Mathematical models of microbial growth and competition in the chemostat regulated by cell-bound extracellular enzymes

A mathematical model of growth and competitive interaction of microorganisms in the chemostat is analyzed. The growth-limiting nutrient is not in a form that can be directly assimilated by the microorganisms, and must first be transformed into an intermediate product by cell-bound extracellular enzymes. General monotone functions, including Michaelis-Menten and sigmoidal response functions, are used to describe nutrient conversion and growth due to consumption of the intermediate product. It is shown that the initial concentration of the species is an important determining factor for survival or washout. When there are two species whose growth is limited by the same nutrient, three different modes of competition are described. Competitive coexistence steady states are shown to be possible in two of them, but they are always unstable. In all of our numerical simulations, the system approaches a steady state corresponding to the washout of one or both of the species from the chemostat.Research supported by NSF grant DMS-90-96279Research supported by NSERC grant A-9358     

A parameter identification problem arising from a model of canalicular bile formation

We develop a simple mathematical model for bile formation and analyze some features of the model that suggest the design for future physiological experiments. The mathematical model results in a boundary value problem for a system of functional differential equations depending on several physical parameters. From the observability of the boundary values we can identify, both qualitatively and quantitatively, some of these physical parameters. This identification then suggests physical experiments from which one could infer some of the bile transport phenomena that are not, at present, directly observable. The mathematical parameter identification problem is solved by converting the boundary value problem to a transition time problem for a quadratic system of ordinary differential equations on the plane where we are able to employ some special properties of quadratic systems in order to obtain a solution.The author was supported by the Air Force Office of Scientific Research and the National Science Foundation under the grants AF-AFOSR-89-0078 and DMS-9022621The author was supported by National Institutes of Health under grant number R37 DK-27623     

A stage structured predator-prey model and its dependence on maturation delay and death rate

Many of the existing models on stage structured populations are single species models or models which assume a constant resource supply. In reality, growth is a combined result of birth and death processes, both of which are closely linked to the resource supply which is dynamic in nature. From this basic standpoint, we formulate a general and robust predator-prey model with stage structure with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and computational study. Our work indicates that if the juvenile death rate (through-stage death rate) is nonzero, then for small and large values of maturation time delays, the population dynamics takes the simple form of a globally attractive steady state. Our linear stability work shows that if the resource is dynamic, as in nature, there is a window in maturation time delay parameter that generates sustainable oscillatory dynamics.Work is partially supported by NSF grant DMS-0077790.Mathamatics Subject Classification (2000):92D25, 35R10Revised version: 26 February 2004     

Positive solutions of an elliptic system arising from a model in evolutionary ecology

We give sufficient and almost necessary conditions for the existence of positive solutions to an elliptic system satisfying various Dirichlet boundary conditions. The elliptic system consists of the steady-state equations of a parabolic system used to model the growth and spread of a particular gene and population living in a bounded region. The model takes into account the fact that the fitness of the individuals in the population may depend on the population size. Some non-existence results are also included.Research partially supported by NSF grant no. DMS-8801968     

Progression age enhanced backward bifurcation in an epidemic model with super-infection

 We consider a model for a disease with a progressing and a quiescent exposed class and variable susceptibility to super-infection. The model exhibits backward bifurcations under certain conditions, which allow for both stable and unstable endemic states when the basic reproduction number is smaller than one. Received: 11 October 2001 / Revised version: 17 September 2002 / Published online: 17 January 2003 Present address: Department of Biological Statistics and Computational Biology, 434 Warren Hall, Cornell University, Ithaca, NY 14853-7801 This author was visiting Arizona State University when most of the research was done. Research partially supported by NSF grant DMS-0137687. This author's research was partially supported by NSF grant DMS-9706787. Key words or phrases: Backward bifurcation – Multiple endemic equilibria – Alternating stability – Break-point density – Super-infection – Dose-dependent latent period – Progressive and quiescent latent stages – Progression age structure – Threshold type disease activation – Operator semigroups – Hille-Yosida operators – Dynamical systems – Persistence – Global compact attractor     

Stability,instability in delay equations modeling human respiration

A system of delay equations describing a simple model of the respiratory control mechanism in humans is considered and conditions guaranteeing stability, instability of steady-state equilibrium solutions of that system are presented.This research was supported in part by an NSF research grant (K. C.), and by the Institute of Mathematics and Its Applications with funds provided by the NSF (K. C. and J. T.)     

Global stability in a chemostat with multiple nutrients

We study a single species in a chemostat, limited by two nutrients, and separate nutrient uptake from growth. For a broad class of uptake and growth functions it is proved that a nontrivial equilibrium may exist. Moreover, if it exists it is unique and globally stable, generalizing a result in [15]. supported in part by NSF grant DMS 0342153. supported in part by NSF grant DEB-0083566 and the Andrew W. Mellon Foundation. supported in part by USAF grant F49620-01-1-0063 and NSF grant CCR-0206789. Part of this work was carried out when P. De Leenheer was a post-doctoral fellow at DIMACS, Rutgers University, supported in part by NSF grant EIA03-331486 and USAF grant F49620-01-1-0063.     

A juvenile-adult model with periodic vital rates

A global branch of positive cycles is shown to exist for a general discrete time, juvenile-adult model with periodically varying coefficients. The branch bifurcates from the extinction state at a critical value of the mean, inherent fertility rate. In comparison to the autonomous system with the same mean fertility rate, the critical bifurcation value can either increase or decrease with the introduction of periodicities. Thus, periodic oscillations in vital parameter can be either advantageous or deleterious. A determining factor is the phase relationship among the oscillations in the inherent fertility and survival rates.Research supported by NSF grant DMS-0414212.     

Variability,vagueness and comparison methods for ecological models

A unified approach is presented for the construction and analysis of models for the dynamics of populations and communities in the presence of temporal variability, vague density dependence, chaos or analytical intractability. The approach is based on comparisons involving simpler models which provide ceilings and floors to the densities predicted by the full models. The method is applied to examples of several types of models, including difference equations, ordinary differential equations, non-linear Leslie matrices and reaction-diffusion equations. The models treated describe various ecological phenomena including self-regulation, competition, predator-prey interactions, age structure and spatial structure. Some results needed for the analysis of matrix models and patch models are given in the Appendix. Research partially supported by NSF grant DMS-93-03708.     

Juvenile versus adult competition

A general class of age-structured models based upon the McKendrick/von Foerster equations are used to study intraspecific competition between juveniles and adults. Criteria for the existence and stability of equilibria are obtained and the dependence of equilibrium stability (i.e. equilibrium resilience) on competition coefficients is analyzed for low inherent net reproductive numbers. The results are applied to the question of whether juvenile vs. adult intraspecific competition is stabilizing or destabilizing. Two types of competition are studied. The first, involving suppressed adult fertility due to competition from juveniles, was found to be destabilizing in that equilibrium levels are lowered and equilibrium resilience weakened by increased competition. The second, involving increased juvenile mortality due to competition from adults, was found to be considerably more complicated. While equilibrium levels were again reduced by increased competition, equilibrium resilience can either be weakened or strengthened. A criterion for determining the effects on resilience is derived and several examples are given to illustrate various possibilities in this case.The author gratefully acknowledges the support of the Applied Mathematics Division and the Population Biology/Ecology Division of the National Science Foundation under NSF grant No. DMS-8902508Research supported by the Department of Energy under contracts W-7405-ENG-36 and KC-07-01-01     

Convergence to constant equilibrium for a density-dependent selection model with diffusion

We consider the classical single locus two alleles selection model with diffusion where the fitnesses of the genotypes are density dependent. Using a theorem of Peter Brown, we show that in a bounded domain with homogeneous Neumann boundary conditions, the allele frequency and population density converge to a constant equilibrium lying on the zero population mean fitness curve. The results agree with the case without diffusion obtained by Selgrade and Namkoong. Frequency and density dependent selection is also considered.Research partially supported by NSF grant DMS-8601585     

Modeling the impact of periodic bottlenecks, unidirectional mutation, and observational error in experimental evolution

Antibiotic resistant bacteria are a constant threat in the battle against infectious diseases. One strategy for reducing their effect is to temporarily discontinue the use of certain antibiotics in the hope that in the absence of the antibiotic the resistant strains will be replaced by the sensitive strains. An experiment where this strategy is employed in vitro [5] produces data which showed a slow accumulation of sensitive mutants. Here we propose a mathematical model and statistical analysis to explain this data.The stochastic model elucidates the trend and error structure of the data. It provides a guide for developing future sampling strategies, and provides a framework for long term predictions of the effects of discontinuing specific antibiotics on the dynamics of resistant bacterial populations.This Research is part of the Initiative in Bioinformatics and Evolutionary Studies (IBEST) at the University of Idaho. Funding was provided by NSF EPSCoR EPS-0080935, NSF EPSCoR, EPS-0132626, and NIH NCRR grant NIH NCRR- 20RR016448. Paul Joyce is also funded by NSF DEB-0089756, and NSF DMS-0072198.     

Persistence in reaction diffusion models with weak allee effect

We study the positive steady state distributions and dynamical behavior of reaction-diffusion equation with weak Allee effect type growth, in which the growth rate per capita is not monotonic as in logistic type, and the habitat is assumed to be a heterogeneous bounded region. The existence of multiple steady states is shown, and the global bifurcation diagrams are obtained. Results are applied to a reaction-diffusion model with type II functional response, and also a model with density-dependent diffusion of animal aggregation. J. S. is partially supported by United States NSF grants DMS-0314736 and EF-0436318, College of William and Mary summer grants, and a grant from Science Council of Heilongjiang Province, China.     

Density-dependent selection migration model with non-monotone fitness functions

This paper studies the classical single locus, diallelic selection model with diffusion for a continuously reproducing population. The phase variables are population density and allele frequency (or allele density). The genotype fitness depend only on population density but include one-hump functions of the density variable. With mild assumptions on genotype fitnesses, we study the geometry of the nullclines and the asymptotic behavior of solutions of the selection model without diffusion. For the diffusion model with zero Neumann boundary conditions, we use this geometric information to show that if the initial data satisfy certain conditions then the corresponding solution to the reaction-diffusion equation converges to the spatially constant stable equilibrium which is closest to the initial data.Research partially supported by NSF grant DMS-8920597Research supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Pioneering (Population Genetics of Forest Trees) Research Unit, Raleigh, North Carolina     

Pattern recognition in several sequences: Consensus and alignment

The comparison of several sequences is central to many problems of molecular biology. Finding consensus patterns that define genetic control regions or that determine structural or functional themes are examples of these problems. Previously proposed methods, such as dynamic programming, are not adequate for solving problems of realistic size. This paper gives a new and practical solution for finding unknown patterns that occur imperfectly above a preset frequency. Algorithms for finding the patterns are given as well as estimates of statistical significance. This author supported by a grant from the System Development Foundation. This author supported by NSF grant MCS-8301960 and by a grant from the System Development Foundation. This author supported by NIH grant GM19036.     

Mathematical models of the early embryonic cell cycle: the role of MPF activation and cyclin degradation

Recent advances in cell biology indicate that the interactions between two proteins, cdc2 and cyclin, together with the activity of the cdc2/cyclin complex called MPF in the cytoplasm form the basis of a universal biochemical control mechanism for the cell division cycle in eukaryotes. Based on experimental facts that total cdc2 level is constant throughout the cell cycle and that onset of mitosis is subsequent to activation of MPF, we propose and analyze two different but related models — an ordinary differential equations model and a delay differential equations model — for the control of the early embryonic cell division cycle. Assuming very general reaction terms in the model equations, it is shown that MPF activation and rapid cyclin degradation triggered by active MPF drive cells to alternate between interphase and mitosis, the two phases of the cell cycle.S. Busenberg passed away on April 3, 1993 from complications of ALS (Lou Gehrig's disease). His research was supported by NSF Grant DMS-9112821Research was carried out at Harvey Mudd College and was supported by NSF Grant HRD-9252994     

On the two-locus sampling distribution

Two methods are discussed for evaluating the distribution of the configuration of unlabeled gametic types in a random sample of size n from the two-locus infinitely-many-neutral-alleles diffusion model at stationarity. Both involve finding systems of linear equations satisfied by the desired probabilities. The first approach, which is due to Golding, is to include additional probabilities in the system that allow some members of the sample to be specified at only one locus. The second approach, which is new, considers the joint distribution of the sample configuration and the number of recombination events since the time of the most recent common ancestor. The first approach is used for numerical computation, whereas the second approach is used to derive a two-locus version of Hoppe's urn model. The latter permits efficient simulation of the two-locus sampling distribution, provided the recombination parameter is not too large.Supported in part by NSF grants DMS-8704369 and DMS-8902991     

Global convergence properties in multilocus viability selection models: the additive model and the Hardy-Weinberg law

A natural coordinate system is introduced for the analysis of the global stability of the Hardy-Weinberg (HW) polymorphism under the general multilocus additive viability model. A global convergence criterion is developed and used to prove that the HW polymorphism is globally stable when each of the loci is diallelic, provided the loci are overdominant and the multilocus recombination is positive. As a corollary the multilocus Hardy-Weinberg law for neutral selection is derived.Research supported in part by NIH grants GM 39907-01, GM 10452-26 and NSF Grant DMS 86-06244Research supported in part by a US-Israel Binational Science Foundation grant 85-00021 and NIH grant GM 28016     

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