Asymptotic behaviour of solutions to abstract logistic equations

We analyze the asymptotic behaviour of solutions of the abstract differential equation u'(t)=Au(t)-F(u(t))u(t)+f. Our results are applicable to models of structured population dynamics in which the state space consists of population densities with respect to the structure variables. In the equation the linear term A corresponds to internal processes independent of crowding, the nonlinear logistic term F corresponds to the influence of crowding, and the source term f corresponds to external effects. We analyze three separate cases and show that for each case the solutions stabilize in a way governed by the linear term. We illustrate the results with examples of models of structured population dynamics -- a model for the proliferation of cell lines with telomere shortening, a model of proliferating and quiescent cell populations, and a model for the growth of tumour cord cell populations.     

六阶滞后差分方程渐近稳定性分析

在这篇论文中,我们应用特征根法、生成函数法等方法,讨论了六阶滞后差分方程的渐近稳定性,得到了其零解渐近稳定的充要条件.     

Asymptotic behaviour of positive solutions of periodic delay logistic equations

The delay logistic equation with periodic coefficients is studied. Under condition (2.1) below the existence and global attractivity of a unique periodic solution is proved by mean of monotonicity methods.Work partially supported by G.N.A.F.A.-C.N.R.     

Inverse bifurcation problem, singular Wiener-Hopf equations, and mathematical models in ecology

A single-species population dynamics with dispersal in a spatially heterogeneous environment is modeled by a nonlinear reaction-diffusion equation with a potential term. To each nonlinear kinetics there corresponds a bifurcation curve that describes the relation between the growth rate and the central density of a steady-state population distribution. Our main concern is an inverse problem for this correspondence. The existence of nonlinear kinetics realizing a prescribed bifurcation curve is established. It is shown that the freedom of such kinetics is of degree finite and even, depending only on the heterogeneity of the environment, and conversely that any nonnegative even integer occurs as the degree of freedom in some environments. A discussion is also made on under what kind of environment the degree is equal to zero or is positive. The mathematical analysis involves the development of a general theory for singular multiplicative Wiener-Hopf integral equations.     

Asymptotic stability in a generalized probabilistic/deterministic model of the cell cycle

A new mathematical model of the cell cycle is presented which generalizes the probabilistic/deterministic model of Lasota-Mackey [1] and the tandem model of Tyson and Hannsgen [7]. By the use of a multiplicative (exponential) Lyapunov function a stability theorem is proved, parallel to the results of Lasota-Mackey. Some open problems related to the tandem model are also solved.     

Linear stability criteria for population models with periodically perturbed delays

We examine some simple population models that incorporate a time delay which is not a constant but is instead a known periodic function of time. We examine what effect this periodic variation has on the linear stability of the equilibrium states of scalar population models and of a simple predator prey system. The case when the delay differs from a constant by a small amplitude periodic perturbation can be treated analytically by using two-timing methods. Of particular interest is the case when the system is initially marginally stable. The introduction of variation in the delay can then have either a stabilising effect or a destabilizing one, depending on the frequency of the periodic perturbation. The case when the periodic perturbation has large amplitude is studied numerically. If the fluctuation is large enough the effect can be stabilising.     

Felinine stability in the presence of selected urine compounds

Summary. The stability of felinine, an amino acid present in feline urine, was investigated. Synthetic felinine was unstable in the urine of a selection of mammals. Felinine was found to stable in feline urine in which urea had been degraded. Synthetic felinine was found to react specifically with urea and did not react with urea analogues such as biuret or thiourea or other nucleophilic compounds such as ammonia which is more nucleophilic or acetamide and water which are less nucleophilic than urea. The reaction of urea and felinine was independent of pH over the range of 3–10. Urea did not react with N-acetyl-felinine suggesting a felinine N-terminal interaction with urea. Mass spectral analysis of the reaction products showed the presence of carbamylated felinine and fragmentation ions derived from carbamyl-felinine. The physiological relevance of felinine carbamylation is yet to be determined.     

Equilibrium structure and stability in a frequency-dependent,two-population diploid model

We investigate the equilibrium structure for an evolutionary genetic model in discrete time involving two monoecious populations subject to intraspecific and interspecific random pairwise interactions. A characterization for local stability of an equilibrium is found, related to the proximity of this equilibrium with evolutionarily stable strategies (ESS). This extends to a multi-population framework a principle initially proposed for single populations, which states that the mean population strategy at a locally stable equilibrium is as close as possible to an ESS.     

Global asymptotic stability for a vector disease model with spatial spread

Summary We analyze the global behaviour of a vector disease model which involves spatial spread and hereditary effects. This model can be applied to investigate growth and spread of malaria. No immunization is considered. We prove that, if the recovery rate is less than or equal to a threshold value, the disease dies out, otherwise the infectious people density tends to a homogeneous distribution. Our results follow using contracting convexes techniques and agree with the results given by K. L. Cooke for the model without diffusion.Work supported by C.N.R., Grant No. 79.00696.01.     

鼠类-天敌系统渐近稳定性的数学分析

天敌能否控制鼠害是鼠害防治实践中最重要的课题之一,通过建立鼠类--天敌系统的捕食--食饵离散模型,推导出自然生态条件下鼠类数量与天敌数量的平衡关系以及鼠类--天敌系统渐近稳定的判定条件,数学分析表明,天敌对鼠的数值反应、功能反应以及鼠类种群繁殖调节是系统能够渐近稳定的决定因素。     

Quality and flavour stability of coffee substitute prepared by extrusion of wheat germ and chicory roots

Summary. A mixture of roasted chicory roots and wheat germ (1:1 w/w) was subjected to extrusion processing for preparation of coffee substitute. Comparative studies concerning sensory characteristics and headspace volatiles were carried out between genuine coffee and a freshly prepared coffee substitute. The sensory evaluation revealed similarities between the two samples. The comparative odour profile analysis showed that the sweetish/caramel-like note scored higher in our coffee substitute sample than in real coffee, whereas the other odour quality attributes showed an opposite trend. The high quality of the fresh coffee substitute was correlated to the presence of volatiles that are responsible for the fresh coffee aroma, such as: 2-methylbutanal, 3-methylbutanal, 2-methylfuran and 2,3-butanedione in high concentration. Storage of coffee substitute samples revealed a noticeable decrease in concentration of the Strecker aldehydes and diketones and a remarkable increase in phenolic compounds, whereas pyrazine and furan derivatives showed no linear changes during storage. The ratio of 2,3-butanedione/2-methylfuran (B/M) was used as an indicator for aging of coffee substitute samples. The variation in this ratio (B/M) during storage for 6 months was consistent with that of the odour profile analysis. Authors’ address: Prof. Dr. Hoda H. M. Fadel, Chemistry of Flavour and Aroma Department, National Research Centre, Al-Behos St., Dokki, Cairo, Egypt     

增长区域上扩散的Logistic方程解的全局渐近性

考虑了一个各向同性增长区域上扩散的Logistic型方程.首先给出了增长区域上的反应扩散模型;然后利用比较原理,通过构造合适的上下解得到了反应扩散问题解的渐近性态;最后给出了数值模拟,以验证理论结果的正确性.     

Asymptotics of the total net direct pharmacological effect for large drug doses

The direct pharmacological effect E is described by the E _max model relating E to the drug plasma concentration C _p . The area under the effect vs. time curve (AUC _E ) is used as the measurement of the total net pharmacological effect. The drug plasma concentrations are solutions of compartmental systems of ordinary differential equations with the input terminated after a finite time and controlled in a proportional manner by a single dose-like parameter. The asymptotics of the time derivative of C _p for large doses are derived and used as conditions which have to be satisfied by functions for which the asymptotics of the integral defining AUC _E are derived. The AUC _E is proportional to the time T _C>EC50 for which the drug concentration stays above the threshold level EC ₅₀. The threshold EC ₅₀ denotes the drug plasma concentration which elicits 50% of the maximum effect. The parameter T _C>EC50 is proportional to the logarithm of drug dose for large doses and its asymptotics is calculated up to the order o(1) as dose increases to infinity. The results are applied to basic pharmacokinetic systems. Received: 7 December 1999 / Revised version: 23 May 2000 / Published online: 23 October 2000     

Numerical bifurcation analysis of delay differential equations arising from physiological modeling

This paper has a dual purpose. First, we describe numerical methods for continuation and bifurcation analysis of steady state solutions and periodic solutions of systems of delay differential equations with an arbitrary number of fixed, discrete delays. Second, we demonstrate how these methods can be used to obtain insight into complex biological regulatory systems in which interactions occur with time delays: for this, we consider a system of two equations for the plasma glucose and insulin concentrations in a diabetic patient subject to a system of external assistance. The model has two delays: the technological delay of the external system, and the physiological delay of the patient's liver. We compute stability of the steady state solution as a function of two parameters, compare with analytical results and compute several branches of periodic solutions and their stability. These numerical results allow to infer two categories of diabetic patients for which the external system has different efficiency.     

A new explicit stability criterion for human periodic breathing<--RID="*"--><--ID="*" This research was supported by grants from DRED (96-920).-->

The aim of this paper is to carry out a stability analysis for periodic breathing in humans that incorporates the dynamic characteristics of ventilation control. A simple CO₂ model that takes into account the main elements of the respiratory system, i.e. the lungs and the ventilatory controller with its dynamic properties, is presented. This model results in a three-dimensional non-linear delay differential system for which there exists a unique equilibrium point. Our stability analysis of this equilibrium point leads to the definition of a new explicit stability criterion and to the demonstration of the existence of a Hopf bifurcation. Numerical simulations illustrate the influence of physiological parameters on the stability of ventilation, and particularly the major role of the dynamic characteristics of the respiratory controller. Received: 2 February 1999 / Revised version: 18 June 1999 / Published online: 23 October 2000     

Logistic型时滞泛函微分方程解的渐近性

本文研究Logistic型时滞微分方程的解的渐近性,获得了其每一个解趋向零的充分条件,改进了已有的结果.     

Existence and stability of equilibria in age-structured population dynamics

The existence of positive equilibrium solutions of the McKendrick equations for the dynamics of an age-structured population is studied as a bifurcation phenomenon using the inherent net reproductive rate n as a bifurcation parameter. The local existence and uniqueness of a branch of positive equilibria which bifurcates from the trivial (identically zero) solution at the critical value n=1 are proved by implicit function techniques under very mild smoothness conditions on the death and fertility rates as functional of age and population density. This first requires the development of a suitable linear theory. The lowest order terms in the Liapunov-Schmidt expansions are also calculated. This local analysis supplements earlier global bifurcation results of the author. The stability of both the trivial and the positive branch equilibria is studied by means of the principle of linearized stability. It is shown that in general the trivial solution losses stability as n increases through one while the stability of the branch solution is stable if and only if the bifurcation is supercritical. Thus the McKendrick equations exhibit, in the latter case, a standard exchange of stability with regard to equilibrium states as they depend on the inherent net reproductive rate. The derived lower order terms in the Liapunov-Schmidt expansions yield formulas which explicitly relate the direction of bifurcation to properties of the age-specific death and fertility rates as functionals of population density. Analytical and numerical results for some examples are given which illustrate these results.     

Estimating equations for spatially correlated data in multi-dimensional space

We use the quasilikelihood concept to propose an estimatingequation for spatial data with correlation across the studyregion in a multi-dimensional space. With appropriate mixingconditions, we develop a central limit theorem for a randomfield under various L_p metrics. The consistency and asymptoticnormality of quasilikelihood estimators can then be derived.We also conduct simulations to evaluate the performance of theproposed estimating equation, and a dataset from East LansingWoods is used to illustrate the method.     

高阶非线性变时滞差分方程解的渐近性

本文研究了一类高阶非线性变滞量差分方程解的渐近性,给出了该类方程的非振动解当n→＋∞时渐近趋于零或超于某有限非零值的几个充分条件。