Mixed-up Trees: the Structure of Phylogenetic Mixtures

In this paper, we apply new geometric and combinatorial methods to the study of phylogenetic mixtures. The focus of the geometric approach is to describe the geometry of phylogenetic mixture distributions for the two state random cluster model, which is a generalization of the two state symmetric (CFN) model. In particular, we show that the set of mixture distributions forms a convex polytope and we calculate its dimension; corollaries include a simple criterion for when a mixture of branch lengths on the star tree can mimic the site pattern frequency vector of a resolved quartet tree. Furthermore, by computing volumes of polytopes we can clarify how “common” non-identifiable mixtures are under the CFN model. We also present a new combinatorial result which extends any identifiability result for a specific pair of trees of size six to arbitrary pairs of trees. Next we present a positive result showing identifiability of rates-across-sites models. Finally, we answer a question raised in a previous paper concerning “mixed branch repulsion” on trees larger than quartet trees under the CFN model. F.A. Matsen’s and M. Steel’s research was supported by the Allan Wilson Centre for Molecular Ecology and Evolution. E. Mossel’s research was supported by a Sloan fellowship in Mathematics, NSF awards DMS 0528488 and DMS 0548249 (CAREER) and by ONR grant N0014-07-1-05-06.     

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.     

Stochastic models for phylogenetic trees on higher-order taxa

Simple stochastic models for phylogenetic trees on species have been well studied. But much paleontology data concerns time series or trees on higher-order taxa, and any broad picture of relationships between extant groups requires use of higher-order taxa. A coherent model for trees on (say) genera should involve both a species-level model and a model for the classification scheme by which species are assigned to genera. We present a general framework for such models, and describe three alternate classification schemes. Combining with the species-level model of Aldous and Popovic (Adv Appl Probab 37:1094–1115, 2005), one gets models for higher-order trees, and we initiate analytic study of such models. In particular we derive formulas for the lifetime of genera, for the distribution of number of species per genus, and for the offspring structure of the tree on genera. David Aldous’s research was supported by NSF Grant DMS-0704159.     

Multiple limit cycles in the standard model of three species competition for three essential resources

We consider the dynamics of the standard model of 3 species competing for 3 essential (non-substitutable) resources in a chemostat using Liebig's law of the minimum functional response. A subset of these systems which possess cyclic symmetry such that its three single-population equilibria are part of a heteroclinic cycle bounding the two-dimensional carrying simplex is examined. We show that a subcritical Hopf bifurcation from the coexistence equilibrium together with a repelling heteroclinic cycle leads to the existence of at least two limit cycles enclosing the coexistence equilibrium on the carrying simplex- the ``inside' one is an unstable separatrix and the ``outside' one is at least semi-stable relative to the carrying simplex. Numerical simulations suggest that there are exactly two limit cycles and that almost every positive solution approaches either the stable limit cycle or the stable coexistence equilibrium, depending on initial conditions. Bifurcation diagrams confirm this picture and show additional features. In an alternative scenario, we show that the subcritical Hopf together with an attracting heteroclinic cycle leads to an unstable periodic orbit separatrix. This research was partially supported by NSF grant DMS 0211614. KY 40292, USA. This author's research was supported in part by NSF grant DMS 0107160     

Does Dormancy Increase Fitness of Bacterial Populations in Time-Varying Environments?

A simple family of models of a bacterial population in a time varying environment in which cells can transit between dormant and active states is constructed. It consists of a linear system of ordinary differential equations for active and dormant cells with time-dependent coefficients reflecting an environment which may be periodic or random, with alternate periods of low and high resource levels. The focus is on computing/estimating the dominant Lyapunov exponent, the fitness, and determining its dependence on various parameters and the two strategies—responsive and stochastic—by which organisms switch between dormant and active states. A responsive switcher responds to good and bad times by making timely and appropriate transitions while a stochastic switcher switches continuously without regard to the environmental state. The fitness of a responsive switcher is examined and compared with fitness of a stochastic switcher, and with the fitness of a dormancy-incapable organism. Analytical methods show that both switching strategists have higher fitness than a dormancy-incapable organism when good times are rare and that responsive switcher has higher fitness than stochastic switcher when good times are either rare or common. Numerical calculations show that stochastic switcher can be most fit when good times are neither too rare or too common. This research was supported by NSF Grant DMS 0414270, Department of Mathematics, Arizona State University, Tempe, AZ.     

An adaptive model for synchrony in the firefly Pteroptyx malaccae

We describe a new model for synchronization of neuronal oscillators that is based on the observation that certain species of fireflies are able to alter their free-running period. We show that by adding adaptation to standard oscillator models it is possible to observe the frequency alteration. One consequence of this is the perfect synchrony between coupled oscillators. Stability and some analytic results are included along with numerical simulations.This work was partially supported by NSF Grant DMS9002028 and the Mathematical Research Branch of The National Institutes of Health     

Equilibrium distribution of species among vessels of a gradostat

The distribution of concentrations of two competing microbial species among the vessels of an n-vessel gradostat at equilibrium is studied for the standard mathematical model of the gradostat. As the equilibrium concentrations cannot be explicitly computed, a continuum limit, as the number of vessels becomes large, is considered which yields a singularly perturbed boundary value problem. Standard singular perturbation techniques yield information on equilibrium species concentration distributions which agree well with numerical calculations for even moderate values of n.Research supported by NSF Grant DMS 8922654     

On the Delayed Ross–Macdonald Model for Malaria Transmission

The feedback dynamics from mosquito to human and back to mosquito involve considerable time delays due to the incubation periods of the parasites. In this paper, taking explicit account of the incubation periods of parasites within the human and the mosquito, we first propose a delayed Ross–Macdonald model. Then we calculate the basic reproduction number R ₀ and carry out some sensitivity analysis of R ₀ on the incubation periods, that is, to study the effect of time delays on the basic reproduction number. It is shown that the basic reproduction number is a decreasing function of both time delays. Thus, prolonging the incubation periods in either humans or mosquitos (via medicine or control measures) could reduce the prevalence of infection. S. Ruan’s research was partially supported by NIH grants P20RR020770-02 and R01GM083607-01, NSF grant DMS-0715772. D. Xiao’s research was supported by the National Natural Science Fund (NNSF) of China.     

Modeling insect dispersal and estimating parameters when mark-release techniques may cause initial disturbances

We consider the problem of quantitatively modeling movements of marked flea beetles in cultivated arrays of the cole crop, collards (Brassica oleraceae). Methods for the estimation of temporally and spatially dependent parameters in general dispersal models are outlined and a summary of our findings using these methods with flea beetle data is given.This research was supported in part by the National Science Foundation under NSF # MCS-8205335 and by the U.S. Air Force Office of Scientific Research under contract AFOSR 81-0198. Part of the research was carried out while this author was a Visiting Professor in the Department of Mathematics at Southern Methodist UniversityThis research was supported in part by the National Science Foundation under NSF # DEB-8207117This research was supported in part by the National Science Foundation under NSF # MCS-8200883. Part of the research was carried out while the first and third authors were visitors at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, Virginia which is operated under NASA contracts NAS1-15810 and NAS1-16394     

A new graphical model for untangling complex relationships among environment, biodiversity, and ecosystem functioning

Recent advances in the research field of ‘biodiversity-ecosystem functioning’ have successfully begun to reconcile the apparent controversy on relationships between productivity and species richness. By unifying new advances into a single framework, I propose a 3D graphical model connecting the relationships among resource availability, species richness, and ‘community productivity.’ An emergent pattern from this model predicts that the effect of species richness on community productivity is maximized at intermediate levels of resource availability. This model will contribute to better understanding the relationships among environment, biodiversity, and ecosystem functioning.     

Trade-offs between reproductive and somatic (storage) investments in animals: a comparative test of the Van Noordwijk and De Jong model

Classical life-history theory predicts ‘trade-offs’ between reproductive and somatic investments. However, empirical studies have shown that intraspecific phenotypic correlations between these two resource investments are often positive or nonsignificant, rather than negative as predicted. The model of Van Noordwijk and De Jong (1986) was proposed to explain these unexpected results. According to their model, positive correlations between reproductive and somatic investments will result if individual variation in resource acquisition exceeds that of resource allocation, whereas negative correlations will result if individual variation in resource allocation exceeds that of resource acquisition. To test this model, I used body storage/condition as an index of somatic investment because it is usually strongly related to level of resource acquisition. I predicted that laboratory studies should more often show negative correlations between reproductive and somatic investments than field studies, because individual variation in resource acquisition is expected to be lower in controlled laboratory environments than in variable natural environments. A literature review revealed that correlations between somatic (storage) investment and reproductive investment (estimated as clutch/litter mass, number of offspring per clutch/litter, or number of clutches/litters) among conspecific breeding female animals are more often positive (15 species) or nonsignificant (17 species) than negative (6 species). Moreover, as expected, five of six negative correlations were observed in laboratory studies, whereas 13 of 15 positive correlations were observed in field studies. It is concluded that future empirical and theoretical work on life histories should consider individual variation in both resource acquisition and allocation and the interaction between the two. This revised version was published online in July 2006 with corrections to the Cover Date.     

Response of a solutivory chain to a nutrient pulse

A microbial community model is proposed that accounts for byproducts of one strain being nutrients for another and for cells passing in and out of states of torpor. It is shown that such models can sustain the propagation of a nutrient pulse as observed, for example, in methanogenesis.This work was supported in part by NSF Grant DMS9206677 and in part by NSF STC Center in Microbial Ecology, Michigan State University.     

Phytoplankton stoichiometry

Because phytoplankton live at the interface between the abiotic and the biotic compartments of ecosystems, they play an important role in coupling multiple nutrient cycles. The quantitative details of how these multiple nutrient cycles intersect is determined by phytoplankton stoichiometry. Here we review some classic work and recent advances on the determinants of phytoplankton stoichiometry and their role in determining ecosystem stoichiometry. First, we use a model of growth with flexible stoichiometry to reexamine Rhee and Goldman’s classic chemostat data. We also discuss a recent data compilation by Hall and colleagues that illustrates some limits to phytoplankton flexibility, and a model of physiological adaptation that can account for these results. Second, we use a model of resource allocation to determine the how the optimal nitrogen-to-phosphorus stoichiometry depends on the ecological conditions under which species grow and compete. Third, we discuss Redfield’s mechanism for the homeostasis of the oceans’ nitrogen-to-phosphorus stoichiometry and show its robustness to additional factors such as iron-limitation and temporal fluctuations. Finally, we suggest areas for future research.     

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     

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     

Phylogeography and conservation of the endemic Hispaniolan Palm-Tanagers (Aves: <Emphasis Type="Italic">Phaenicophilus</Emphasis>)

The Gray-crowned Palm-Tanager (Phaenicophilus poliocephalus), sometimes considered conspecific with its more widespread congener P. palmarum, is restricted to Haiti’s Tiburon Peninsula, a biodiversity hotspot threatened by extensive habitat loss. We used a multilocus phylogeographic approach to identify evolutionarily distinct populations of Phaenicophilus. Mitochondrial haplotypes formed two reciprocally monophyletic groups separated by 5% uncorrected divergence. Genealogical patterns of differentiation at nuclear intron alleles were congruent with those of mtDNA, and the two species also differed in body size and shape. An ancient sea channel between the Tiburon Peninsula and mainland Haiti was likely a dispersal barrier that led to allopatric divergence, a hypothesis supported by our estimates of divergence times. Our results support the recognition of two Palm-Tanager species, confirming P. poliocephalus as Haiti’s only endemic bird species and underscoring the need to protect the Tiburon Peninsula’s single primary forest reserve.     

Competition in the gradostat: the role of the communication rate

In this paper we study a mathematical model of competition between two species of microorganisms for a single limiting nutrient in a laboratory device called a gradostat. A gradostat consists of several (we consider only two) chemostats (CSTR's) connected together so that material can flow between the vessels in such a way that a nutrient gradient is established. Our model is a slightly modified version of one considered recently by Jäger et al. [3], in that the rate of exchange of material between the two vessels (the communication rate) is allowed to differ from the dilution rate. The outcome of competition turns out to be surprisingly sensitive to variation of the communication rate. We identify several coexistence regimes in parameter space and describe a method for obtaining operating diagrams for given pairs of competing microorganisms.Research supported in part by NSF Grant DMS 8521605     

Breeding migration and population stability

We have modeled habitat shift for reproduction to examine the relationship between the timing of migration and population stability, by modifying Takimoto’s (Am Nat 162:93–109, 2003) consumer–resource model with a consumer’s ontogenetic niche shift. We found that equilibrium was always locally unstable if migration occurs at a fixed time or level of energy storage, whereas it could be stable if the timing of migration was adaptively flexible to maximize reproductive output. The general conditions for stability were safer breeding rather than feeding habitat and abundant resources at the feeding habitat. These results imply that both adopting an adaptive plastic strategy in the timing of migration and choosing to migrate from a rich feeding habitat to a safe breeding habitat can contribute to population stability. We also found that reduced reproductive success with delays in migration, and the survival rate after reproduction, had complicated effects on stability, depending on resource availability at the feeding habitat. The equilibrium was more likely to be stable when reproduction success was only slightly (or greatly) reduced or survival rate was high (or low) if the feeding habitat was rich (or poor). These are significant predictions for ecological study of migrating animals.     

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