A simple SIS epidemic model with a backward bifurcation

It is shown that an SIS epidemic model with a non-constant contact rate may have multiple stable equilibria, a backward bifurcation and hysteresis. The consequences for disease control are discussed. The model is based on a Volterra integral equation and allows for a distributed infective period. The analysis includes both local and global stability of equilibria.     

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 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     

Life and death near a windy oasis

We propose a simple experiment to study delocalization and extinction in inhomogeneous biological systems. The nonlinear steady state for, say, a bacteria colony living on and near a patch of nutrient or favorable illumination (“oasis”) in the presence of a drift term (“wind”) is computed. The bacteria, described by a simple generalization of the Fisher equation, diffuse, divide A→A + A, die A→ 0, and annihilate A + A→ 0. At high wind velocities all bacteria are blown into an unfavorable region (“desert”), and the colony dies out. At low velocity a steady state concentration survives near the oasis. In between these two regimes there is a critical velocity at which bacteria first survive. If the “desert” supports a small nonzero population, this extinction transition is replaced by a delocalization transition with increasing velocity. Predictions for the behavior as a function of wind velocity are made for one and two dimensions. Received: 3 August 1998 / Revised version: 17 July 1999 / Published online: 4 July 2000     

Archaeobotany of the Old Prague Town defence system, Czech Republic: archaeology, macro-remains, pollen, and diatoms

The stone fortification of Prague Old Town was constructed shortly after A.D. 1230. Large-scale archaeological and archaeobotanical research has been carried out in this area in recent years. The deep moat and the drainage ditch have been excavated at several places in front of a line of former ramparts. During the initial period of their existence, the moat and the ditch contained only minimal waste. A sudden change in the management of these features occurred at the beginning of the 14th cent., when they started to be filled with large amounts of rubbish. This article concentrates on the evaluation of these rubbish strata at three profiles through the drainage ditch and one profile through the dry moat. The results of plant macro-remains (including wood), pollen, and diatom analyses clearly demonstrate the development of the local (aquatic) environment from being relatively clean to highly polluted. An actualistic (phytosociological) approach was adopted to reconstruct the local environment of the town periphery in detail, and to distinguish a variety of habitats that were situated in the economic hinterland of the city. Some species from the rich assemblages of plant remains are interpreted in terms of their use in High Medieval households. Diatom analysis completes the picture by providing the proxy data on water quality history in the drainage ditch. Received September 15, 2001 / Accepted May 21, 2002     

Coevolutionary interactions between a haploid species and a diploid species

We investigate a general model describing coevolutionary interaction between a haploid population and a diploid population, each with two alleles at a single locus. Both species are allowed to evolve, with the fitness of the genotypes of each species assumed to depend linearly on the frequencies of the genotypes of the other species. We explore the resulting outcomes of these interactions, in particular determining the location of equilibria under various conditions. The coevolution here is much more complex than that between two haploid populations and allows for the possibility of two polymorphic equilibria. To allow for further analysis, we construct a semi-symmetric model. The variety of outcomes possible even in this second model provides support for the geographic mosaic theory of coevolution by suggesting the possibility of small local populations coevolving to very different outcomes, leading to a shifting geographic mosaic as neighboring populations interact with each other through migration.     

Oscillations in a refractory neural net

A functional differential equation that arises from the classic theory of neural networks is considered. As the length of the absolute refractory period is varied, there is, as shown here, a super-critical Hopf bifurcation. As the ratio of the refractory period to the time constant of the network increases, a novel relaxation oscillation occurs. Some approximations are made and the period of this oscillation is computed.     

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.     

The interaction of thin-film flow, bacterial swarming and cell differentiation in colonies of Serratia liquefaciens

 The rate of expansion of bacterial colonies of S. liquefaciens is investigated in terms of a mathematical model that combines biological as well as hydrodynamic processes. The relative importance of cell differentiation and production of an extracellular wetting agent to bacterial swarming is explored using a continuum representation. The model incorporates aspects of thin film flow with variable suspension viscosity, wetting, and cell differentiation. Experimental evidence suggests that the bacterial colony is highly sensitive to its environment and that a variety of mechanisms are exploited in order to proliferate on a variety of surfaces. It is found that a combination of effects are required to reproduce the variation of bacterial colony motility over a large range of nutrient availability and medium hardness. Received: 29 April 1999     

Stone Age to recent land-use history at Pegrema, northern Lake Onega, Russian Karelia

The village of Pegrema in Karelia may be regarded as a Stone Age innovation centre in the large Lake Onega area. Two pollen and plant macrofossil diagrams are presented which represent the first contribution to the study of human impact in the area using anthropogenic pollen indicators. A continuous but sporadic human presence from the Mesolithic onwards is demonstrated. While there is no archaeological evidence relating to the period 4200–3000 B.P., the pollen data suggest continuous, though rather sparse human presence. The data do not support any natural catastrophes in Pegrema as has been suggested elsewhere. Cerealia pollen is recorded earlier than expected (c. 5000 B.P.=. In the Bronce Age and Iron Age, the settlement of the Zaonezhye peninsula is reflected by a slight increase in herb pollen representation, sporadic Cerealia pollen and several periods of regression in Picea. The long introductory period of agriculture to the area, as well as the similarities and discrepancies between different sources of evidence (palaeoecological, archaeological and historical) are discussed at some length. The start of land clearance for permanent cultivation in the profile Pegrema S was dated to the late 13^th century. The beginning of more intensive field cultivation in the 15^th century is clearly seen in the pollen succession at both localities. The fluctuation in anthropogenic indicators can be related to population density based on historical data. The village of Pegrema was depopulated in 1956 which is reflected in a distinct decline in settlement indicators. Received February 17 / Accepted May 19, 2000     

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     

Traveling waves and pulses in a one-dimensional network of excitable integrate-and-fire neurons

 We study the existence and stability of traveling waves and pulses in a one-dimensional network of integrate-and-fire neurons with synaptic coupling. This provides a simple model of excitable neural tissue. We first derive a self-consistency condition for the existence of traveling waves, which generates a dispersion relation between velocity and wavelength. We use this to investigate how wave-propagation depends on various parameters that characterize neuronal interactions such as synaptic and axonal delays, and the passive membrane properties of dendritic cables. We also establish that excitable networks support the propagation of solitary pulses in the long-wavelength limit. We then derive a general condition for the (local) asymptotic stability of traveling waves in terms of the characteristic equation of the linearized firing time map, which takes the form of an integro-difference equation of infinite order. We use this to analyze the stability of solitary pulses in the long-wavelength limit. Solitary wave solutions are shown to come in pairs with the faster (slower) solution stable (unstable) in the case of zero axonal delays; for non-zero delays and fast synapses the stable wave can itself destabilize via a Hopf bifurcation. Received: 27 October 1998     

Cinnamomin: separation,crystallization and preliminary X-ray diffraction study

Cinnamomin from Cinnamonum camphora seeds, a type II ribosome-inactivating protein that interferes with protein biosynthesis in mammalian cells, can induce the apoptosis of carcinoma cells and be used as an insecticide. A rapid and improved method has been developed for the extraction and purification of cinnamomin from camphora seed. Purification of cinnamomin is achieved with two successive steps of hydrophobic interaction chromatography carried out on a fast protein liquid chromatography (FPLC) system. Crystals suitable for X-ray diffraction analysis were obtained by vapor diffusion method. A complete data set at 2.8 A resolution has been collected. Data indexation and refinement indicate that the crystal is orthorhombic with space group P2(1)2(1)2(1) and unit cell dimensions a = 52.39 A, b = 126.33 A, c = 161.45 A. There are two molecules per asymmetric unit. Initial phasing by molecular replacement method yielded a solution, which will contribute to the structure determination. A molecular model will further the understanding of the mechanism of cinnamomin function. The latter will be combined with bio-informatics to facilitate the medical and other applications of cinnamomin.     

Two SIS epidemiologic models with delays

 The SIS epidemiologic models have a delay corresponding to the infectious period, and disease-related deaths, so that the population size is variable. The population dynamics structures are either logistic or recruitment with natural deaths. Here the thresholds and equilibria are determined, and stabilities are examined. In a similar SIS model with exponential population dynamics, the delay destabilized the endemic equilibrium and led to periodic solutions. In the model with logistic dynamics, periodic solutions in the infectious fraction can occur as the population approaches extinction for a small set of parameter values. Received: 10 January 1997 / 18 November 1997     

Lysine catabolism, an effective versatile regulator of lysine level in plants

Lysine is a nutritionally important essential amino acid, whose synthesis in plants is strongly regulated by the rate of its synthesis. Yet, lysine level in plants is also finely controlled by a super-regulated catabolic pathway that catabolizes lysine into glutamate and acetyl Co-A. The first two enzymes of lysine catabolism are synthesized from a single LKR/SDH gene. Expression of this gene is subject to compound developmental, hormonal and stress-associated regulation. Moreover, the LKR/SDH gene of different plant species encodes up to three distinct polypeptides: (i) a bifunctional enzyme containing the linked lysine-ketoglutarate (LKR) and saccharopine dehydrogenase (SDH) whose LKR activity is regulated by its linked SDH enzyme; (ii) a monofunctional SDH encoded by an internal promoter, which is a part of the coding DNA region of the LKR/SDH gene; and (iii) a monofunctional, highly potent LKR that is formed by polyadenylation within an intron. LKR activity in the bifunctional LKR/SDH polypeptide is also post-translationally regulated by phosphorylation by casein kinase-2 (CK2), but the consequence of this regulation is still unknown. Why is lysine metabolism super-regulated by synthesis and catabolism? A hypothesis addressing this important question is presented, suggesting that lysine may serve as a regulator of plant growth and interaction with the environment.     

An explicit representation of the Luria–Delbrück distribution

The probability distribution of the number of mutant cells in a growing single-cell population is presented in explicit form. We use a discrete model for mutation and population growth which in the limit of large cell numbers and small mutation rates reduces to certain classical models of the Luria-Delbrück distribution. Our results hold for arbitrarily large values of the mutation rate and for cell populations of arbitrary size. We discuss the influence of cell death on fluctuation experiments and investigate a version of our model that accounts for the possibility that both daughter cells of a non-mutant cell might be mutants. An algorithm is presented for the quick calculation of the distribution. Then, we focus on the derivation of two essentially different limit laws, the first of which applies if the population size tends to infinity while the mutation rate tends to zero such that the product of mutation rate times population size converges. The second limit law emerges after a suitable rescaling of the distribution of non-mutant cells in the population and applies if the product of mutation rate times population size tends to infinity. We discuss the distribution of mutation events for arbitrary values of the mutation rate and cell populations of arbitrary size, and, finally, consider limit laws for this distribution with respect to the behavior of the product of mutation rate times population size. Thus, the present paper substantially extends results due to Lea and Coulson (1949), Bartlett (1955), Stewart et al. (1990), and others.     

The long history of Cannabis and its cultivation by the Romans in central Italy, shown by pollen records from Lago Albano and Lago di Nemi

The cores from the Albano and Nemi lakes, near Rome, were studied within the European Union funded PALICLAS project and provided high resolution records of the Late-glacial and Holocene. Pollen evidence of increasing human influence on vegetation was recorded in the Holocene parts of both diagrams, and the Cannabis (hemp) curve was one of the major signs. In this paper we present unambiguous pollen evidence from the Cannabaceae records for the cultivation of hemp in central Italy by the Romans. The oldest records of Cannabis and Humulus (hop) date from to the Late-glacial. Hop pollen values rise during the mid Holocene, while hemp pollen becomes more abundant from ca. 3000 cal B.P. onwards. The highest earliest hemp peak (21%) is dated to the 1st century A.D. This ‘Cannabis phase’, with the abrupt rise of hemp pollen soon after the rise of cultivated trees (Castanea, Juglans and Olea) is associated with the increase in cereals and ruderal plants. This unambiguous proof of cultivation by Romans around 2000 B.P. occurs as well as a long lasting pre-Roman presence of hemp in the area, which is natural and possibly also anthropogenic. Subsequent clear episodes of cultivation in the medieval period were found. Received February 4, 2002 / Accepted September 13, 2002 Correspondence to: Anna Maria Mercuri, e-mail: mercuri.annamaria@unimo.it     

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.     

A generalized transport model for biased cell migration in an anisotropic environment

 A generalized transport model is derived for cell migration in an anisotropic environment and is applied to the specific cases of biased cell migration in a gradient of a stimulus (taxis; e.g., chemotaxis or haptotaxis) or along an axis of anisotropy (e.g., contact guidance). The model accounts for spatial or directional dependence of cell speed and cell turning behavior to predict a constitutive cell flux equation with drift velocity and diffusivity tensor (termed random motility tensor) that are explicit functions of the parameters of the underlying random walk model. This model provides the connection between cell locomotion and the resulting persistent random walk behavior to the observed cell migration on longer time scales, thus it provides a framework for interpreting cell migration data in terms of underlying motility mechanisms. Received: 8 April 1999