prove the existence of solutions, which exist globally, and are L∞-bounded onfinite time intervals. The hypotheses require nonlocal conditions on the species-inducedproduction of the chemoattractant.

In this paper we present an epidemic model affecting an age-structured population. Weshow by numerical simulations that this demographic structure can induce persistentoscillations in the epidemic. The model is then extended to encompass a stage-structureddisease within an age-dependent population. In this case as well, persistent oscillationsare observed in the infected as well as in the whole population.

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.

A small vicinity of a contact line, with well-defined (micro)scales (henceforth the“microstructure”), is studied theoretically for a system of a perfectly wetting liquid,its pure vapor and a superheated flat substrate. At one end, the microstructure terminatesin a non-evaporating microfilm owing to the disjoining-pressure-induced Kelvin effect. Atthe other end, for motionless contact lines, it terminates in a constant film slope(apparent contact angle as seen on a larger scale), the angle being non-vanishing despitethe perfect wetting due to an overall dynamic situation engendered by evaporation. Here wego one step beyond the standard one-sided model by incorporating the effect of vaporpressure non-uniformity as caused by a locally intense evaporation flow, treated in theStokes approximation. Thereby, the film dynamics is primarily affected throughthermodynamics (shift of the local saturation temperature and evaporation rate), thedirect mechanical impact being rather negligible. The resulting integro-differentiallubrication film equation is solved by handling the newly introduced effect (giving riseto the “integro” part) as a perturbation. In the ammonia (at 300   K) example dealt withhere, it proves to be rather weak indeed: the contact angle decreases while the integralevaporation flux increases just by a few percent for a superheat of  ~1   K.However, the numbers grow (roughly linearly) with the superheat.

About twenty five years ago the first discrete mathematical model of the immune systemwas proposed. It was very simple and stylized. Later, many other computational models havebeen proposed each one adding a certain level of sophistication and detail to thedescription of the system. One of these, the Celada-Seiden model published back in 1992,was already mature at its birth, setting apart from the topic-specific nature of the othermodels. This one was not just a model but rather a framework with which one couldimplement his own immunological theories.

Here we describe this computational framework, developed to perform simulations ofdifferent pathologies that are directly or indirectly connected to the immune system. Webriefly describe the system first, then we report on few applications so to give thereader a clear idea of its practical utility in clinical research problems.

Infectious diseases ranging from the common cold to cholera affect our society physically, emotionally, ecologically, and economically. Yet despite their importance and impact, there remains a lack of effective teaching materials for epidemiology and disease ecology in K-12, undergraduate, and graduate curricula [2]. To address this deficit, we’ve developed a classroom lesson with three instructional goals: (1) Familiarize students on basic concepts of infectious disease ecology; (2) Introduce students to a classic compartmental model and its applications in epidemiology; (3) Demonstrate the application and importance of mathematical modeling as a tool in biology. The instructional strategy uses a game-based mathematical manipulative designed to engage students in the concepts of infectious disease spread. It has the potential to be modified for target audiences ranging from Kindergarten to professional schools in science, public health, policy, medical, and veterinarian programs. In addition, we’ve provided variations of the activity to enhance the transfer of fundamental concepts covered in the initial lesson to more complex concepts associated with vaccination and waning immunity. While 10 variations are presented here, the true number of directions in which the game might extend will only be limited by the imagination of its students [6].

The interplay between intrinsic and network dynamics has been the focus of manyinvestigations. Here we use a combination of theoretical and numerical approaches to studythe effects of delayed global feedback on the information transmission properties ofneural networks. Specifically, we compare networks of neurons that display intrinsicinterspike interval correlations (nonrenewal) to networks that do not (renewal). We findthat excitatory and inhibitory delays can tune information transmission by single neuronsbut not by the entire network. Most surprisingly, addition of a delay can change thedependence of the information on the coupling strength for renewal neurons and not fornonrenewal neurons. Our results show that intrinsic ISI correlations can have nontrivialinteractions with network-induced phenomena.

Increasing experimental evidence suggests that epigenetic and microenvironmental factors play a key role in cancer progression. In this respect, it is now generally recognized that the immune system can act as an additional selective pressure, which modulates tumor development and leads, through cancer immunoediting, to the selection for resistance to immune effector mechanisms. This may have serious implications for the design of effective anti-cancer protocols. Motivated by these considerations, we present a mathematical model for the dynamics of cancer and immune cells under the effects of chemotherapy and immunity-boosters. Tumor cells are modeled as a population structured by a continuous phenotypic trait, that is related to the level of resistance to receptor-induced cell death triggered by effector lymphocytes. The level of resistance can vary over time due to the effects of epigenetic modifications. In the asymptotic regime of small epimutations, we highlight the ability of the model to reproduce cancer immunoediting. In an optimal control framework, we tackle the problem of designing effective anti-cancer protocols. The results obtained suggest that chemotherapeutic drugs characterized by high cytotoxic effects can be useful for treating tumors of large size. On the other hand, less cytotoxic chemotherapy in combination with immunity-boosters can be effective against tumors of smaller size. Taken together, these results support the development of therapeutic protocols relying on combinations of less cytotoxic agents and immune-boosters to fight cancer in the early stages.

For an SI type endemic model with one host and two parasite strains, we study thestability of the endemic coexistence equilibrium, where the host and both parasite strainsare present. Our model, which is a system of three ordinary differential equations,assumes complete cross-protection between the parasite strains and reduced fertility andincreased mortality of infected hosts. It also assumes that one parasite strain isexclusively vertically transmitted and cannot persists just by itself. We give severalsufficient conditions for the equilibrium to be locally asymptotically stable. One of themis that the horizontal transmission is of density-dependent (mass-action) type. If thehorizontal transmission is of frequency-dependent (standard) type, we show that, undercertain conditions, the equilibrium can be unstable and undamped oscillations can occur.We support and extend our analytical results by numerical simulations and bytwo-dimensional plots of stability regions for various pairs of parameters.

The paper is devoted to optimization of resonances in a 1-D open optical cavity. Thecavity’s structure is represented by its dielectric permittivity functionε(s). It is assumed thatε(s) takes values in the range1 ≤ ε1 ≤ ε(s) ≤ ε2.The problem is to design, for a given (real) frequency α, a cavity havinga resonance with the minimal possible decay rate. Restricting ourselves to resonances of agiven frequency α, we define cavities and resonant modes with locallyextremal decay rate, and then study their properties. We show that such locally extremalcavities are 1-D photonic crystals consisting of alternating layers of two materials withextreme allowed dielectric permittivities ε1 andε2. To find thicknesses of these layers, a nonlineareigenvalue problem for locally extremal resonant modes is derived. It occurs thatcoordinates of interface planes between the layers can be expressed via arg-function ofcorresponding modes. As a result, the question of minimization of the decay rate isreduced to a four-dimensional problem of finding the zeroes of a function of twovariables.