The paper proposes a general framework allowing the analysis of wetting problems in thesituation when interfacial tensions depend on external fields. An equation predictingapparent contact angles of sessile droplets deposited on rough surfaces in the presence ofexternal fields is derived. The problem of wetting is discussed in the framework of thevariational approach. Derivation of a general equation generalizing the Cassie and Wenzelapproaches is presented. The effects related to the line tension which are important fornano-structured surfaces are considered.

The effective dynamics of interacting waves for coupled Schrödinger-Korteweg-de Vriesequations over a slowly varying random bottom is rigorously studied. One motivation forstudying such a system is better understanding the unidirectional motion of interactingsurface and internal waves for a fluid system that is formed of two immiscible layers. Itwas shown recently by Craig-Guyenne-Sulem [1] thatin the regime where the internal wave has a large amplitude and a long wavelength, thedynamics of the surface of the fluid is described by the Schrödinger equation, while thatof the internal wave is described by the Korteweg-de Vries equation. The purpose of thisletter is to show that in the presence of a slowly varying random bottom, the coupledwaves evolve adiabatically over a long time scale. The analysis covers the cases when thesurface wave is a stable bound state or a long-lived metastable state.

We illustrate the appearance of oscillating solutions in delay differential equationsmodeling hematopoietic stem cell dynamics. We focus on autonomous oscillations, arising asconsequences of a destabilization of the system, for instance through a Hopf bifurcation.Models of hematopoietic stem cell dynamics are considered for their abilities to describeperiodic hematological diseases, such as chronic myelogenous leukemia and cyclicalneutropenia. After a review of delay models exhibiting oscillations, we focus on threeexamples, describing different delays: a discrete delay, a continuous distributed delay,and a state-dependent delay. In each case, we show how the system can have oscillatingsolutions, and we characterize these solutions in terms of periods and amplitudes.

In this paper we analyze the stochastic version of a minimalistic multi-strain model,which captures essential differences between primary and secondary infections in denguefever epidemiology, and investigate the interplay between stochasticity, seasonality andimport. The introduction of stochasticity is needed to explain the fluctuations observedin some of the available data sets, revealing a scenario where noise and complexdeterministic skeleton strongly interact. For large enough population size, the stochasticsystem can be well described by the deterministic skeleton gaining insight on the relevantparameter values purely on topological information of the dynamics, rather than classicalparameter estimation of which application is in general restricted to fairly simpledynamical scenarios.

Cancer has recently overtaken heart disease as the world’s biggest killer. Cancer isinitiated by gene mutations that result in local proliferation of abnormal cells and theirmigration to other parts of the human body, a process called metastasis. The metastasizedcancer cells then interfere with the normal functions of the body, eventually leading todeath. There are two hundred types of cancer, classified by their point of origin. Most ofthem share some common features, but they also have their specific character. In thisarticle we review mathematical models of such common features and then proceed to describemodels of specific cancer diseases.

We study the spectral stability of solitary wave solutions to the nonlinear Diracequation in one dimension. We focus on the Dirac equation with cubic nonlinearity, knownas the Soler model in (1+1) dimensions and also as the massive Gross-Neveu model.Presented numerical computations of the spectrum of linearization at a solitary wave showthat the solitary waves are spectrally stable. We corroborate our results by findingexplicit expressions for several of the eigenfunctions. Some of the analytic results holdfor the nonlinear Dirac equation with generic nonlinearity.

Motivated by recent experiments on the electro-hydrodynamic instability of spin-castpolymer films, we study the undulation instability of a thin viscoelastic polymer filmunder in-plane stress and in the presence of either a close by contactor or an electricfield, both inducing a normal stress on the film surface. We find that the in-plane stressaffects both the typical timescale of the instability and the unstable wavelengths. Thefilm stability is also sensitive to the boundary conditions used at the film-substrateinterface. We have considered two conditions, either rigidly attaching the film to thesubstrate or allowing for slip.

We present a model for describing the spread of an infectious disease with publicscreening measures to control the spread. We want to address the problem of determining anoptimal screening strategy for a disease characterized by appreciable duration of theinfectiveness period and by variability of the transmission risk. The specific disease wehave in mind is the HIV infection. However the model will apply to a disease for whichclass-age structure is significant and should not be disregarded.

Recently, Wang and Xiao studied a four-dimensional competitive Lotka-Volterra systemwithin a deterministic environment in [11]. Withthe help of numerical example they showed the existence of a chaotic attractor through theperiod doubling route. In this paper, we are interested to study the dynamics of the samemodel in presence of environmental driving forces. To incorporate the environmentaldriving force into the deterministic system, we perturb the growth rates of each speciesby white noise terms. Then we prove that the unique positive global solution exists forthe noise added system and the general p-th order moment of it is boundedfor p ≥ 1, which ensures that the solution is stochasticallybounded. It is also shown that the solution of the stochastic system is stochasticallypermanent under some simple conditions. Finally , we demonstrate the noise inducedoscillation for the concerned model with the help of numerical example..

The dynamic model of plant growth GreenLab describes plant architecture and functionalgrowth at the level of individual organs. Structural development is controlled by formalgrammars and empirical equations compute the amount of biomass produced by the plant, andits partitioning among the growing organs, such as leaves, stems and fruits. The number oforgans initiated at each time step depends on the trophic state of the plant, which isevaluated by the ratio of biomass available in plant to the demand of all the organs. Thecontrol of the plant organogenesis by this variable induces oscillations in the simulatedplant behaviour. The mathematical framework of the GreenLab model allows to compute theconditions for the generation of oscillations and the value of the period according to theset of parameters. Two case-studies are presented, corresponding to emergence ofoscillations associated to fructification and branching.

Similar alternating patterns arecommonly reported by botanists. In this article, two examples were selected: alternatepatterns of fruits in cucumber plants and alternate appearances of branches inCecropia trees. The model was calibrated from experimental datacollected on these plants. It shows that a simple feedback hypothesis of trophic controlon plant structure allows the emergence of cyclic patterns corresponding to the observedones.

We consider two atomic transitions excited by two variable laser fields in a three-level system. We study the soliton-pair propagation out of resonance and under thermal bath effect. We present general analytical implicit expression of the soliton-pair shape. Furthermore, we show that when the coupling to the environment exceeds a critical value, the soliton-pair propagation through three-level atomic system will be prohibited.

The impacts of the two-beam interference heating on the number of core-shell and embeddednanoparticles and on nanostructure coarsening are studied numerically based on thenon-linear dynamical model for dewetting of the pulsed-laser irradiated, thin (< 20nm) metallic bilayers. The model incorporates thermocapillary forces and disjoiningpressures, and assumes dewetting from the optically transparent substrate atop of thereflective support layer, which results in the complicated dependence of lightreflectivity and absorption on the thicknesses of the layers. Stabilizing thermocapillaryeffect is due to the local thickness-dependent, steady-state temperature profile in theliquid, which is derived based on the mean substrate temperature estimated from theelaborate thermal model of transient heating and melting/freezing. Linear stabilityanalysis of the model equations set for Ag/Co bilayer predicts the dewetting length scalesin the qualitative agreement with experiment.

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.

When invading the tissue, malignant tumour cells (i.e. cancer cells) need to detach fromneighbouring cells, degrade the basement membrane, and migrate through the extracellularmatrix. These processes require loss of cell-cell adhesion and enhancement of cell-matrixadhesion. In this paper we present a mathematical model of an intracellular pathway forthe interactions between a cancer cell and the extracellular matrix. Cancer cells usesimilar mechanisms as with normal cells for their interactions with the extracellularmatrix. We develop a model of cell-matrix adhesion that accounts for reactions between thecell surface receptor integrins, the matrix glycoprotein fibronectin, and the actinfilaments in the cytoskeleton. Each represents components for an intermediate compartment,the extracellular compartment, and the intracellular compartment, respectively. Binding offibronectin with integrins triggers a clustering of protein complexes, which thenactivates and phosphorylates regulatory proteins that are involved in actin reorganisationcausing actin polymerization and stress fibre assembly. Rearrangement of actin filamentswith integrin/fibronectin complexes near adhesion sites and interaction with fibrillarfibronectin produces the force necessary for cell migration, accounting for cell-matrixadhesion.

In this review paper we consider physiologically structured population models that havebeen widely studied and employed in the literature to model the dynamics of a wide varietyof populations. However in a number of cases these have been found inadequate to describesome phenomena arising in certain real-world applications such as dispersion in thestructure variables due to growth uncertainty/variability. Prompted by this, we describedtwo recent approaches that have been investigated in the literature to describe thisgrowth uncertainty/variability in a physiologically structured population. One involvesformulating growth as a Markov diffusion process while the other entails imposing aprobabilistic structure on the set of possible growth rates across the entire population.Both approaches lead to physiologically structured population models with nontrivialdispersion. Even though these two approaches are conceptually quite different, they werefound in [17] to have a close relationship: in somecases with properly chosen parameters and coefficient functions, the resulting stochasticprocesses have the same probability density function at each time.

A linear, uniformly stratified ocean model is used to investigate propagation of baroclinic Kelvin waves in a cylindrical basin. It is found that smaller wave amplitudes are inherent to higher mode individual terms of the obtained solutions that are also evanescent away of a costal line toward the center of the circular basin. It is also shown that the individual terms if the obtained solutions can be visualized as spinning patterns in rotating stratified fluid confined in a circular basin. Moreover, the fluid patterns look rotating in an anticlockwise sense looking above the North Pole and that spinning is more intensive for smaller mode numbers. Finally, we observe the existence of the oceanic region where the pressure increases relatively rapidly with the depth.

Flow cytometric analysis using intracellular dyes such as CFSE is a powerful experimentaltool which can be used in conjunction with mathematical modeling to quantify the dynamicbehavior of a population of lymphocytes. In this survey we begin by providing an overviewof the mathematically relevant aspects of the data collection procedure. We then presentan overview of the large body of mathematical models, along with their assumptions anduses, which have been proposed to describe the dynamics of proliferating cell populations.While much of this body of work has been aimed at modeling the generation structure (cellsper generation) of the proliferating population, several recent models have considered themore fundamental task of modeling CFSE histogram data directly. Such models are analyzedand recent results are discussed. Finally, directions for future research aresuggested.