Plasmid-bearing,plasmid-free organisms competing for two complementary nutrients in a chemostat

A model of competition for two complementary nutrients between plasmid-bearing and plasmid-free organisms in a chemostat is proposed. A rigorous mathematical analysis of the global asymptotic behavior of the model is presented. The work extends the model of competition for a single-limited nutrient studied by Stephanopoulos and Lapidus [Chem. Engng. Sci. 443 (1988) 49] and Hsu, Waltman and Wolkowicz [J. Math. Biol. 32 (1994) 731].     

Competition between plasmid-bearing and plasmid-free organisms in a chemostat with nutrient recycling and an inhibitor

The asymptotic behavior of solutions of a model for competition between plasmid-bearing and plasmid-free organisms in the chemostat with two distributed delays and an external inhibitor is considered. The model presents a refinement of the one considered by Lu and Hadeler [Z. Lu, K.P. Hadeler, Model of plasmid-bearing plasmid-free competition in the chemostat with nutrient recycling and an inhibitor, Math. Biosci. 167 (2000) p. 177]. The delays model the fact that the nutrient is partially recycled after the death of the biomass by bacterial decomposition. Furthermore, it is assumed that there is inter-specific competition between the plasmid-bearing and plasmid-free organisms as well as intra-specific competition within each population. Conditions for boundedness of solutions and existence of non-negative equilibrium are given. Analysis of the extinction of the organisms, including plasmid-bearing and plasmid-free organisms, and the uniform persistence of the system are also carried out. By constructing appropriate Liapunov-like functionals, some sufficient conditions of global attractivity to the extinction equilibria are obtained and the combined effects of the delays and the inhibitor are studied.     

Multiple limit cycles in the chemostat with variable yield

The global asymptotic behavior of solutions of the variable yield model is determined. The model generalizes the classical Monod model and it assumes that the yield is an increasing function of the nutrient concentration. In contrast to the Monod model, it is demonstrated that the variable yield model exhibits sustained oscillations. Moreover, it is shown that the variable yield model may undergo a subcritical Hopf bifurcation and feature at least two distinct limit cycles. Implications for the coexistence of competing populations are discussed.     

An impulsively controlled pest management model with n predator species and a common prey

This paper investigates the dynamics of a competitive single-prey n-predators model of integrated pest management, which is subject to periodic and impulsive controls, from the viewpoint of finding sufficient conditions for the extinction of prey and for prey and predator permanence. The per capita death rates of prey due to predation are given in abstract, unspecified forms, which encompass large classes of death rates arising from usual predator functional responses, both prey-dependent and predator-dependent. The stability and permanence conditions are then expressed as balance conditions between the cumulative death rate of prey in a period, due to predation from all predator species and to the use of control, and to the cumulative birth rate of prey in the same amount of time. These results are then specialized for the case of prey-dependent functional responses, their biological significance being also discussed.     

Dynamic multipopulation and density dependent evolutionary games related to replicator dynamics. A metasimplex concept

This paper contains the basic extensions of classical evolutionary games (multipopulation and density dependent models). It is shown that classical bimatrix approach is inconsistent with other approaches because it does not depend on proportion between populations. The main conclusion is that interspecific proportion parameter is important and must be considered in multipopulation models. The paper provides a synthesis of both extensions (a metasimplex concept) which solves the problem intrinsic in the bimatrix model. It allows us to model interactions among any number of subpopulations including density dependence effects. We prove that all modern approaches to evolutionary games are closely related. All evolutionary models (except classical bimatrix approaches) can be reduced to a single population general model by a simple change of variables. Differences between classic bimatrix evolutionary games and a new model which is dependent on interspecific proportion are shown by examples.     

A discrete, size-structured model of microbial growth and competition in the chemostat

A nonlinear matrix model of a size-structured microbial population growing on a scarce nutrient in a chemostat, derived by Gage et al. [6], is modified to include two competing populations. It is shown that competitive exclusion results. The winner is the population able to grow at the lower nutrient concentration.Research partially supported by NSF Grant DMS 9300974     

Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression

Mathematical analysis is carried out that completely determines the global dynamics of a mathematical model for the transmission of human T-cell lymphotropic virus I (HTLV-I) infection and the development of adult T-cell leukemia (ATL). HTLV-I infection of healthy CD4(+) T cells takes place through cell-to-cell contact with infected T cells. The infected T cells can remain latent and harbor virus for several years before virus production occurs. Actively infected T cells can infect other T cells and can convert to ATL cells, whose growth is assumed to follow a classical logistic growth function. Our analysis establishes that the global dynamics of T cells are completely determined by a basic reproduction number R(0). If R(0)< or =1, infected T cells always die out. If R(0)>1, HTLV-I infection becomes chronic, and a unique endemic equilibrium is globally stable in the interior of the feasible region. We also show that the equilibrium level of ATL-cell proliferation is higher when the HTLV-I infection of T cells is chronic than when it is acute.     

Estimation of effective vaccination rate: pertussis in New Zealand as a case study

In some cases vaccination is unreliable. For example vaccination against pertussis has comparatively high level of primary and secondary failures. To evaluate efficiency of vaccination we introduce the idea of effective vaccination rate and suggest an approach to estimate it. We consider pertussis in New Zealand as a case study. The results indicate that the level of immunity failure for pertussis is considerably higher than was anticipated.     

Impulsive control strategies in biological control of pesticide

By presenting and analyzing the pest-predator model under insecticides used impulsively, two impulsive strategies in biological control are put forward. The first strategy: the pulse period is fixed, but the proportional constant E(1) changes, which represents the fraction of pests killed by applying insecticide. For this scheme, two thresholds, E(1)(**) and E(1)(*) for E(1) are obtained. If E(1)>or=E(1)(*), both the pest and predator (natural enemies) populations go to extinction. If E(1)(**)相似文献   

Cluster formation for multi-strain infections with cross-immunity

Many infectious diseases exist in several pathogenic variants, or strains, which interact via cross-immunity. It is observed that strains tend to self-organise into groups, or clusters. The aim of this paper is to investigate cluster formation. Computations demonstrate that clustering is independent of the model used, and is an intrinsic feature of the strain system itself. We observe that an ordered strain system, if it is sufficiently complex, admits several cluster structures of different types. Appearance of a particular cluster structure depends on levels of cross-immunity and, in some cases, on initial conditions. Clusters, once formed, are stable, and behave remarkably regularly (in contrast to the generally chaotic behaviour of the strains themselves). In general, clustering is a type of self-organisation having many features in common with pattern formation.     

Phylogenetic distances for neighbour dependent substitution processes

We consider models of nucleotidic substitution processes where the rate of substitution at a given site depends on the state of the neighbours of the site. We first estimate the time elapsed between an ancestral sequence at stationarity and a present sequence. Second, assuming that two sequences are issued from a common ancestral sequence at stationarity, we estimate the time since divergence. In the simplest non-trivial case of a Jukes-Cantor model with CpG influence, we provide and justify mathematically consistent estimators in these two settings. We also provide asymptotic confidence intervals, valid for nucleotidic sequences of finite length, and we compute explicit formulas for the estimators and for their confidence intervals. In the general case of an RN model with YpR influence, we extend these results under a proviso, namely that the equation defining the estimator has a unique solution.     

Mathematical modeling and qualitative analysis of insulin therapies

Several insulin therapies are widely in clinical use with the basic strategy that mimics insulin secretion in a normal glucose-insulin endocrine metabolic regulatory system. In this paper, we model the insulin therapies using a delay differential equation model. We study the dynamics of the model both qualitatively and quantitatively. The analytical results show the existence and uniqueness of a stable periodic solution that corresponds to ultradian insulin secretion oscillations. Numerically we simulate the insulin administration based on our model. The numerical simulation results are in agreement with findings of clinical studies.     

Oscillations in a patchy environment disease model

For a single patch SIRS model with a period of immunity of fixed length, recruitment-death demographics, disease related deaths and mass action incidence, the basic reproduction number R(0) is identified. It is shown that the disease-free equilibrium is globally asymptotically stable if R(0)<1. For R(0)>1, local stability of the endemic equilibrium and Hopf bifurcation analysis about this equilibrium are carried out. Moreover, a practical numerical approach to locate the bifurcation values for a characteristic equation with delay-dependent coefficients is provided. For a two patch SIRS model with travel, it is shown that there are several threshold quantities determining its dynamic behavior and that travel can reduce oscillations in both patches; travel may enhance oscillations in both patches; or travel can switch oscillations from one patch to another.     

Stability analysis for HIV infection delay model with protease inhibitor

In this article, we considered a model of HIV-1 infection with a protease inhibitor therapy and three delays. The frequency of the bifurcating periodic solution as well as the threshold value is approximated numerically using realistic parameter. The estimated threshold value is realistic and the frequency of the oscillations is consistent with that of the observed viral blips.     

A theoretical study on mathematical modelling of an infectious disease with application of optimal control

In this paper, we propose and analyze an epidemic problem which can be controlled by vaccination as well as treatment. In the first part of our analysis we study the dynamical behavior of the system with fixed control for both vaccination and treatment. Basic reproduction number is obtained in all possible cases and it is observed that the simultaneous use of vaccination and treatment control is the most favorable case to prevent the disease from being epidemic. In the second part, we take the controls as time dependent and obtain the optimal control strategy to minimize both the infected populations and the associated costs. All the analytical results are verified by simulation works. Some important conclusions are given at the end of the paper.     

Function-valued adaptive dynamics and the calculus of variations

Adaptive dynamics has been widely used to study the evolution of scalar-valued, and occasionally vector-valued, strategies in ecologically realistic models. In many ecological situations, however, evolving strategies are best described as function-valued, and thus infinite-dimensional, traits. So far, such evolution has only been studied sporadically, mostly based on quantitative genetics models with limited ecological realism. In this article we show how to apply the calculus of variations to find evolutionarily singular strategies of function-valued adaptive dynamics: such a strategy has to satisfy Euler's equation with environmental feedback. We also demonstrate how second-order derivatives can be used to investigate whether or not a function-valued singular strategy is evolutionarily stable. We illustrate our approach by presenting several worked examples.     

On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment

In this paper we extend the deterministic model for the epidemics induced by virulent phages on bacteria in marine environment introduced by Beretta and Kuang [Math. Biosci. 149 (1998) 57], allowing random fluctuations around the positive equilibrium. The stochastic stability properties of the model are investigated both analytically and numerically suggesting that the deterministic model is robust with respect to stochastic perturbations.     

Sequence length bounds for resolving a deep phylogenetic divergence

In evolutionary biology, genetic sequences carry with them a trace of the underlying tree that describes their evolution from a common ancestral sequence. The question of how many sequence sites are required to recover this evolutionary relationship accurately depends on the model of sequence evolution, the substitution rate, divergence times and the method used to infer phylogenetic history. A particularly challenging problem for phylogenetic methods arises when a rapid divergence event occurred in the distant past. We analyse an idealised form of this problem in which the terminal edges of a symmetric four-taxon tree are some factor (λ) times the length of the interior edge. We determine an order λ² lower bound on the growth rate for the sequence length required to resolve the tree (independent of any particular branch length). We also show that this rate of sequence length growth can be achieved by existing methods (including the simple ‘maximum parsimony’ method), and compare these order λ² bounds with an order λ growth rate for a model that describes low-homoplasy evolution. In the final section, we provide a generic bound on the sequence length requirement for a more general class of Markov processes.     

A basic limitation on inferring phylogenies by pairwise sequence comparisons

Distance-based approaches in phylogenetics such as Neighbor-Joining are a fast and popular approach for building trees. These methods take pairs of sequences, and from them construct a value that, in expectation, is additive under a stochastic model of site substitution. Most models assume a distribution of rates across sites, often based on a gamma distribution. Provided the (shape) parameter of this distribution is known, the method can correctly reconstruct the tree. However, if the shape parameter is not known then we show that topologically different trees, with different shape parameters and associated positive branch lengths, can lead to exactly matching distributions on pairwise site patterns between all pairs of taxa. Thus, one could not distinguish between the two trees using pairs of sequences without some prior knowledge of the shape parameter. More surprisingly, this can happen for any choice of distinct shape parameters on the two trees, and thus the result is not peculiar to a particular or contrived selection of the shape parameters. On a positive note, we point out known conditions where identifiability can be restored (namely, when the branch lengths are clocklike, or if methods such as maximum likelihood are used).