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.

In this paper, we consider one-dimensional wave equation with real-valued square-summablepotential. We establish the long-time asymptotics of solutions by, first, studying thestationary problem and, second, using the spectral representation for the evolutionequation. In particular, we prove that part of the wave travels ballistically ifq ∈ L 2(ℝ+) and this result issharp.

Since Rosenzweig showed the destabilisation of exploited ecosystems, the so called Paradox of enrichment, severalmechanisms have been proposed to resolve this paradox. In this paper we will show that a feeding threshold in the functional response for predators feeding on a prey population stabilizes the system and that there exists a minimumthreshold value above which the predator-prey system is unconditionally stable with respect to enrichment. Two models areanalysed, the first being the classical Rosenzweig-MacArthur (RM) model with an adapted Holling type-II functional response to include a feeding threshold. This mathematical model can be studied using analytical tools, which gives insight into the mathematical properties of the two dimensional ode system and reveals underlying stabilisation mechanisms. The second model is a mass-balance (MB) model for a predator-prey-nutrient system with complete recycling ofthe nutrient in a closed environment. In this model a feeding threshold is also taken into account for the predator-prey trophic interaction. Numerical bifurcation analysis is performed on both models. Analysis results are compared between models and are discussed in relation to the analytical analysis of the classical RM model. Experimental data from the literature of a closed system with ciliates, algae and a limiting nutrient are used to estimate parameters for the MB model. This microbial system forms the bottom trophic levels of aquatic ecosystems and therefore a complete overview of its dynamics is essential for understanding aquatic ecosystem dynamics.

A computer-aided method for accurately carrying out the Chapman-Enskog expansion of the Boltzmann equation, including its inelastic variant, is presented and employed to derive a hydrodynamic description of a dilute binary mixture of smooth inelastic spheres. Constitutive relations, formally valid for all physical values of the coefficients of restitution, are calculated by carrying out the pertinent Chapman-Enskog expansion to sufficient high orders in the Sonine polynomials to ensure numerical convergence. The resulting hydrodynamic description is applied to the analysis of a vertically vibrated binary mixture of particles (under gravity) differing only in their respective coefficients of restitution. It is shown that even with this “minor”difference the mixture partly segregates, its steady state exhibiting a sandwich-like configuration.

We perform a numerical study of the fluctuations of the rescaled hydrodynamic transversevelocity field during the cooling state of a homogeneous granular gas. We are interestedin the role of Molecular Chaos for the amplitude of the hydrodynamic noise and itsrelaxation in time. For this purpose we compare the results of Molecular Dynamics (MD,deterministic dynamics) with those from Direct Simulation Monte Carlo (DSMC, randomprocess), where Molecular Chaos can be directly controlled. It is seen that the large timedecay of the fluctuation’s autocorrelation is always dictated by the viscosity coefficientpredicted by granular hydrodynamics, independently of the numerical scheme (MD or DSMC).On the other side, the noise amplitude in Molecular Dynamics, which is known toviolate the equilibrium Fluctuation-Dissipation relation, is not alwaysaccurately reproduced in a DSMC scheme. The agreement between the two models improves ifthe probability of recollision (controlling Molecular Chaos) is reduced by increasing thenumber of virtual particles per cells in the DSMC. This result suggests that DSMC is notnecessarily more efficient than MD, if the real number of particles is small(~103 ± 104) and if one is interested in accurately reproducefluctuations. An open question remains about the small-times behavior of theautocorrelation function in the DSMC, which in MD and in kinetic theory predictions is nota straight exponential.

We study pattern-forming instabilities in reaction-advection-diffusion systems. Wedevelop an approach based on Lyapunov-Bloch exponents to figure out the impact of aspatially periodic mixing flow on the stability of a spatially homogeneous state. We dealwith the flows periodic in space that may have arbitrary time dependence. We propose adiscrete in time model, where reaction, advection, and diffusion act as successiveoperators, and show that a mixing advection can lead to a pattern-forming instability in atwo-component system where only one of the species is advected. Physically, this can beexplained as crossing a threshold of Turing instability due to effective increase of oneof the diffusion constants.

The emergence of multidrug resistance among gram-negative bacilli is complex. Numerousfactors need to be considered, including the biological fitness cost of resistance, fitnesscompensatorymutations and frequency and type of antibiotic exposure. A mathematical modelevaluating these complex relationships was developed in an individual colonized with strains ofpan-susceptible, single-, two- and multidrug-resistant (MDR) gram-negative bacilli (GN). The effectof bacterial fitness, compensatory mutations and the frequency of three-antimicrobial regimenexposure to predominance of multidrug-resistant strains were quantified. The model predicts thatinitially, in the absence of antibiotic exposure, the biologically fitter pan-susceptible strain predominatesover the resistant strains. Over time, the fitness of the MDR strains increases fasterwith repeated antimicrobial exposure, through compensatory-fitness mutations. Increasing thefrequency of exposure to the three-antimicrobial regimen or, increasing the initial fitness of theresistant strains, substantially decreases the time to MDR-GN predominance. The model impliesthat when MDR-GN strains evolve into strains that are fitter than susceptible strains, a reductionin antimicrobial exposure may not result in a decrease of MDR-GN, since the absence of selectiveantimicrobial pressure would no longer favor susceptible strains. The model also implies thatantimicrobial cycling may promote the emergence of MDR-GN.

We discuss the zero-controllability and the zero-stabilizability for the nonnegative solutions to some Fisher-like models with nonlocal terms describing the dynamics of biological populations with diffusion, logistic term and migration. A necessary and sufficient condition for the nonnegative zero-stabilizabiity for a linear integro-partial differential equation is obtained in terms of the sign of the principal eigenvalue to a certain non-selfadjoint operator. For a related semilinear problem a necessary condition and a sufficient condition for the local nonnegative zero-stabilizability are also derived in terms of the magnitude of the above mentioned principal eigenvalue. The rate of stabilization corresponding to a simple feedback stabilizing control is dictated by the principal eigenvalue. A large principal eigenvalue leads to a fast stabilization to zero. A necessary condition and a sufficient condition for the stabilization to zero of the predator population in a predator-prey system is also investigated. Finally, a method to approximate the above mentioned principal eigenvalues is indicated.

Spatial heterogeneity greatly affects the population spread. Although the theory forbiological invasion in heterogeneous spatially continuous habitats have receivedconsiderable attention, spatially discrete models have remained outside of the mainstream.In this study, we formulate and analyze a Coupled Map Lattice model for a single speciespopulation invading a two dimensional heterogeneous environment. The population growthrate and dispersal coefficient depend on the site quality. We first find an analyticalcriterium for the spread success in terms of the population growth rate and the dispersalcoefficient in unfavorable regions. We then implemented our model for two distinct spatialconfigurations: periodical stripe-like and randomized environments. The spread rate iscomputed numerically and it shows a decrease with an increase of the fraction of thehostile sites. However, we observed that invasion success does not depend on the fractionof favorable sites but crucially depends on the connectivity of favorable regions.

We construct the heat kernels of the sub-Laplacian and the Laplacian on theh-Heisenberg group and compute the limits as h → 0 ofthe heat kernels.

The CMV matrices are unitary analogues of the discrete one-dimensional Schrödingeroperators. We review spectral properties of a few classes of CMV matrices and describefamilies of random and deterministic CMV matrices which exhibit a transition in thedistribution of their eigenvalues.

In order to study the impact of fishing on a grouper population, we propose in this paperto model the dynamics of a grouper population in a fishing territory by using structuredmodels. For that purpose, we have integrated the natural population growth, the fishing,the competition for shelter and the dispersion. The dispersion was considered as aconsequence of the competition. First we prove, that the grouper stocks may be lesssensitive to the removal of large male individuals if female population are totallyprotected. Second, we show that fishing does not disturb the demographic structure of thepopulation. Finally, we prove that female selective fisheries have the potential ofdrastically reduce reproductive rates. We also prove that male fishing decreasescompetition and then increases the total population number.

We discuss the hull of a multi-dimensional limit-periodic potential and show that such ahull is an inverse limit of product cyclic groups. We present the result in an explicitway, which will be useful for a future study of multi-dimensional limit-periodicSchrödinger operators.

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

The influence of a global delayed feedback control which acts on a system governed by asubcritical complex Ginzburg-Landau equation is considered. The method based on avariational principle is applied for the derivation of low-dimensional evolution models.In the framework of those models, one-pulse and two-pulse solutions are found, and theirlinear stability analysis is carried out. The application of the finite-dimensional modelallows to reveal the existence of chaotic oscillatory regimes and regimes withdouble-period and quadruple-period oscillations. The diagram of regimes resembles thosefound in the damped-driven nonlinear Schrödinger equation. The obtained results arecompared with the results of direct numerical simulations of the original problem.

Biological rhythms occur at different levels in the organism. In single cells, the celldivision cycle shows rhythmicity in the way its molecular regulators, the cyclin dependantkinases (CDKs), modulate their activity periodically to ensure a healthy progression. Intissues, cell proliferation is driven by the circadian clock, which modulates theprogression through the cell cycle along the day. The circadian clock shows endogenousrhythmicity through a robust network of transcription-translation feedback loops thatcreate sustained oscillations. Rhythmicity is preserved in cell populations by thecoordination of the clocks among cells, through rhythmic synchronization signals. Here wediscuss mechanisms for generating rhythmic activities in cell populations by reviewingsome of the mathematical models that deal with them. We discuss the implication ofbiological rhythms for tissue growth and the possible application to chronomodulatedcancer treatments.

New conservation laws bifurcating from the classical form of conservation laws are constructed to the nonlinear Boussinesq model describing internal Kelvin waves propagating in a cylindrical wave field of an uniformly stratified water affected by the earth’s rotation. The obtained conservation laws are different from the well known energy conservation law for internal waves and they are associated with symmetries of the Boussinesq model. Particularly, it is shown that application of Lie group analysis provide three infinite sets of nontrivial integral conservation laws depending on two arbitrary functions, namely a(t, θ), b(t, r) and an arbitrary function c(t, θ, r) which is given implicitly as a nontrivial solution of a partial differential equation involving a(t, θ) and b(t, r).

In this article a novel model framework to simulate cells and their internal structure is described. The model is agent-based and suitable to simulate single cells with a detailed internal structure as well as multi-cellular compounds. Cells are simulated as a set of many interacting particles, with neighborhood relations defined via a Delaunay triangulation. The interacting sub-particles of a cell can assume specific roles – i.e., membrane sub-particle, internal sub-particle, organelles, etc –, distinguished by specific interaction potentials and, eventually, also by the use of modified interaction criteria. For example, membrane sub-particles may interact only on a two-dimensional surface embedded on three-dimensional space, described via a restricted Delaunay triangulation. The model can be used not only to study cell shape and movement, but also has the potential to investigate the coupling between internal space-resolved movement of molecules and determined cell behaviors.

This paper is devoted to solving of dynamic problems in biomechanics that requiredetailed study of fast processes. Numerical method of characteristics is used to model thetemporal development of the processes with high accuracy.