Mathematical models and simulations of bacterial growth and chemotaxis in a diffusion gradient chamber

The diffusion gradient chamber (DGC) is a novel device developed to study the response of chemotactic bacteria to combinations of nutrients and attractants [7]. Its purpose is to characterize genetic variants that occur in many biological experiments. In this paper, a mathematical model which describes the spatial distribution of a bacterial population within the DGC is developed. Mathematical analysis of the model concerning positivity and boundedness of the solutions are given. An ADI (Alternating Direction Implicit) method is constructed for finding numerical solutions of the model and carrying out computer simulations. The numerical results of the model successfully reproduced the patterns that were observed in the experiments using the DGC. Received: 3 June 1997 / Revised version: 15 August 2000 / Published online: 20 December 2000     

Periodic solutions in a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor

We obtain necessary and sufficient conditions on the existence of a unique positive equilibrium point and a set of sufficient conditions on the existence of periodic solutions for a 3-dimensional system which arises from a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor. Our results improve the corresponding results obtained by Hsu, Luo, and Waltman [1]. Received: 20 November 1997 / Revised version: 12 February 1999 / Published online: 20 December 2000     

The effect of dispersal on single-species nonautonomous dispersal models with delays

In this paper, single-species nonautonomous dispersal models with delays are considered. An interesting result on the effect of dispersal for persistence and extinction is obtained. That is, if the species is persistent in a patch then it is also persistent in all other patches; if the species is permanent in a patch then it is also permanent in all other patches; if the species is extinct in a patch then it is also extinct in all other patches. Furthermore, some new sufficient conditions for the permanence and extinction of the species in a patch are established. The existence of positive periodic solutions is obtained in the periodic case by employing Teng and Chen's results on the existence of positive periodic solutions for functional differential equations. Received: 26 June 2000 / Revised version: 6 October 2000 / Published online: 10 April 2001     

Stochastic epidemics in dynamic populations: quasi-stationarity and extinction

Empirical evidence shows that childhood diseases persist in large communities whereas in smaller communities the epidemic goes extinct (and is later reintroduced by immigration). The present paper treats a stochastic model describing the spread of an infectious disease giving life-long immunity, in a community where individuals die and new individuals are born. The time to extinction of the disease starting in quasi-stationarity (conditional on non-extinction) is exponentially distributed. As the population size grows the epidemic process converges to a diffusion process. Properties of the limiting diffusion are used to obtain an approximate expression for τ, the mean-parameter in the exponential distribution of the time to extinction for the finite population. The expression is used to study how τ depends on the community size but also on certain properties of the disease/community: the basic reproduction number and the means and variances of the latency period, infectious period and life-length. Effects of introducing a vaccination program are also discussed as is the notion of the critical community size, defined as the size which distinguishes between the two qualitatively different behaviours. Received: 14 February 2000 / Revised version: 5 June 2000 / Published online: 24 November 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.     

Structured population on two patches: modeling dispersal and delay

We derive from the age-structured model a system of delay differential equations to describe the interaction of spatial dispersal (over two patches) and time delay (arising from the maturation period). Our model analysis shows that varying the immature death rate can alter the behavior of the homogeneous equilibria, leading to transient oscillations around an intermediate equilibrium and complicated dynamics (in the form of the coexistence of possibly stable synchronized periodic oscillations and unstable phase-locked oscillations) near the largest equilibrium.     

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.     

A deterministic size-structured population model for the worm Naidis elinguis

In this paper we model the population dynamics of the worm Nais elinguis, which reproduces by division into two unequal parts. By using renewal theory we derive the asymptotic behaviour of a Naidis elinguis population. In particular we prove a certain relation between the fraction of the population that was born small (respectively the fraction that was born large) and the inter-division times. Received 20 January 1999 / Revised version: 1 August 1999?Published online: 10 April 2001     

An analysis of the coexistence of two host species with a shared pathogen

Population dynamics of two-host species under direct transmission of an infectious disease or a pathogen is studied based on the Holt–Pickering mathematical model, which accounts for the influence of the pathogen on the population of the two-host species. Through rigorous analysis and a numerical scheme of study, circumstances are specified under which the shared pathogen leads to the coexistence of the two-host species in either a persistent or periodic form. This study shows the importance of intrinsic growth rates or the differences between birth rates and death rates of the two host susceptibles in controlling these circumstances. It is also demonstrated that the periodicity may arise when the positive intrinsic growth rates are very small, but the periodicity is very weak which may not be observed in an empirical investigation.      

On periodic cohort solutions of a size-structured population model

 We consider a size-structured population model with discontinuous reproduction and feedback through the environmental variable ‘substrate’. The model admits solutions with finitely many cohorts and in that case the problem is described by a system of ODEs involving a bifurcation parameter β. Existence of nontrivial periodic n-cohort solutions is investigated. Moreover, we discuss the question whether n cohorts (n≧2) with small size differences will tend to a periodic one-cohort solution as t→∞. Received 16 March 1995; received in revised form 7 January 1997     

The dynamics of infectious diseases in orchards with roguing and replanting as control strategy

 Roguing and replanting is a widely adopted control strategy of infectious diseases in orchards. Little is known about the effect of this type of management on the dynamics of the infectious disease. In this paper we analyze a structured population model for the dynamics of an S-I-R type epidemic under roguing and replanting management. The model is structured with respect to the total number of infections and the number of post-infectious infections on a tree. Trees are assumed to be rogued, and replaced by uninfected trees, when the total number of infections on the tree reaches a threshold value. Stability analysis and numerical exploration of the model show that for specific parameter combinations the internal equilibrium can become unstable and large amplitude periodic fluctuations arise. Several hypothesis on the mechanism causing the destabilisation of the steady-state are considered. The mechanism leading to the large amplitude fluctuations is identified and biologically interpreted. Received 2 September 1994     

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     

Chaos and population disappearances in simple ecological models

A class of truncated unimodal discrete-time single species models for which low or high densities result in extinction in the following generation are considered. A classification of the dynamics of these maps into five types is proven: (i) extinction in finite time for all initial densities, (ii) semistability in which all orbits tend toward the origin or a semi-stable fixed point, (iii) bistability for which the origin and an interval bounded away from the origin are attracting, (iv) chaotic semistability in which there is an interval of chaotic dynamics whose compliment lies in the origin’s basin of attraction and (v) essential extinction in which almost every (but not every) initial population density leads to extinction in finite time. Applying these results to the Logistic, Ricker and generalized Beverton-Holt maps with constant harvesting rates, two birfurcations are shown to lead to sudden population disappearances: a saddle node bifurcation corresponding to a transition from bistability to extinction and a chaotic blue sky catastrophe corresponding to a transition from bistability to essential extinction. Received: 14 February 2000 / Revised version: 15 August 2000 / Published online: 16 February 2001     

A simple parasite model with complicated dynamics

 During their first year of life sheep acquire parasites through grazing, and simultaneously build up an immunity to infection. At the beginning of each year non-immune lambs are introduced onto contaminated pasture. We represent this process by differential equations describing the within-year dynamics, and defining a difference equation that describes the between-year dynamics. An example with two system parameters is analysed in detail. It is shown that regions exist in parameter space where periodic (between-year) or aperiodic solutions occur. Parasite control schemes could change the system dynamics from a stable equilibrium to complicated long-term fluctuations. Received: 11 August 1997 / Revised version: 4 December 1997     

Overcompensatory recruitment and generation delay in discrete age-structured population models

 The effect of overcompensatory recruitment and the combined effect of overcompensatory recruitment and generation delay in discrete nonlinear age-structured population models is studied. Considering overcompensatory recruitment alone, we present formal proofs of the supercritical nature of bifurcations (both flip and Hopf) as well as an extensive analysis of dynamics in unstable parameter regions. One important finding here is that in case of small and moderate year to year survival probabilities there are large regions in parameter space where the qualitative behaviour found in a general n+1 dimensional model is retained already in a one-dimensional model. Another result is that the dynamics at or near the boundary of parameter space may be very complicated. Generally, generation delay is found to act as a destabilizing effect but its effect on dynamics is by no means unique. The most profound effect occurs in the n-generation delay cases. In these cases there is no stable equilibrium X ^* at all, but whenever X ^* small, a stable cycle of period n+1 where the periodic points in the cycle are on a very special form. In other cases generation delay does not alter the dynamics in any substantial way. Received 25 April 1995; received in revised form 21 November 1995     

Teratogenicity of 3-hydroxynorvaline in chicken and mouse embryos

Summary. 3-Hydroxynorvaline (HNV; 2-amino-3-hydroxypentanoic acid), a microbial L-threonine analogue, is toxic to mammalian cells and displays antiviral properties. In view of this, we investigated the toxicity and/or potential teratogenicity of HNV in developing chicken and mouse embryos. HNV was administered to chicken embryos (in ovo; dose 75–300 μmole/egg; 48 h post-incubation) and pregnant Hanover NMRI mice (per os; total dose 900–1800 mg/kg body mass; gestation days 7–9). Control animals received sterile saline solutions. Harvested embryos (chicken embryos, 10 days post-incubation; mouse embryos; gestation day 18) were fixed in glutaraldehyde and stereomicroscopically inspected for signs of dysmorphogenesis. Body mass, body and toe length and mortality of chicken embryos, and the body mass and mortality of mouse embryos were recorded. HNV exposure significantly increased the incidence of embryotoxic (growth retardation, toxic mortality) and congenital defects in both chicken and mouse embryos. All the observed effects were dose-dependent. In conclusion, HNV is an embryotoxic and teratogenic compound, which caused significant developmental delay and congenital defects in developing chicken and mouse embryos.     

Global dynamics of a chemostat competition model with distributed delay

 We study the global dynamics of n-species competition in a chemostat with distributed delay describing the time-lag involved in the conversion of nutrient to viable biomass. The delay phenomenon is modelled by the gamma distribution. The linear chain trick and a fluctuation lemma are applied to obtain the global limiting behavior of the model. When each population can survive if it is cultured alone, we prove that at most one competitor survives. The winner is the population that has the smallest delayed break-even concentration, provided that the orders of the delay kernels are large and the mean delays modified to include the washout rate (which we call the virtual mean delays) are bounded and close to each other, or the delay kernels modified to include the washout factor (which we call the virtual delay kernels) are close in L ¹-norm. Also, when the virtual mean delays are relatively small, it is shown that the predictions of the distributed delay model are identical with the predictions of the corresponding ODEs model without delay. However, since the delayed break-even concentrations are functions of the parameters appearing in the delay kernels, if the delays are sufficiently large, the prediction of which competitor survives, given by the ODEs model, can differ from that given by the delay model. Received: 9 August 1997 / Revised version: 2 July 1998     

Mathematical modelling of the intravenous glucose tolerance test

 Several attempts at building a satisfactory model of the glucose-insulin system are recorded in the literature. The minimal model, which is the model currently mostly used in physiological research on the metabolism of glucose, was proposed in the early eighties for the interpretation of the glucose and insulin plasma concentrations following the intravenous glucose tolerance test. It is composed of two parts: the first consists of two differential equations and describes the glucose plasma concentration time-course treating insulin plasma concentration as a known forcing function; the second consists of a single equation and describes the time course of plasma insulin concentration treating glucose plasma concentration as a known forcing function. The two parts are to be separately estimated on the available data. In order to study glucose-insulin homeostasis as a single dynamical system, a unified model would be desirable. To this end, the simple coupling of the original two parts of the minimal model is not appropriate, since it can be shown that, for commonly observed combinations of parameter values, the coupled model would not admit an equilibrium and the concentration of active insulin in the “distant” compartment would be predicted to increase without bounds. For comparison, a simple delay-differential model is introduced, is demonstrated to be globally asymptotically stable around a unique equilibrium point corresponding to the pre-bolus conditions, and is shown to have positive and bounded solutions for all times. The results of fitting the delay-differential model to experimental data from ten healthy volunteers are also shown. It is concluded that a global unified model is both theoretically desirable and practically usable, and that any such model ought to undergo formal analysis to establish its appropriateness and to exclude conflicts with accepted physiological notions. Received: 22 June 1998 / Revised version: 24 February 1999     

Brucellosis, botflies, and brainworms: the impact of edge habitats on pathogen transmission and species extinction

Ecological interactions between species that prefer different habitat types but come into contact in edge regions at the interfaces between habitat types are modeled via reaction-diffusion systems. The primary sort of interaction described by the models is competition mediated by pathogen transmission. The models are somewhat novel because the spatial domains for the variables describing the population densities of the interacting species overlap but do not coincide. Conditions implying coexistence of the two species or the extinction of one species are derived. The conditions involve the principal eigenvalues of elliptic operators arising from linearizations of the model system around equilibria with only one species present. The conditions for persistence or extinction are made explicit in terms of the parameters of the system and the geometry of the underlying spatial domains via estimates of the principal eigenvalues. The implications of the models with respect to conservation and refuge design are discussed. Received: 10 June 1999 / Revised version: 7 July 2000 / Published online: 20 December 2000     

Simulation of ‘hitch-hiking’ genealogies

An ancestral influence graph is derived, an analogue of the coalescent and a composite of Griffiths' (1991) two-locus ancestral graph and Krone and Neuhauser's (1997) ancestral selection graph. This generalizes their use of branching-coalescing random graphs so as to incorporate both selection and recombination into gene genealogies. Qualitative understanding of a ‘hitch-hiking’ effect on genealogies is pursued via diagrammatic representation of the genealogical process in a two-locus, two-allele haploid model. Extending the simulation technique of Griffiths and Tavaré (1996), computational estimation of expected times to the most recent common ancestor of samples of n genes under recombination and selection in two-locus, two-allele haploid and diploid models are presented. Such times are conditional on sample configuration. Monte Carlo simulations show that ‘hitch-hiking’ is a subtle effect that alters the conditional expected depth of the genealogy at the linked neutral locus depending on a mutation-selection-recombination balance. Received: 21 July 2000 / Published online: 5 December 2000