Two-dimensional standing gradient osmotic flow: a generalization of the “isotonic convection approximation”

A linear two-dimensional model of a flow system for solute and fluid transport in intercellular spaces has been obtained by using the so called isotonic convection approximation, already employed in the one-dimensional case (Segel (1970)). This is equivalent to ignoring the convective components in the relevant differential equation. The solutions found by means of the eigenfunctions of a Sturm-Liouville system have been compared with the numerical solutions of the general non-linear model, in which also the convective terms are present.The results of the linear model agree fairly well with those of the non-linear one, in the range of interest of parameters. This fact shows that the ignorance of the profile of fluid velocity does not introduce significant errors in the evaluation of solute and fluid fluxes.The differences of results between the one-dimensional and the twodimensional, the linear and non-linear models, give a way to evaluate the relative effects of the diffusive and convective terms in the differential equation and to estimate the errors introduced by the one-dimensional approximation.     

具体长结构的鱼类种群的数学模型及其应用

本文报道了一个新的鱼类种群的数学模型,它是一阶变系数线性偏微分方程。该模型可用于预测某一水域中某种鱼类种群不同体长的鱼的数量随时间变化的情况。       

Age structure in predator-prey systems: Intraspecific carnivore interaction,passive diffusion,and the paradox of enrichment

An existing arthropod predator-prey model incorporating age structure in the carnivore through the use of the von Foerster equation is extended to include the effects of intraspecific carnivore interaction and passive diffusion or migration. A linear stability analysis of the community equilibrium point of that differential-integral equation system is performed and the resulting secular equation analyzed by the method of D-partitions. These stability results are then compared to those obtained by employing an analogous differential equation model without age structure, in particular as they relate to the so-called paradox of enrichment. In the absence of passive diffusion, it is shown that, unlike for a differential equation model, the paradox of enrichment can occur even with a carnivore which exhibits intraspecific competition. This destabilizing effect of age structure is seen to occur most dramatically when interspecific interactions are large, while the effect of passive diffusion is to offset that tendency and restabilize the system. These predictions are in accordance with relevant experimental evidence involving mites.     

Deterministic Three-Half-Order Kinetic Model for Microbial Degradation of Added Carbon Substrates in Soil

The kinetics of mineralization of carbonaceous substrates has been explained by a deterministic model which is applicable to either growth or nongrowth conditions in soil. The mixed-order nature of the model does not require a priori decisions about reaction order, discontinuity period of lag or stationary phase, or correction for endogenous mineralization rates. The integrated equation is simpler than the integrated form of the Monod equation because of the following: (i) only two, rather than four, interdependent constants have to be determined by nonlinear regression analysis, (ii) substrate or product formation can be expressed explicitly as a function of time, (iii) biomass concentration does not have to be known, and (iv) the required initial estimate for the nonlinear regression analysis can be easily obtained from a linearized form rather than from an interval estimate of a differential equation. ¹⁴CO₂ evolution data from soil have been fitted to the model equation. All data except those from irradiated soil gave better fits by residual sum of squares (RSS) by assuming growth in soil was linear (RSS = 0.71) as opposed to exponential (RSS = 2.87). The underlying reasons for growth (exponential versus linear), no growth, and relative degradation rates of substrates are consistent with the basic mechanisms from which the model is derived.     

Membrane transport models with fast and slow reactions: General analytical solution for a single relaxation

Summary Membrane transport models are usually expressed on the basis of chemical kinetics. The states of a transporter are related by rate constants, and the time-dependent changes of these states are given by linear differential equations of first order. To calculate the time-dependent transport equation, it is necessary to solve a system of differential equations which does not have a general analytical solution if there are more than five states. Since transport measurements in a complex system rarely provide all the time constants because some of them are too rapid, it is more appropriate to obtain approximate analytical solutions, assuming that there are fast and slow reaction steps. The states of the fast steps are related by equilibrium constants, thus permitting the elimination of their differential equations and leaving only those for the slow steps. With a system having only two slow steps, a single differential equation is obtained and the state equations have a single relaxation. Initial conditions for the slow reactions are determined after the perturbation which redistribute the states related by fast reactions. Current and zero-trans uptake equations are calculated. Curve fitting programs can be used to implement the general procedure and obtain the model parameters.     

Response of Tumor Spheroids to Radiation: Modeling and Parameter Estimation

We propose a spatially distributed continuous model for the spheroid response to radiation, in which the oxygen distribution is represented by means of a diffusion-consumption equation and the radiosensitivity parameters depend on the oxygen concentration. The induction of lethally damaged cells by a pulse of radiation, their death, and the degradation of dead cells are included. The compartments of lethally damaged cells and of dead cells are subdivided into different subcompartments to simulate the delays that occur in cell death and cell degradation, with a gain in model flexibility. It is shown that, for a single irradiation and under the hypothesis of a sufficiently small spheroid radius, the model can be reformulated as a linear stationary ordinary differential equation system. For this system, the parameter identifiability has been investigated, showing that the set of unknown parameters can be univocally identified by exploiting the response of the model to at least two different radiation doses. Experimental data from spheroids originated from different cell lines are used to identify the unknown parameters and to test the predictive capability of the model with satisfactory results.     

Muscle dynamics: Dependence of muscle length on changes in external load

A phenomenological theory of muscle dynamics has been elaborated on the basis of data obtained in experiments on hind limb extensor muscles of narcotized cat. Functional dependence of muscle length on external load was explored in conditions of a constant frequency of the efferent stimulation. It was shown that the system under study could be presented for a rather wide class of input signals as a system with nonlinear statics and linear dynamics. The nonlinear statics was shown to be determined mainly by the hysteretical effects of muscle contraction, whereas dynamic element was described by the first order linear differential equation corresponding to the traditional three-component mechanical model of the muscle. A hypothesis was proposed to explain the hysteresis in active muscle on the basis of functioning of the troponin-tropomyosin regulatory complex. Elaborated mathematical model of muscle dynamics can be used to predict and evaluate changes in the muscle length evoked by arbitrary changes in the external load.     

Stochastic models for an open biochemical system

This paper uses the theory of Markov processes to derive stochastic models for a single open biochemical system at st?ady state under 3 sets of assumptions. The system is a one substrate, one product reaction. Each set of assumptions results in a separate solution for the probability functions. A system of linear equations in the probability function as well as an equivalent differential equation in its generating function are derived. The assumption of no flux leads to the first (exact) solution of the linear equations. The form agrees with that of the closed systems. Making assumptions that simplify the system to model active transport results in the second (exact) solution to the linear equations. Assuming the presence of a large number of molecules in the system facilitates obtaining the third (approximate) solution to the differential equations.     

Gene-culture waves of advance

Major genetic and cultural changes may have been coupled during hominid evolution. Since hominids have had a wide geographical distribution for about one million years, any mutant gene or cultural innovation that became established had to spread from its origin. A pair of nonlinear diffusion equations is derived which models the propagation of a mutant gene and a cultural innovation. Both are assumed to originate in the same locality along a linear habitat. The mutant gene and its allele are semidominant, and the two cultural choices are transmitted according to what I call the logistic attraction-repulsion model. The genes influence cultural choice, and the two interact to determine fitness. Of particular interest is the case in which mutant gene and cultural innovation are mutually dependent, neither being able to spread without the other. Each equation of the pair is similar in form to Fisher's equation, with a linear function of the other dependent variable replacing the constant coefficient in the reaction term. The partial differential equations are solved numerically to obtain the asymptotic speeds. Their form also suggests an heuristic argument which has proved useful, but I have been unable to obtain any analytic results. The waves of the system are shown to be of two types, synchronous and asynchronous. When genes and culture are mutually dependent, synchronous travelling waves can exist. However, their existence is dependent on initial conditions, and the speed of propagation is slow.     

Modeling epithelial cell homeostasis: Assessing recovery and control mechanisms

Critical to epithelial cell viability is prompt and direct recovery, following a perturbation of cellular conditions. Although a number of transporters are known to be activated by changes in cell volume, cell pH, or cell membrane potential, their importance to cellular homeostasis has been difficult to establish. Moreover, the coordination among such regulated transporters to enhance recovery has received no attention in mathematical models of cellular function. In this paper, a previously developed model of proximal tubule (Weinstein, 1992, Am. J. Physiol. 263, F784–F798), has been approximated by its linearization about a reference condition. This yields a system of differential equations and auxiliary linear equations, which estimate cell volume and composition and transcellular fluxes in response to changes in bath conditions or membrane transport coefficients. Using the singular value decomposition, this system is reduced to a linear dynamical system, which is stable and reproduces the full model behavior in a useful neighborhood of the reference. Cost functions on trajectories formulated in the model variables (e.g., time for cell volume recovery) are translated into cost functions for the dynamical system. When the model is extended by the inclusion of linear dependence of membrane transport coefficients on model variables, the impact of each such controller on the recovery cost can be estimated with the solution of a Lyapunov matrix equation. Alternatively, solution of an algebraic Riccati equation provides the ensemble of controllers that constitute optimal state feedback for the dynamical system. When translated back into the physiological variables, the optimal controller contains some expected components, as well as unanticipated controllers of uncertain significance. This approach provides a means of relating cellular homeostasis to optimization of a dynamical system.     

A stochastic approach to predator-prey models

A deterministic investigation of a linear differential equation system which describes predator vs prey behavior as a function of equilibrium densities and reproductive rates is given. A more realistic structure of this model in a stochastic framework is presented. The reproductive rates and initial population sizes are considered to be random variables and their probabilistic behavior characterized by various joint probability distributions. The deterministic behaviors of the prey and predator species as functions of time are compared with the mean behaviors in the stochastic model.     

Multiplicatively interacting point processes and applications to neural modeling

We introduce a nonlinear modification of the classical Hawkes process allowing inhibitory couplings between units without restrictions. The resulting system of interacting point processes provides a useful mathematical model for recurrent networks of spiking neurons described as Wiener cascades with exponential transfer function. The expected rates of all neurons in the network are approximated by a first-order differential system. We study the stability of the solutions of this equation, and use the new formalism to implement a winner-takes-all network that operates robustly for a wide range of parameters. Finally, we discuss relations with the generalised linear model that is widely used for the analysis of spike trains.     

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.     

Infection-age structured epidemic models with behavior change or treatment

We formulate infection-age structured susceptible-infective-removed (SIR) models with behavior change or treatment of infections. Individuals change their behavior or have treatment after they are infected. Using infection age as a continuous variable, and dividing infectives into discrete groups with different infection stages, respectively, we formulate a partial differential equation model and an ordinary differential equation model with behavior change or treatment. We derive explicit formulas for the reproductive number by linear stability analysis of the infection-free equilibrium, and explicit formulas for the unique endemic equilibrium, when it exists, for both models. These formulas provide mathematical theoretical frameworks for analysis of impact of behavior change or treatment of infection to the transmission dynamics of infectious diseases. We study several special cases and provide sensitivity analysis for the reproductive numbers with respect to model parameters based on those formulas.     

Infection-age structured epidemic models with behavior change or treatment

We formulate infection-age structured susceptible-infective-removed (SIR) models with behavior change or treatment of infections. Individuals change their behavior or have treatment after they are infected. Using infection age as a continuous variable, and dividing infectives into discrete groups with different infection stages, respectively, we formulate a partial differential equation model and an ordinary differential equation model with behavior change or treatment. We derive explicit formulas for the reproductive number by linear stability analysis of the infection-free equilibrium, and explicit formulas for the unique endemic equilibrium, when it exists, for both models. These formulas provide mathematical theoretical frameworks for analysis of impact of behavior change or treatment of infection to the transmission dynamics of infectious diseases. We study several special cases and provide sensitivity analysis for the reproductive numbers with respect to model parameters based on those formulas.     

A nonlinear,second-order population model

Delay differential, difference, and partial differential equation models are being used more extensively to explain single-species population oscillations and limit cycle behavior. Ordinary differential equation (ODE) models have been largely ignored. This is because first-order ODE models are inherently monotonic. Certainly this is not usual population behavior in the real world. If it is assumed that the per capita growth rate of a population changes over time as a result of regulating factors impinging on it, then a more realistic biological model results. The model translates into a second-order nonlinear ODE. Such a model can exhibit oscillatory and limit cycle as well as monotonic solutions, i.e., behavior for which non-ODE models have been used to explain. Although first-order ODE models are gross simplifications of real phenomena, ODE models in general should not be disregarded as important analytical tools.