Dynamics of maturing populations and their asymptotic behaviour

Summary It is well known that the partial differential equation of the traditional model describing the dynamics of an age-dependent population is of the first order hyperbolic type. An equation of that type cannot simultaneously accommodate a renewal type birth boundary condition and a death boundary condition by old age (accumulation of aging injury) and thus lacks biological realism (mortality by old age). In this paper a governing equation of a parabolic type is derived to represent the expected size of a stochastically maturing population. Using techniques well known for the solution of parabolic partial differential and Volterra integral equations, the asymptotic behaviour of such a maturing population is discussed. Due to a non-local boundary condition, the boundary value problem encountered appears to be new.     

Analytical solutions of a minimal model of species migration in a bounded domain

 A minimal model of species migration is presented which takes the form of a parabolic equation with boundary conditions and initial data. Solutions to the differential problem are obtained that can be used to describe the small- and large-time evolution of a species distribution within a bounded domain. These expressions are compared with the results of numerical simulations and are found to be satisfactory within appropriate temporal regimes. The solutions presented can be used to describe existing observations of nematode distributions, can be used as the basis for further work on nematode migration, and may also be interpreted more generally. Received: 15 August 1999     

Electrodiffusion model simulation of rectangular current pulses in a voltage-biased biological channel

Numerical methods are presented for simulating stochastic-in-time current pulses for an electrodiffusion model of the biological channel, with a fixed applied voltage across the channel. The electrodiffusion model consists of the parabolic advection-diffusion equation coupled either to Gauss' law or Poisson's equation, depending on the choice of boundary conditions. The TRBDF2 method is employed for the advection-diffusion equation. The rectangular wave shape of previously simulated traveling wave current pulses is preserved by the full set of partial differential equations for electrodiffusion.     

A nonlinear treatment of the protocell model by a boundary layer approximation

The “protocell” is a mathematical model of a self-maintaining unity based on the dynamics of simple reaction-diffusion processes and a self-controlled dynamics of the surface. In this paper its spatio-temporal behaviour far from the stationary structure is investigated by means of a boundary layer approximation. It is shown in detail how a simplified and mathematically feasible equation can be derived from the original parabolic problem. It turns out that the known instability which is initiated in the linear region around the stationary structure is continued further in the direction to a division by nonlinear dynamics.     

Numerical simulation for a neurotransmitter transport model in the axon terminal of a presynaptic neuron

Neurotransmitters in the terminal bouton of a presynaptic neuron are stored in vesicles, which diffuse in the cytoplasm and, after a stimulation signal is received, fuse with the membrane and release its contents into the synaptic cleft. It is commonly assumed that vesicles belong to three pools whose content is gradually exploited during the stimulation. This article presents a model that relies on the assumption that the release ability is associated with the vesicle location in the bouton. As a modeling tool, partial differential equations are chosen as they allow one to express the continuous dependence of the unknown vesicle concentration on both the time and space variables. The model represents the synthesis, concentration-gradient-driven diffusion, and accumulation of vesicles as well as the release of neuroactive substances into the cleft. An initial and boundary value problem is numerically solved using the finite element method (FEM) and the simulation results are presented and discussed. Simulations were run for various assumptions concerning the parameters that govern the synthesis and diffusion processes. The obtained results are shown to be consistent with those obtained for a compartment model based on ordinary differential equations. Such studies can be helpful in gaining a deeper understanding of synaptic processes including physiological pathologies. Furthermore, such mathematical models can be useful for estimating the biological parameters that are included in a model and are hard or impossible to measure directly.     

Derivation and analysis of a system modeling the chemotactic movement of hematopoietic stem cells

It has been shown that hematopoietic stem cells migrate in vitro and in vivo following the gradient of a chemotactic factor produced by stroma cells. In this paper, a quantitative model for this process is presented. The model consists of chemotaxis equations coupled with an ordinary differential equation on the boundary of the domain and subjected to nonlinear boundary conditions. The existence and uniqueness of a local solution is proved and the model is simulated numerically. It turns out that for adequate parameter ranges, the qualitative behavior of the stem cells observed in the experiment is in good agreement with the numerical results. Our investigations represent a first step in the process of elucidating the mechanism underlying the homing of hematopoietic stem cells quantitatively.      

Exact fundamental solutions of linear parabolic equations with spatially varying coefficients

The exact general solution is obtained to a linear second order ordinary differential equation which has quite general coefficients depending on an arbitrary function of the independent variable. From this, the exact fundamental solution is derived for the corresponding linear parabolic partial differential equation with coefficients depending on the single space coordinate. In a special case this latter equation reduces to one of the Fokker-Planck type. These coefficients are then generalised and the appropriate fundamental solution is obtained. Extensions are given to linear parabolic equations in two andn space dimensions. The paper provides a collection of basic examples which illustrate and develop the theory for the generation of the exact fundamental solutions. Reduction to, and the corresponding fundamental solutions of the Fokker-Planck equations is presented, where appropriate.     

Analytical solution to the initial-value problem for traveling bands of chemotactic bacteria

The governing parabolic partial differential equations for the diffusion and chemotactic transport of a distribution of bacteria and for the diffusion and bacterial degradation of a distribution of chemotactic agent are supplemented with boundary and initial conditions that model the recent capillary tube experiments on the formation and propagation of traveling bands of chemotactic bacteria. An iteration procedure that takes the exact solution to the “diffusionless” problem as a first approximation is applied to solve the equations of the complete theoretical model. It is shown that satisfactory agreement with experiment obtains for the analytical results of the first approximation which relate the velocity of propagation and total number of bacteria cells per unit cross-sectional area in a traveling band to the constant parameters in the governing equations and supplementary conditions. The second approximation is shown to yield approximate analytical expressions for the solution functions which are in close correspondence with previously derived traveling band solutions for values of time after the initial period of formation.     

Peristaltic Transport of Carreau-Yasuda Fluid in a Curved Channel with Slip Effects

The wide occurrence of peristaltic pumping should not be surprising at all since it results physiologically from neuro-muscular properties of any tubular smooth muscle. Of special concern here is to predict the rheological effects on the peristaltic motion in a curved channel. Attention is focused to develop and simulate a nonlinear mathematical model for Carreau-Yasuda fluid. The progressive wave front of peristaltic flow is taken sinusoidal (expansion/contraction type). The governing problem is challenge since it has nonlinear differential equation and nonlinear boundary conditions even in the long wavelength and low Reynolds number regime. Numerical solutions for various flow quantities of interest are presented. Comparison for different flow situations is also made. Results of physical quantities are interpreted with particular emphasis to rheological characteristics.     

A mathematical description of a pump/leak/buffer model for plant nitrate uptake

A pump/leak/buffer model of nitrate in plants is presented in the form of a differential equation, based on the computer simulation model of Scaife (1989). The equation is solved, and the long-term behaviour of the rate of nitrate uptake of the plants described.     

Monte Carlo simulation of miniature endplate current generation in the vertebrate neuromuscular junction.

A Monte Carlo method for modeling the neuromuscular junction is described in which the three-dimensional structure of the synapse can be specified. Complexities can be introduced into the acetylcholine kinetic model used with only a small increase in computing time. The Monte Carlo technique is shown to be superior to differential equation modeling methods (although less accurate) if a three-dimensional representation of synaptic geometry is desired. The conceptual development of the model is presented and the accuracy estimated. The consequences of manipulations such as varying the spacing of secondary synaptic folds or that between the release of multiple quantal packets of acetylcholine, are also presented. Increasing the spacing between folds increases peak current. Decreased spacing of adjacent quantal release sites increases the potentiation of peak current.     

Vesicular transport and apotransferrin in intestinal iron absorption, as shown in the Caco-2 cell model

The potential roles of vesicular transport and apotransferrin (entering from the blood) in intestinal Fe absorption were investigated using Caco-2 cell monolayers with tight junctions in bicameral chambers as a model. As shown previously, addition of 39 microM apotransferrin (apoTf) to the basolateral fluid during absorption studies markedly stimulated overall transport of 1 microM (59)Fe from the apical to the basal chamber and stimulated its basolateral release from prelabeled cells, implicating endo- and exocytosis. Rates of transport more than doubled. Uptake was also stimulated, but only 20%. Specific inhibitors of aspects of vesicular trafficking were applied to determine their potential effects on uptake, retention, and basolateral (overall) transport of (59)Fe. Nocodazole and 5'-(4-fluorosulfonylbenzoyl)-adenosine each reduced uptake and basolateral transport up to 50%. Brefeldin A inhibited about 10%. Tyrphostin A8 (AG10) reduced uptake 35% but markedly stimulated basolateral efflux, particularly that dependent on apoTf. Cooling of cells to 4 degrees C (which causes depolymerization of microtubules and lowers energy availability) profoundly inhibited uptake and basolateral transfer of Fe (7- to 12-fold). Apical efflux (which was substantial) was not temperature affected. Our results support the involvement of apoTf cycling in intestinal Fe absorption and indicate that as much as half of the iron uses apoTf and non-apoTf-dependent vesicular pathways to cross the basolateral membrane and brush border of enterocytes.     

A mathematical theory of enamel solubility and the onset of dental caries: III. Development and computer simulation of a model of caries formation

Part III attempts to develop a diffusion controlled model of caries in the intact enamel employing the kinetic results of the previous two parts. A model of the enamel as a granular bed with a diffusible organic matrix filling the interstices is considered. The basic equations of diffusion and simultaneous reaction are developed under the assumption that all the reactions are so rapid as compared with the diffusion rate, that they are in a quasi-equilibrium state. The resultant system of seven coupled, non-linear parabolic partial differential equations is of such complexity that only numerical solutions could be attempted. Stability restrictions inherent in the problem dictated the use of the DuFort-Frankel numerical solution for parabolic boundary problems. Numerical solutions giving the concentration of all reactants, the rate of mineral loss, and the enamel porosity were obtained for a variety of boundary conditions. It is found that departure from the equilibrium condition expressed in part II is necessary for the occurrence of an attack on the enamel. The rate and pattern of penetration is then determined primarily by the concentrations of undissociated buffer, and salts, together with the rate of diffusion in the surrounding medium. The possibility of a relatively intact surface layer persisting over a demineralized subsurface region due solely to the composition of the demineralizing medium is noted. Remineralization behavior in portions of the carious lesion occurs in the model under certain boundary conditions.     

A model for the analysis of growth data from designed experiments

A model for growth data from designed experiments is presented which extends the stochastic differential equation of Sandland and McGilchrist (1979, Biometrics 35, 255-272). Residual maximum likelihood (REML) is used to estimate the parameters of the model. The model is easily extended to incomplete data and is shown to overcome some of the practical difficulties encountered with the profile model. The procedure is applied to data from experiments on pigs and sheep.     

Bionomics dynamic model of a class of competition systems ☆

Differential equation problem is an important research topic in the international academia. In accordance with certain ecological phenomena, previous research was conducted based on simple observational and statistical data. But this approach does not effectively study the essence of the ecological phenomena. Recently, one dynamic approach has been proposed for the study of ecology in the international academia. According to this approach, first of all, the ecology is reduced to the differential equation model which represents the essential phenomenon, and then the dynamic law and rules of mathematics and biology will be studied. Currently, an extensive research is conducted on the differential equation problem. This paper primarily explores a type of competitive ecological model, which is a system of differential equation with infinite integral. we first study the existence of positive periodic solution to this model, and then present sufficient conditions for the global attractivity of positive periodic solutions.     

Modeling and Analysis of Unsteady Axisymmetric Squeezing Fluid Flow through Porous Medium Channel with Slip Boundary

The aim of this article is to model and analyze an unsteady axisymmetric flow of non-conducting, Newtonian fluid squeezed between two circular plates passing through porous medium channel with slip boundary condition. A single fourth order nonlinear ordinary differential equation is obtained using similarity transformation. The resulting boundary value problem is solved using Homotopy Perturbation Method (HPM) and fourth order Explicit Runge Kutta Method (RK4). Convergence of HPM solution is verified by obtaining various order approximate solutions along with absolute residuals. Validity of HPM solution is confirmed by comparing analytical and numerical solutions. Furthermore, the effects of various dimensionless parameters on the longitudinal and normal velocity profiles are studied graphically.     

Numerical methods and parameter estimation of a structured population model with discrete events in the life history

We consider two numerical methods for the solution of a physiologically structured population (PSP) model with multiple life stages and discrete event reproduction. The model describes the dynamic behaviour of a predator-prey system consisting of rotifers predating on algae. The nitrate limited algal prey population is modelled unstructured and described by an ordinary differential equation (ODE). The formulation of the rotifer dynamics is based on a simple physiological model for their two life stages, the egg and the adult stage. An egg is produced when an energy buffer reaches a threshold value. The governing equations are coupled partial differential equations (PDE) with initial and boundary conditions. The population models together with the equation for the dynamics of the nutrient result in a chemostat model. Experimental data are used to estimate the model parameters. The results obtained with the explicit finite difference (FD) technique compare well with those of the Escalator Boxcar Train (EBT) method. This justifies the use of the fast FD method for the parameter estimation, a procedure which involves repeated solution of the model equations.     

Transient bioheat simulation of the laser-tissue interaction in human skin using hybrid finite element formulation

This paper presents a hybrid finite element model for describing quantitatively the thermal responses of skin tissue under laser irradiation. The model is based on the boundary integral-based finite element method and the Pennes bioheat transfer equation. In this study, temporal discretization of the bioheat system is first performed and leads to the well-known modified Helmholtz equation. A radial basis function approach and the boundary integral based finite element method are employed to obtain particular and homogeneous solutions of the laser-tissue interaction problem. In the boundary integral based finite element formulation, two independent fields are assumed: intra-element field and frame field. The intra-element field is approximated through a linear combination of fundamental solutions at a number of source points outside the element domain. The frame temperature field is expressed in terms of nodal temperature and the corresponding shape function. Numerical examples are considered to verify and assess the proposed numerical model. Sensitivity analysis is performed to explore the thermal effects of various control parameters on tissue temperature and to identify the degree of burn injury due to laser heating.     

Equations of heat distribution within the body

A vector integral equation describing heat distribution within the body has been derived. The factors considered are heat conduction, forced convection via the circulatory system, environmental exchange, metabolic heat production, and change in heat content. The vector partial differential equation and alternative forms incorporating boundary conditions were also developed. A difference equation based on a first-order approximation to the fundamental equations was derived to form the basis of a model for heat distribution within the body. It has been shown that factors involving conduction and convection must be considered independently unless the temperature of the blood flowing from a region of the body is equal to the average temperature of the tissue in that region. If this relation between tissue and blood temperature does exist, only a single temperature from each eleeent is needed to describe the heat distribution. In this latter case, models which ascribe all heat transfer to “equivalent” conduction or to convection can give valid predictions.     

Model of the impulse activity of a neuron receiving a stationary impulse imput]

The stationary model of spike activity of a single neuron is considered. This model exponential decay and constant threshold can be described as discontinuous one. The problem of calculation of Laplas' transform the probability density function (PDF) of interspike interval (ISI) can be reduced to the boundary problem for the usual differential equation. For the exponential PDF of the magnitude of the PSP the Laplas' transform of ISI PDF is expressed by special functions. The mean and standard deviation of ISI for different combinations of parameters were plotted with the aid of computer. The experiment without intracellular recoding, allowing to estimate the values of certain physiological parameters of a neuron on the basis of the model is proposed.