On the kinetics of the Langmuir-type heterogeneous reactions

In this paper we investigate three two-dimensional in space mathematical models of the kinetics of unimolecular heterogeneous reactions proceeding onto planar surfaces. All models include the diffusion of the reactant from a bounded vessel towards an adsorbent, adsorption of the molecules of the reactant, their desorption, conversion (reaction) of the adsorbate into a product, instantaneous product desorption, and the diffusion of the product from the adsorbent into the same vessel. One of these models is based on the Langmuir-type kinetics of the surface reactions, the other one is based on the local steady-state value of the surface coverage, and the last one, in addition to the first model, involves the diffusion of the adsorbate along the adsorbent. Diffusivity of all species is assumed to be constant. Models were solved numerically by using the finite difference technique. By changing input parameters the effects of the rate constants of the reactant adsorption, desorption, and reaction and the influence of the surface diffusion of the adsorbate and approximation of the surface coverage by its steady-state value on the kinetics of surface reactions were studied numerically

Modelling of catalytic reactivity of inhomogeneous surfaces in monomer-monomer reactions

The kinetics of a A1 + A2 -> A1A2 reaction on inhomogeneous surfaces with continuously distributed adsorption sites is investigated numerically using two phenomenological models. One of them includes: the bulk diffusion of reactants from a bounded vessel towards the adsorbent and the product bulk one from the adsorbent into the same vessel, adsorption and desorption of molecules of both reactants, and surface diffusion of adsorbed and product particles before their desorption. The other model describes surface reaction provided that concentrations of both reactants at the surface are given. Both models are based on the Langmuir–Hinshelwood reaction mechanism coupled with the Eley–Rideal step. Two surface diffusion mechanisms are used. According to one of them, the diffusion flux of the adsorbed and product particles is described by the standard Fick law, while in the other one the surface diffusion flux is based on the particle jumping into a nearest vacant adsorption site. Simulations were performed using the finite difference technique. The kinetic rate constants, Eley–Rideal steps, and surface diffusion mechanisms influence on the catalytic reactivity of surf aces is studied

On asymptotics of a population model with random mating

This paper deals with a model for an age‐sex structured population consisting of male, single and fertilized female subclasses taking into account a random coupling of sexes (for a period of mating only) and females’ pregnancy. For certain forms of the demographic rates there are presented separable solutions, and the asymptotic behaviour of the general solution is demonstrated. First Published Online: 14 Oct 201

An immunity-structured SEIRS epidemic model with variable infectivity and spatial heterogeneity

A mathematical model is proposed for the spread of an epidemic disease of agedependent infectivity through an asexual population with spatial heterogeneity, assuming that some individuals recover from the disease with temporary immunity, another part recover with permanent immunity, and the last part recover with no immunity. The demographic changes such as births and deaths due to natural causes and the chronological age of individuals are not taken into account. The model is based on a system of partial integro-differential equations including a differential equation to describe the evolution of individuals who have recovered with temporary immunity. The existence and uniqueness of the globally defined solution is proved, and its long-time behaviour is studied

On age‐space structure of an autosomal diploid population dynamics model

We discuss an age‐structured autosomal polylocal multiallelic diploid population dynamics deterministic model taking into account random mating of sexes, females’ pregnancy and its dispersal in whole space. Dispersal mechanism is described by the diffusion one with constant dispersal moduli while the birth moduli depend on the spatial density of the total population with a time delay. It is assumed that the population consists of male, single (nonfertilized) female, and fertilized female subclasses. Using the method of the fundamental solution for the uniformly parabolic second‐order differential operator with bounded Hölder continuous coefficients we prove the existence and uniqueness theorem for the classic solution of the Cauchy problem for this model. We analyze population's growth and decay, too. Mutation is not considered in this paper. First Published Online: 14 Oct 201

The robust finite volume schemes for modeling non-classical surface reactions

A coupled system of nonlinear parabolic PDEs arising in modeling of surface reactions with piecewise continuous kinetic data is studied. The nonclassic conjugation conditions are used at the surface of the discontinuity of the kinetic data. The finite-volume technique and the backward Euler method are used to approximate the given mathematical model. The monotonicity, conservativity, positivity of the approximations are investigated by applying these finite-volume schemes for simplified subproblems, which inherit main new nonstandard features of the full mathematical model. Some results of numerical experiments are discussed

A reaction-diffusion model of the receptor-toxin-antibody interaction

<p>Abstract</p> <p>Background</p> <p>It was recently shown that the treatment effect of an antibody can be described by a consolidated parameter which includes the reaction rates of the receptor-toxin-antibody kinetics and the relative concentration of reacting species. As a result, any given value of this parameter determines an associated range of antibody kinetic properties and its relative concentration in order to achieve a desirable therapeutic effect. In the current study we generalize the existing kinetic model by explicitly taking into account the diffusion fluxes of the species.</p> <p>Results</p> <p>A refined model of receptor-toxin-antibody (RTA) interaction is studied numerically. The protective properties of an antibody against a given toxin are evaluated for a spherical cell placed into a toxin-antibody solution. The selection of parameters for numerical simulation approximately corresponds to the practically relevant values reported in the literature with the significant ranges in variation to allow demonstration of different regimes of intracellular transport.</p> <p>Conclusions</p> <p>The proposed refinement of the RTA model may become important for the consistent evaluation of protective potential of an antibody and for the estimation of the time period during which the application of this antibody becomes the most effective. It can be a useful tool for <it>in vitro </it>selection of potential protective antibodies for progression to <it>in vivo </it>evaluation.</p

The robust finite-volume schemes for modeling nonclassical surface reactions

A coupled system of nonlinear parabolic PDEs arising in modeling of surface reactions with piecewise continuous kinetic data is studied. The nonclassic conjugation conditions are used at the surface of the discontinuity of the kinetic data. The finite-volume technique and the backward Euler method are used to approximate the given mathematical model. The monotonicity, conservativity, positivity of the approximations are investigated by applying these finite-volume schemes for simplified subproblems, which inherit main new nonstandard features of the full mathematical model. Some results of numerical experiments are discussed

Bakterijų dinamikos kompiuterinis modeliavimas

The system of partial differential equations, describing the dynamics of the oxygen consuming bacteria in a chamber with an open top, is investigated. The finite difference method is used for solving the problem. Solutions of two different numerical schemes are compared and analyzed when values of oxygen concentration become negative in the case of a shallow layer.Nagrinėjama netiesinių diferencialinių lygčių sistema, kuri aprašo deguonimi mintančių bakterijų dinamiką atvirame inde. Uždavinys sprendžiamas baigtinių skirtumų metodu. Analizuojami skaitiniai sprendiniai, gaunami taikant skirtingus skaičiavimo būdus, kai deguonies koncentracijos reikšmės sekliame sluoksnyje mažesnės už leistiną ribinę reikšmę

Phenomenological model of bacterial aerotaxis with a negative feedback

A phenomenological model for the suspension of the aerotactic swimming microorganisms placed in a chamber with its upper surface open to air is presented. The model was&nbsp;constructed to embody some complexity of the aerotaxis phenomenon, especially, changes in&nbsp;the average bacteria drift velocity under changing environmental conditions. It was assumed that&nbsp;effective forces applied to the cell (gravitational, drag, and thrust) should be essential for the&nbsp;overall system dynamics; and that bacterial propulsion force, but not their swimming velocity, is&nbsp;proportional to the gradient of the oxygen concentration. Mathematically, the model consists of&nbsp;three coupled equations for the oxygen dynamics; for the cell conservation; and for the balance of&nbsp;forces acting on bacteria. An analytical steady-state solution is given for the shallow and deep layers&nbsp;and numerical results are given for the steady-state and initial value problems which are compared&nbsp;with corresponding ones to the Keller–Segel model