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.     

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     

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.     

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     

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.     

Interaction of maturation delay and nonlinear birth in population and epidemic models

 A population with birth rate function B(N) N and linear death rate for the adult stage is assumed to have a maturation delay T>0. Thus the growth equation N′(t)=B(N(t−T)) N(t−T) e⁻ d ₁ T−dN(t) governs the adult population, with the death rate in previous life stages d ₁≧0. Standard assumptions are made on B(N) so that a unique equilibrium N _e exists. When B(N) N is not monotone, the delay T can qualitatively change the dynamics. For some fixed values of the parameters with d ₁>0, as T increases the equilibrium N _e can switch from being stable to unstable (with numerically observed periodic solutions) and then back to stable. When disease that does not cause death is introduced into the population, a threshold parameter R ₀ is identified. When R ₀<1, the disease dies out; when R ₀>1, the disease remains endemic, either tending to an equilibrium value or oscillating about this value. Numerical simulations indicate that oscillations can also be induced by disease related death in a model with maturation delay. Received: 2 November 1998 / Revised version: 26 February 1999     

Allometric scaling in animals and plants

In this paper we give a derivation for the allometric scaling relation between the metabolic rate and the mass of animals and plants. We show that the characteristic scaling exponent of 3/4 occurring in this relation is a result of the distribution of sources and sinks within the living organism. We further introduce a principle of least mass and discuss the kind of flows that arise from it.     

Mutation and recombination with tight linkage

An exact solution of the mutation-recombination equation in continuous time is presented, with linear ordering of the sites and at most one mutation or crossover event taking place at every instant of time. The differential equation may be obtained from a mutation-recombination model with discrete generations, in the limit of short generations, or weak mutation and recombination. The solution relies on the multilinear structure of the dynamical system, and on the commuting properties of the mutation and recombination operators. It is obtained through diagonalization of the mutation term, followed by a transformation to certain measures of linkage disequilibrium that simultaneously linearize and diagonalize the recombination dynamics. The collection of linkage disequilibria, as well as their decay rates, are given in closed form. Received: 26 January 1999 / Revised version: 20 October 2000 / Published online: 10 April 2001     

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.     

The influence of drug treatment on the maintenance of schistosome genetic diversity

Drug treatment of patients with schistosomiasis may select for drug-resistant parasites. In this article, we formulate a deterministic model with multiple strains of schistosomes (helminth parasites with a two-host life cycles) in order to explore the role of drug treatment in the maintenance of a polymorphism of parasite strains that differ in their resistance levels. The basic reproductive numbers for all strains are computed, and are shown to determine the stabilities of equilibria of the model and consequently the distribution of parasite phenotypes with different levels of drug tolerance. Analysis of our model shows that the likelihood that resistant strains will increase in frequency depends on the interplay between their relative fitness, the cost of resistance, and the degree of selection pressure exerted by the drug treatments.     

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.     

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     

The effect of density-dependent treatment and behavior change on the dynamics of HIV transmission

In this work, we propose a model for heterosexual transmission of HIV/AIDS in a population of varying size with an intervention program in which treatment and/or behavior change of the infecteds occur as an increasing function of the density of the infected class in the population. This assumption has socio-economic implications which is important for public health considerations since density-dependent treatment/behavior change may be more cost-saving than a program where treatment/behavior change occurs linearly with respect to the number of infecteds. We will make use of the conservation law of total sexual contacts which enables us to reduce the two-sex model to a simpler one-sex formulation. Analytical results will be given. Unlike a similar model with linear treatment/behavior change in Hsieh (1996) where conditions were obtained for the eradication of disease, we will show that density-dependent treatment/behavior change cannot eradicate the disease if the disease is able to persist without any treatment/behavior change. This work demonstrates the need to further understand how treatment/behavior change occurs in a society with varying population.     

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     

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     

Influence of nitric oxide synthase activity on amino acid concentration in the quinolinate lesioned rat striatum: A microdialysis and histochemical study

Summary. The influence of nitric oxide synthase (NOS) activity on the KCl-evoked amino acid concentrations was investigated by in vivo microdialysis in the striatum in a rat model of excitotoxic lesion. Basal microdialysate levels of amino acids decreased during the quinolinic acid-induced neurodegeneration process, except for glutamine that increased initially and returned to control values 30 days after quinolinic acid exposure. KCl-evoked increase of extracellular amino acid concentration was reduced due to NOS activity in the striatum of both controls and lesioned animals, except for 120 days after quinolinic acid injection. These changes of amino acid concentrations in microdialysates correlated with the known biochemistry of the consecutive domineered cell types during the lesion process as revealed by histochemistry for NOS, NADPH-diaphorase, GFAP and isolectin B4. The present data provide direct evidence that NOS activity can modulate extracellular amino acid concentrations in the striatum not only under physiological conditions, but also during a pharmacologically induced lesion process and, thus, suggests that nitric oxide affects neurodegeneration via this pathway. Received October 20, 1999; Accepted February 25, 2000     

Effects of endogenous glutamate on extracellular concentrations of taurine in striatum and nucleus accumbens of the awake rat: Involvement of NMDA and AMPA/kainate receptors

Summary. Using microdialysis, the effects of endogenous glutamate on extracellular concentrations of taurine in striatum and nucleus accumbens of the awake rat were investigated. The glutamate uptake inhibitor L-trans-pyrrolidine-2,4-dicarboxylic acid (PDC) was used to increase the extracellular concentration of glutamate. PDC (1, 2 and 4 mM) produced a dose-related increase of extracellular concentrations of glutamate and taurine in striatum and nucleus accumbens. Increases of extracellular taurine were significantly correlated with increases of extracellular glutamate, but not with PDC doses, which suggests that endogenous glutamate produced the observed increases of extracellular taurine in striatum and nucleus accumbens. The role of ionotropic glutamate receptors on the increases of taurine was also studied. In striatum, perfusion of the antagonists of NMDA and AMPA/kainate glutamate receptors attenuated the increases of extracellular taurine. AMPA/kainate, but not NMDA receptors, also reduced the increases of extracellular taurine in nucleus accumbens. These results suggest that glutamate-taurine interactions exist in striatum and nucleus accumbens of the awake rat. Received March 5, 1999/Accepted September 22, 1999     

Control of gravimorphogenesis by auxin: accumulation pattern of CS-IAA1 mRNA in cucumber seedlings grown in space and on the ground

Cucumber (Cucumis sativus L.) seedlings grown in microgravity developed a peg on each side of the transition zone between hypocotyl and root, whereas seedlings grown in a horizontal position on the ground developed a peg on the concave side of the gravitropically bending transition zone. The morphological features of the space-grown seedlings were similar to those of seedlings grown in a vertical position on the ground with their radicles pointing down: both became two-pegged seedlings. Morphogenesis of cucumber seedlings is thus inhibited by gravity. Analysis by in-situ hybridization of an auxin-inducible gene, CS-IAA1, showed that its mRNA accumulated to a much greater extent on the lower side of the transition zone in the horizontally placed seedlings on the ground just prior to and during the initiation period of peg formation. On the other hand, when seedlings were grown in microgravity or in a vertical position on the ground, accumulation of CS-IAA1 mRNA occurred all around the transition zone. Accumulation of CS-IAA1 mRNA in horizontally grown seedlings appreciably decreased on the upper side of the transition zone and increased on the lower side upon gravistimulation, compared with the two-pegged seedlings. Application of IAA to seedlings in a horizontal position caused the development of a peg on each side of the transition zone, or a collar-like protuberance, depending on the concentration used. These results suggest that upon gravistimulation the auxin concentration on the upper side of the horizontally placed transition zone is reduced to a level below the threshold value necessary for peg formation. Space-grown seedlings of cucumber might develop two pegs symmetrically because the auxin level in the entire transition zone is maintained above the threshold. This spaceflight experiment verified for the first time that auxin does not redistribute in microgravity. Received: 10 February 2000 / Accepted: 15 March 2000     

Butterfly seed predation: effects of landscape characteristics,plant ploidy level and population structure

Polyploidization has been suggested as one of the most common mechanisms for plant diversification. It is often associated with changes in several morphological, phenological and ecological plant traits, and therefore has the potential to alter insect–plant interactions. Nevertheless, studies evaluating the effect of plant polyploidy on interspecific interactions are still few. We investigated pre-dispersal seed predation by the butterfly Anthocharis cardamines in 195 populations of two ploidy levels of the herb Cardamine pratensis (tetraploid ssp. pratensis, 2n = 30 vs. octoploid ssp. paludosa, 2n = 56–64). We asked if differences in incidence and intensity of predation among populations were related to landscape characteristics, plant ploidy level and population structure. The incidence of the seed predator increased with increasing plant population size and decreasing distance to nearest population occupied by A. cardamines. The intensity of predation decreased with increasing plant population size and was not affected by isolation. Probability of attack decreased with increasing shading, and intensity of predation was higher in grazed than in non-grazed habitats. The attack intensity increased with increasing mean flower number of plant population, but was not affected by flowering phenology. Individuals in tetraploid populations suffered on average from higher levels of seed predation, had higher mean flower number, were less shaded and occurred more often in grazed habitats than octoploid populations. When accounting for differences in habitat preferences between ploidy levels there was no longer a difference in intensity of predation, suggesting that the observed differences in attack rates among populations of the two ploidy levels are mediated by the habitat. Overall, our results suggest that polyploidization is associated with differentiation in habitat preferences and phenotypic traits leading to differences in interspecific interaction among plant populations. This, in turn, may facilitate further divergence of ploidy levels.     

The fruit of their labour: plants and plant processing at EeRb 140 (860±60 uncal B.P. to 160±50 uncal B.P.), a late prehistoric hunter-gatherer-fisher site on the southern Interior Plateau, British Columbia, Canada

Palaeoethnobotanical analysis of Late Period contexts (860±60 uncal B.P. to 160±50 uncal B.P.) at the site of EeRb 140, indicates that it probably served as a seasonal work area, utilised during the mid to late summer by the residents of an adjacent pit-house winter village. Food processing was evidently the primary plant-related activity at this open-air hunter-gatherer-fisher site on the British Columbia Plateau in Canada. One feature appears to have functioned as both an open hearth for the drying and preservation of berries and as a pit-oven, possibly for preparing foods for immediate consumption. Comparisons of the archaeobotany of EeRb 140 with Plateau ethnographies suggest that women's task group activities are represented here. Most significantly, the identification of a specialised plant-processing site in such close proximity to a winter village contrasts strongly with existing ethnoarchaeological models for the British Columbia Plateau and introduces a type of site not previously identified archaeologically in this region. Received August 14, 2001 / Accepted March 13, 2002