On spreading speeds and traveling waves for growth and migration models in a periodic habitat

 It is shown that the methods previously used by the author [Wei82] and by R. Lui [Lui89] to obtain asymptotic spreading results and sometimes the existence of traveling waves for a discrete-time recursion with a translation invariant order preserving operator can be extended to a recursion with a periodic order preserving operator. The operator can be taken to be the time-one map of a continuous time reaction-diffusion model, or it can be a more general model of time evolution in population genetics or population ecology in a periodic habitat. Methods of estimating the speeds of spreading in various directions will also be presented. Received: 12 July 2001 / Revised version: 19 July 2002 / Published online: 17 October 2002 Mathematics Subject Classification (2000): 92D40, 92D25, 35K55, 35K57, 35B40 Keywords or phrases: Periodic – Spreading speed – Traveling wave     

Limits of a multi-patch SIS epidemic model

 We start from a stochastic SIS model for the spread of epidemics among a population partitioned into M sites, each containing N individuals; epidemic spread occurs through within-site (`local') contacts and global contacts. We analyse the limit behaviour of the system as M and N increase to ∞. Two limit procedures are considered, according to the order in which M and N go to ∞; independently of the order, the limiting distribution of infected individuals across sites is a probability measure, whose evolution in time is governed by the weak form of a PDE. Existence and uniqueness of the solutions to this problem is shown. Finally, it is shown that the infected distribution converges, as time goes to infinity, to a Dirac measure at the value x ^* , the equilibrium of a single-patch SIS model with contact rate equal to the sum of local and global contacts. Received: 18 July 2001 / Revised version: 16 March 2002 / Published online: 26 September 2002 Mathematics Subject Classification (2000): 92D30, 60F99 Key words or phrases: SIS epidemic – Metapopulation – Markov population processes – Weak convergence of measures     

Monotone travelling fronts in an age-structured reaction-diffusion model of a single species

 We consider a partially coupled diffusive population model in which the state variables represent the densities of the immature and mature population of a single species. The equation for the mature population can be considered on its own, and is a delay differential equation with a delay-dependent coefficient. For the case when the immatures are immobile, we prove that travelling wavefront solutions exist connecting the zero solution of the equation for the matures with the delay-dependent positive equilibrium state. As a perturbation of this case we then consider the case of low immature diffusivity showing that the travelling front solutions continue to persist. Our findings are contrasted with recent studies of the delayed Fisher equation. Travelling fronts of the latter are known to lose monotonicity for sufficiently large delays. In contrast, travelling fronts of our equation appear to remain monotone for all values of the delay. Received: 1 November 2001 / Revised version: 10 May 2002 / Published online: 23 August 2002 Mathematics Subject Classification (2000): 35K57, 92D25 Key words or phrases: Age-structure – Time-delay – Travelling Fronts – Reaction-diffusion     

Change in frequency of a rare mutant allele: A general formula and applications to partial inbreeding models

 We deduce and prove a general formula to approximate the change in frequency of a mutant allele under weak selection, when this allele is introduced in small frequency into a population which was previously at a fixation state. We apply the formula to autosomal genes in partial selfing models and to autosomal as well as sex-linked genes in partial sib mating models. It is shown that the fate of a rare mutant allele depends not only on the selection parameters, the inbreeding coefficient and the reproductive values of the sexes in sex-differentiated populations, but also on coefficients of relatedness between mates. This is interpreted as a kin selection effect caused by inbreeding per se. Received: 3 December 2001 / Revised version: 10 April 2002 / Published online: 19 November 2002 Research supported in part by NSERC of Canada and FCAR of Québec. Mathematics Subject Classification (2000): Primary 60J80, Secondary 92D10, 92D25 Keywords or phrases: Adaptive topography – Partial selfing – Partial sib mating – Kin selection     

On the dynamics of discrete food chains: low- and high-frequency behavior and optimality of chaos

 In this paper we derive and analyze a discrete version of Rosenzweig's (Am. Nat. 1973) food-chain model. We provide substantial analytical and numerical evidence for the general dynamical patterns of food chains predicted by De Feo and Rinaldi (Am. Nat. 1997) remaining largely unaffected by this discretization. Our theoretical analysis gives rise to a classification of the parameter space into various regions describing distinct governing dynamical behaviors. Predator abundance has a local optimum at the edge of chaos. Received: 13 August 1999 / Revised version: 12 March 2002 / Published online: 17 October 2002 Mathematics Subject Classification (1991): 92D40 Keywords or phrases: Discrete food-chain – Discrete Hopf (Neimark-Sacker) bifurcation – Pulsewise birth processes – Mean yield maximization – Nicholson-Bailey model     

A numerical method for renal models that represent tubules with abrupt changes in membrane properties

 The urine concentrating mechanism of mammals and birds depends on a counterflow configuration of thousands of nearly parallel tubules in the medulla of the kidney. Along the course of a renal tubule, cell type may change abruptly, resulting in abrupt changes in the physical characteristics and transmural transport properties of the tubule. A mathematical model that faithfully represents these abrupt changes will have jump discontinuities in model parameters. Without proper treatment, such discontinuities may cause unrealistic transmural fluxes and introduce suboptimal spatial convergence in the numerical solution to the model equations. In this study, we show how to treat discontinuous parameters in the context of a previously developed numerical method that is based on the semi-Lagrangian semi-implicit method and Newton's method. The numerical solutions have physically plausible fluxes at the discontinuities and the solutions converge at second order, as is appropriate for the method. Received: 13 November 2001 / Revised version: 28 June 2002 / Published online: 26 September 2002 This work was supported in part by the National Institutes of Health (National Institute of Diabetes and Digestive and Kidney Diseases, grant DK-42091.) Mathematics Subject Classification (2000): 65-04, 65M12, 65M25, 92-04, 92C35, 35-04, 35L45 Keywords or phrases: Mathematical models – Differential equations – Mathematical biology – Kidney – Renal medulla – Semi-Lagrangian semi-implicit     

A nonlinear dynamical mechanism for bruit generation by an intracranial saccular aneurysm

 Intracranial saccular aneurysms have been clinically observed to emit a transient sound, a bruit, on each heartbeat. The mechanism causing the bruits has been a matter of contention. A qualitative analysis of the nonlinear dynamical properties of the Shah-Humphrey model for periodic pressure forcing of a thin-necked saccular aneurysm, using the Fung nonlinear constitutive model for the aneurysm material, shows that a small blood pressure jump on each beat, whether the pressure is weakly aperiodic or periodic, induces transients in the radial deformation response of the aneurysmal wall on each heartbeat. These transient vibrations, which have a component with frequency near the natural frequency of the system but are not resonant phenomena and which decay rapidly to a limit cycle during each distinct forcing pressure cycle, can generate the bruits. Received: 21 November 2000 / Revised version: 9 August 2001 / Published online: 23 August 2002 Mathematics Subject Classification (2000): 92B99, 70K40, 70K05 Key words or phrases: Intracranial saccular aneurysm – Bruit – Spectrum – Nonlinear dynamics – Transients – Vortex shedding – Fung model     

Development of a biologically-based controlled growth and differentiation model for developmental toxicology

 A mathematical model is developed with a highly controlled birth and death process for precursor cells. This model is both biologically- and statistically-based. The controlled growth and differentiation (CGD) model limits the number of replications allowed in the development of a tissue or organ and thus, more closely reflects the presence of a true stem cell population. Leroux et al. (1996) presented a biologically-based dose-response model for developmental toxicology that was derived from a partial differential equation for the generating function. This formulation limits further expansion into more realistic models of mammalian development. The same formulae for the probability of a defect (a system of ordinary differential equations) can be derived through the Kolmogorov forward equations due to the nature of this Markov process. This modified approach is easily amenable to the expansion of more complicated models of the developmental process such as the one presented here. Comparisons between the Leroux et al. (1996) model and the controlled growth and differentiation (CGD) model as developed in this paper are also discussed. Received: 8 June 2001 / Revised version: 15 June 2002 / Published online: 26 September 2002 Keywords or phrases: Teratology – Multistate process – Cellular kinetics – Numerical simulation     

Words in DNA sequences: some case studies based on their frequency statistics

 One of the critical requirements of data analysis involving large DNA sequences is an effective statistical summarization of those sequences. In this article DNA sequences have been analyzed based on word frequencies. Our analysis focuses on the detection of structural signature of a genome reflected in word frequencies and identification of phylogenetic relationships among different species reflected in the variation of word distributions in their DNA sequences. We have carried out a statistical study of the complete genome of baker's yeast, of various ribosomal RNA sequences from different prokaryotic and eukaryotic organisms and of the full genomes of some bacteriophages. Our exploratory analysis amply demonstrates the usefulness of DNA word frequencies in reducing the dimensionality of large sequences while retaining some of the structural information there that can have biological significance. Some conceptual issues that arise in course of our investigation have been addressed. A few interesting problems related to the statistics of DNA words have been pointed out with some indication of their possible solutions. The work has been partially motivated by the fact that sequence alignment and homology techniques that are quite popular for comparing and analyzing relatively smaller DNA sequences of nearly equal sizes are not applicable to data consisting of large sequences with widely varying sizes, which may contain segments with unknown or no biological functions, and consequently their comparison through functional homology is either impossible or extremely difficult. Received: 15 October 2000 / Revised version: 8 October 2002 Published online: 28 February 2003 Current address: CF186, Salt lake, Calcutta 700064, India Research presented here was supported in part by a grant from Indian Statistical Institute. Key words or phrases: Average linkage clustering – Chernoff's faces – Dendrograms – DNA words – F-ranks of words – F-ratios of words – l ₁-distance – Phylogenetic relationships – Rank correlation – Single linkage clustering     

Branched chain amino acids as source of specific branched chain volatile fatty acids during the fermentation process of fish sauce

Summary.  The source of the formation of branched chain volatile fatty acids (VFA) in fish sauce was investigated. Certain branched VFA were derived from the degradation of specific amino acids as iso-butyric acid from valine and iso-valeric acid from leucine. Short and long straight chain VFA were significantly higher in the linoleic acid added sample than in the control but did not significantly bring changes to the branched chain VFA. It is suggested that straight chain VFA developed from fish fats. Alanine and isoleucine did not have a clear influence on the production of volatile fatty acids. Received November 23, 2001 Accepted June 20, 2002 Published online December 18, 2002 RID="*" ID="*" Part of this paper was presented in the 7th International Congress on Amino Acids and Proteins in Vienna, Austria from August 6–10, 2001. Authors' address: Norlita G. Sanceda, Ph.D., Institute of Environmental Science for Human Life, 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112, Japan, Fax: + 81-3-5978-5805, E-mail: lita@cc.ocha.ac.jp     

A discrete,size-structured model of phytoplankton growth in the chemostat: introduction of inhomogeneous cell division size

 We introduce inhomogeneous, substrate dependent cell division in a time discrete, nonlinear matrix model of size-structured population growth in the chemostat, first introduced by Gage et al. [8] and later analysed by Smith [13]. We show that mass conservation is verified, and conclude that our system admits one non zero globally stable equilibrium, which we express explicitly. Then we run numerical simulations of the system, and compare the predictions of the model to data related to phytoplankton growth, whose obtention we discuss. We end with the identification of several parameters of the system. Received: 9 February 2000 / Revised version: 10 October 2001 / Published online: 23 August 2002 RID="*" ID="*" Present address: Department of Mathematics and Statistics, University of Victoria, B.C., Canada. e-mail: jarino@math.uvic.ca Key words or phrases: Chemostat – Structured population models – Discrete model – Inhomogeneous division size     

Transmission of mitochondrial DNA heteroplasmy in normal pedigrees

The presence of multiple mitochondrial genotypes (heteroplasmy) has been studied in normal individuals. Six multigenerational normal families were screened for heteroplasmy by PCR of the mitochondrial control region and the cytochrome c oxidase intergenic regions. Two individuals from different families exhibited multiple length polymorphisms in a homopolymeric tract at positions 16 184–16 193 and a grandmother in a third family was heteroplasmic for both cytosine and thymidine at position 15 945. Although the 15 945 T variant comprised 28% of the grandmother’s mitochondrial DNA, this sequence was not present in any of her descendants. Heteroplasmy was detected in 2.5% of the 96 mother-offspring pairs, consistent with the possibility that it may not be rare. Received: 18 August 1997 / Accepted: 10 November 1997     

Dynamics of annual influenza A epidemics with immuno-selection

 The persistence of Influenza A in the human population relies on continual changes in the viral surface antigens allowing the virus to reinfect the same hosts every few years. The epidemiology of such a drifting virus is modeled by a discrete season-to-season map. During the epidemic season only one strain is present and its transmission dynamics follows a standard epidemic model. After the season, cross-immunity to next year's virus is determined from the proportion of hosts that were infected during the season. A partial analysis of this map shows the existence of oscillations where epidemics occur at regular or irregular intervals. Received: 16 February 2001 / Revised version: 11 June 2002 / Published online: 28 February 2003 Key words or phrases: Infectious disease – Influenza drift – Cross-immunity – Seasonal epidemics – Iterated map     

Production of transgenic gentian plants by particle bombardment of suspension-culture cells

 Cell suspension cultures were established from leaf explants of gentian (Gentiana triflora×G. scabra) for the generation of transgenic plants by particle bombardment. The parameters for the bombardment of suspension culture cells with a particle gun were examined by monitoring the transient expression of a gene for β-glucuronidase driven by the cauliflower mosaic virus (CaMV) 35S promoter. We found that prior culture of suspension culture cells for 5 days on solid medium was optimum for successful particle bombardment. Putative transformed calli were obtained from bombarded cells after a two-step selection procedure. Cells were cultured first with 30 mg l^–1 hygromycin in liquid MS medium that contained 10 mg l^–1 N-phenyl-N′-1,2,3-thiadiazol-5-yl urea, 1 mg/l 1-naphthaleneacetic acid and 30 g l^–1 sucrose and then on solid medium prepared from the same liquid medium plus 2 g l^–1 gellan gum. After 12 weeks of selection on solid medium that contained 30 mg l^–1 hygromycin, two transgenic gentian plants were regenerated from each selected callus. Analysis by the polymerase chain reaction and Southern blotting revealed the stable integration of transferred DNA. Received: 3 June 1999 / Revision received: 21 September 1999 / Accepted: 20 September 1999     

Feedback control for chemostat models

 It is shown that a chemostat with two organisms can be made coexistent by means of feedback control of the dilution rate. Remaining freedom in the feedback law can be used to guarantee robustness or improve particular performance indices. Unfortunately a topological property prevents coexistence by feedback control for chemostats with more than two organisms. We apply our results to control bioreactors aimed at producing commercial products through genetically altered organisms. In all our results the coexistence takes its simplest form: a global asymptotically stable equilibrium point in the interior of the non-negative orthant. Received: 1 November 2001 / Revised version: 19 August 2002 / Published online: 19 November 2002 Mathematics Subject Classification (2000): 34H05, 34D23, 93C10 Keywords or phrases: Chemostat – Feedback control – Coexistence     

A temperature-sensitive DNA adenine methyltransferase mutant of Salmonella typhimurium

A temperature-sensitive mutant of Salmonella typhimurium was isolated earlier after transposon mutagenesis with Tn10d Tet. The mutant D220 grows well at 28 °C but has a lower growth rate and forms filaments at 37 °C. Transposon-flanking fragments of mutant D220 DNA were cloned and sequenced. The transposon was inserted in the dam gene between positions 803 and 804 (assigned allele number: dam-231 : : Tn10d Tet) and resulted in a predicted ten-amino-acid-shorter Dam protein. The insertion created a stop codon that led to a truncated Dam protein with a temperature-sensitive phenotype. The insertion dam-231 : : Tn10d Tet resulted in a dam“leaky” phenotype since methylated and unmethylated adenines in GATC sequences were present. In addition, the dam-231 : : Tn10d Tet insertion rendered dam mutants temperature-sensitive for growth depending upon the genetic background of the S. typhimurium strain. The wild-type dam gene of S. typhimurium exhibited 82% identity with the Escherichia coli dam gene.     

Cattaneo models for chemosensitive movement

 We derive models for chemosensitive movement based on Cattaneo's law of heat propagation with finite speed. We apply the model to pattern formation as observed in experiments with Dictyostelium discoideum, with Salmonella typhimurium and with Escherichia coli. For Salmonella typhimurium we make predictions on pattern formation which can be tested in experiments. We discuss the relations of the Cattaneo models to classical models and we develop an effective numerical scheme. Received: 8 October 2001 / Revised version: 2 August 2002 / Published online: 19 November 2002 Key words or phrases: Chemotaxis – Aggregation – Cattaneo model – Numerical schemes Acknowledgements. We are very grateful for comments of S. Noelle concerning the numerical scheme. We thank K.P. Hadeler and C. Schmeiser for helpful remarks. The research was supported by the Deutsche Forschungsgemeinschaft, research project ANumE and the Austrian Science Foundation, grant no. W008.     

The onset of oscillatory dynamics in models of multiple disease strains

 We examine a generalised SIR model for the infection dynamics of four competing disease strains. This model contains four previously-studied models as special cases. The different strains interact indirectly by the mechanism of cross-immunity; individuals in the host population may become immune to infection by a particular strain even if they have only been infected with different but closely related strains. Several different models of cross-immunity are compared in the limit where the death rate is much smaller than the rate of recovery from infection. In this limit an asymptotic analysis of the dynamics of the models is possible, and we are able to compute the location and nature of the Takens–Bogdanov bifurcation associated with the presence of oscillatory dynamics observed by previous authors. Received: 5 December 2001 / Revised version: 5 May 2002 / Published online: 17 October 2002 Keywords or phrases: Infection – Pathogen – Epidemiology – Multiple strains – Cross-immunity – Oscillations – Dynamics – Bifurcations     

Deterministic extinction effect of parasites on host populations

 Experimental studies have shown that parasites can reduce host density and even drive host population to extinction. Conventional mathematical models for parasite-host interactions, while can address the host density reduction scenario, fail to explain such deterministic extinction phenomena. In order to understand the parasite induced host extinction, Ebert et al. (2000) formulated a plausible but ad hoc epidemiological microparasite model and its stochastic variation. The deterministic model, resembles a simple SI type model, predicts the existence of a globally attractive positive steady state. Their simulation of the stochastic model indicates that extinction of host is a likely outcome in some parameter regions. A careful examination of their ad hoc model suggests an alternative and plausible model assumption. With this modification, we show that the revised parasite-host model can exhibit the observed parasite induced host extinction. This finding strengthens and complements that of Ebert et al. (2000), since all continuous models are likely break down when all population densities are small. This extinction dynamics resembles that of ratio-dependent predator-prey models. We report here a complete global study of the revised parasite-host model. Biological implications and limitations of our findings are also presented. Received: 30 October 2001 / Revised version: 11 February 2002 / Published online: 17 October 2002 Work is partially supported by NSF grant DMS-0077790 Mathematics Subject Classification (2000): 34C25, 34C35, 92D25. Keywords or phrases: Microparasite model – Ratio-dependent predator-prey model – Host extinction – Global stability – Biological control     

Bifurcation analysis of an orientational aggregation model

 We consider an integro-differential equation for the evolution of a function f on the circle, describing an orientational aggregation process. In the first part we analyze generic bifurcations of steady-state solutions when a single eigenvalue changes sign. Lyapunov-Schmidt reduction leads to the bifurcation equation which is solved explicitly by formal power series. We prove that these series have positive radius of convergence. Two examples exhibit forward and backward bifurcations, respectively. In the second part we assume that the first and second eigenvalues become positive. Again we use Lyapunov-Schmidt reduction to arrive at the reduced bifurcation system from which we get the bifurcating branches as power series. We calculate the two most important parameters of the reduced system for two examples; one of them has interesting mode interactions which lead to various kinds of time-periodic solutions. Received: 23 April 2001 / Revised version: 29 October 2002 / Published online: 28 February 2003 Key words or phrases: Actin – Cytoskeleton – Orientational Aggregation – Bifurcation Analysis – Mode Interaction – Power Series Expansion