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.

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.

We consider the problem of state and parameter estimation for a class of nonlinearoscillators defined as a system of coupled nonlinear ordinary differential equations.Observable variables are limited to a few components of state vector and an input signal.This class of systems describes a set of canonic models governing the dynamics of evokedpotential in neural membranes, including Hodgkin-Huxley, Hindmarsh-Rose, FitzHugh-Nagumo,and Morris-Lecar models. We consider the problem of state and parameter reconstruction forthese models within the classical framework of observer design. This framework offerscomputationally-efficient solutions to the problem of state and parameter reconstructionof a system of nonlinear differential equations, provided that these equations are in theso-called adaptive observer canonic form. We show that despite typical neural oscillatorsbeing locally observable they are not in the adaptive canonic observer form. Furthermore,we show that no parameter-independent diffeomorphism exists such that the originalequations of these models can be transformed into the adaptive canonic observer form. Wedemonstrate, however, that for the class of Hindmarsh-Rose and FitzHugh-Nagumo models,parameter-dependent coordinate transformations can be used to render these systems intothe adaptive observer canonical form. This allows reconstruction, at least partially andup to a (bi)linear transformation, of unknown state and parameter values with exponentialrate of convergence. In order to avoid the problem of only partial reconstruction and atthe same time to be able to deal with more general nonlinear models in which the unknownparameters enter the system nonlinearly, we present a new method for state and parameterreconstruction for these systems. The method combines advantages of standardLyapunov-based design with more flexible design and analysis techniques based on thenotions of positive invariance and small-gain theorems. We show that this flexibilityallows to overcome ill-conditioning and non-uniqueness issues arising in this problem.Effectiveness of our method is illustrated with simple numerical examples.

Propagation of polymerization fronts with liquid monomer and liquid polymer is consideredand the influence of vibrations on critical conditions of convective instability isstudied. The model includes the heat equation, the equation for the concentration and theNavier-Stokes equations considered under the Boussinesq approximation. Linear stabilityanalysis of the problem is fulfilled, and the convective instability boundary is founddepending on the amplitude and on the frequency of vibrations

Background: Community-acquired methicillin-resistant Staphylococcusaureus (CA-MRSA), a novel strain of MRSA, has recently emerged and rapidlyspread in the community. Invasion into the hospital setting with replacement of thehospital-acquired MRSA (HA-MRSA) has also been documented. Co-colonization with bothCA-MRSA and HA-MRSA would have important clinical implications given differences inantimicrobial susceptibility profiles and the potential for exchange of geneticinformation.

Methods: A deterministic mathematical model was developed to characterizethe transmission dynamics of HA-MRSA and CA-MRSA in the hospital setting and to quantifythe emergence of co-colonization with both strains

Results: The model analysis shows that the state of co-colonization becomesendemic over time and that typically there is no competitive exclusion of either strain.Increasing the length of stay or rate of hospital entry among patients colonized withCA-MRSA leads to a rapid increase in the co-colonized state. Compared to MRSAdecolonization strategy, improving hand hygiene compliance has the greatest impact ondecreasing the prevalence of HA-MRSA, CA-MRSA and the co-colonized state.

Conclusions: The model predicts that with the expanding community reservoirof CA-MRSA, the majority of hospitalized patients will become colonized with both CA-MRSAand HA-MRSA.

In this paper, we investigate the complex dynamics of a spatial plankton-fish system withHolling type III functional responses. We have carried out the analytical study for bothone and two dimensional system in details and found out a condition for diffusiveinstability of a locally stable equilibrium. Furthermore, we present a theoreticalanalysis of processes of pattern formation that involves organism distribution and theirinteraction of spatially distributed population with local diffusion. The results ofnumerical simulations reveal that, on increasing the value of the fish predation rates,the sequences spots → spot-stripe mixtures → stripes → hole-stripe mixtures holes → wavepattern is observed. Our study shows that the spatially extended model system has not onlymore complex dynamic patterns in the space, but also has spiral waves.

The oriented movement of biological cells or organisms in response to a chemical gradientis called chemotaxis. The most interesting situation related to self-organizationphenomenon takes place when the cells detect and response to a chemical which is secretedby themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) manyparticularized models have been proposed to describe the aggregation phase of thisprocess. Most of efforts were concentrated, so far, on mathematical models in which theformation of aggregate is interpreted as finite time blow-up of cell density. In recentlyproposed models cells are no more treated as point masses and their finite volume isaccounted for. Thus, arbitrary high cell densities are precluded in such description and athreshold value for cells density is a priori assumed. Different modelingapproaches based on this assumption lead to a class of quasilinear parabolic systems withstrong nonlinearities including degenerate or singular diffusion. We give a survey ofanalytical results on the existence and uniqueness of global-in-time solutions, theirconvergence to stationary states and on a possibility of reaching the density threshold bya solution. Unsolved problems are pointed as well.

A simple model of biological evolution of community food webs is introduced. This modelis based on the niche model, which is known to generate model food webs that are verysimilar to empirical food webs. The networks evolve by speciation and extinction.Co-extinctions due to the loss of all prey species are found to play a major role indetermining the longterm shape of the food webs. The central aim is to design the modelsuch that the characteristic parameters of the niche model food webs remain in realisticintervals. When the mutation rule is chosen accordingly, it is found that food webs with acomplex, biologically meaningful structure emerge and that the statistics of extinctionevents agrees well with that observed in the paleontological data.

In this short note, we apply the technique developed in [Math. Model. Nat. Phenom., 5 (2010), No. 4, 122-149] to study the long-time evolution for Schrödinger equation with slowly decayingpotential.

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 classify the hulls of different limit-periodic potentials and show that the hull of alimit-periodic potential is a procyclic group. We describe how limit-periodic potentialscan be generated from a procyclic group and answer arising questions. As an expositorypaper, we discuss the connection between limit-periodic potentials and profinite groups ascompletely as possible and review some recent results on Schrödinger operators obtained inthis context.

Bacillus subtilis swarms rapidly over the surface of a synthetic mediumcreating remarkable hyperbranched dendritic communities. Models to reproduce such effectshave been proposed under the form of parabolic Partial Differential Equations representingthe dynamics of the active cells (both motile and multiplying), the passive cells(non-motile and non-growing) and nutrient concentration. We test the numerical behavior ofsuch models and compare them to relevant experimental data together with a criticalanalysis of the validity of the models based on recent observations of the swarmingbacteria which show that nutrients are not limitating but distinct subpopulations growingat different rates are likely present.

Evolution of cell populations can be described with dissipative particle dynamics, whereeach cell moves according to the balance of forces acting on it, or with partialdifferential equations, where cell population is considered as a continuous medium. Wecompare these two approaches for some model examples

The aim of this paper is to study the effect of vibrations on convective instability ofreaction fronts in porous media. The model contains reaction-diffusion equations coupledwith the Darcy equation. Linear stability analysis is carried out and the convectiveinstability boundary is found. The results are compared with direct numericalsimulations.

Recent technological advances including brain imaging (higher resolution in space andtime), miniaturization of integrated circuits (nanotechnologies), and acceleration ofcomputation speed (Moore’s Law), combined with interpenetration between neuroscience,mathematics, and physics have led to the development of more biologically plausiblecomputational models and novel therapeutic strategies. Today, mathematical models ofirreversible medical conditions such as Parkinson’s disease (PD) are developed andparameterised based on clinical data. How do these evolutions have a bearing on deep brainstimulation (DBS) of patients with PD? We review how the idea of DBS, a standardtherapeutic strategy used to attenuate neurological symptoms (motor, psychiatric), hasemerged from past experimental and clinical observations, and present how, over the lastdecade, computational models based on different approaches (phase oscillator models,spiking neuron network models, population-based models) have started to shed light ontoDBS mechanisms. Finally, we explore a new mathematical modelling approach based on neuralfield equations to optimize mechanisms of brain stimulation and achieve finer control oftargeted neuronal populations. We conclude that neuroscience and mathematics are crucialpartners in exploring brain stimulation and this partnership should also include otherdomains such as signal processing, control theory and ethics.

Motivated by structured parasite populations in aquaculture we consider a class ofsize-structured population models, where individuals may be recruited into the populationwith distributed states at birth. The mathematical model which describes the evolution ofsuch a population is a first-order nonlinear partial integro-differential equation ofhyperbolic type. First, we use positive perturbation arguments and utilise results fromthe spectral theory of semigroups to establish conditions for the existence of a positiveequilibrium solution of our model. Then, we formulate conditions that guarantee that thelinearised system is governed by a positive quasicontraction semigroup on the biologicallyrelevant state space. We also show that the governing linear semigroup is eventuallycompact, hence growth properties of the semigroup are determined by the spectrum of itsgenerator. In the case of a separable fertility function, we deduce a characteristicequation, and investigate the stability of equilibrium solutions in the general case usingpositive perturbation arguments.

We consider the zigzag half-nanotubes (tight-binding approximation) in a uniform magneticfield which is described by the magnetic Schrödinger operator with a periodic potentialplus a finitely supported perturbation. We describe all eigenvalues and resonances of thisoperator, and theirs dependence on the magnetic field. The proof is reduced to theanalysis of the periodic Jacobi operators on the half-line with finitely supportedperturbations.

In proteomics study, Imaging Mass Spectrometry (IMS) is an emerging and very promisingnew technique for protein analysis from intact biological tissues. Though it has showngreat potential and is very promising for rapid mapping of protein localization and thedetection of sizeable differences in protein expression, challenges remain in dataprocessing due to the difficulty of high dimensionality and the fact that the number ofinput variables in prediction model is significantly larger than the number ofobservations. To obtain a complete overview of IMS data and find trace features based onboth spectral and spatial patterns, one faces a global optimization problem. In thispaper, we propose a weighted elastic net (WEN) model based on IMS data processing needs ofusing both the spectral and spatial information for biomarker selection andclassification. Properties including variable selection accuracy of the WEN model arediscussed. Experimental IMS data analysis results show that such a model not only reducesthe number of side features but also helps new biomarkers discovery.

It is shown that a real Hankel matrix admits an approximate block diagonalization inwhich the successive transformation matrices are upper triangular Toeplitz matrices. Thestructure of this factorization was first fully discussed in [1]. This approach isextended to obtain the quotients and the remainders appearing in the Euclidean algorithmapplied to two polynomials u(x) andv(x) of degree n andm, respectively, whith m <n

An optimal control problem is studied for a predator-prey system of PDE, with a logisticgrowth rate of the prey and a general functional response of the predator. The controlfunction has two components. The purpose is to maximize a mean density of the two speciesin their habitat. The existence of the optimal solution is analyzed and some necessaryoptimality conditions are established. The form of the optimal control is found in someparticular cases.