Modeling vital rates improves estimation of population projection matrices

Population projection matrices are commonly used by ecologists and managers to analyze the dynamics of stage-structured populations. Building projection matrices from data requires estimating transition rates among stages, a task that often entails estimating many parameters with few data. Consequently, large sampling variability in the estimated transition rates increases the uncertainty in the estimated matrix and quantities derived from it, such as the population multiplication rate and sensitivities of matrix elements. Here, we propose a strategy to avoid overparameterized matrix models. This strategy involves fitting models to the vital rates that determine matrix elements, evaluating both these models and ones that estimate matrix elements individually with model selection via information criteria, and averaging competing models with multimodel averaging. We illustrate this idea with data from a population of Silene acaulis (Caryophyllaceae), and conduct a simulation to investigate the statistical properties of the matrices estimated in this way. The simulation shows that compared with estimating matrix elements individually, building population projection matrices by fitting and averaging models of vital-rate estimates can reduce the statistical error in the population projection matrix and quantities derived from it.     

Comparative statics and stochastic dynamics of age-structured populations

Arguments from the comparative statics of populations with fixed vital rates are of limited use in studying age-structured populations subject to stochastically varying vital rates. In an age-structured population that experiences a sequence of independently and identically distributed Leslie matrices, the expectation of the Malthusian parameters of the Leslie matrices has no exact interpretation either as the ensemble average of the long-run rate of growth of each sample path of the population (Eq. (3)) or as the long-run rate of growth of the ensemble average of total population size (Eq. (4)). On the other hand, the Malthusian parameter of the expectation of a sequence of Leslie matrices is exactly the logarithm of the finite growth rate of the ensemble average of total population size when Leslie matrices are independently and identically distributed (though not in general when Leslie matrices are sequentially dependent). These observations appear to contradict the claims of a recent study using computer simulation of age-structured populations with stochastically varying vital rates.     

Stochastic stable population growth in integral projection models: theory and application

Stochastic matrix projection models are widely used to model age- or stage-structured populations with vital rates that fluctuate randomly over time. Practical applications of these models rest on qualitative properties such as the existence of a long term population growth rate, asymptotic log-normality of total population size, and weak ergodicity of population structure. We show here that these properties are shared by a general stochastic integral projection model, by using results in (Eveson in D. Phil. Thesis, University of Sussex, 1991, Eveson in Proc. Lond. Math. Soc. 70, 411-440, 1993) to extend the approach in (Lange and Holmes in J. Appl. Prob. 18, 325-344, 1981). Integral projection models allow individuals to be cross-classified by multiple attributes, either discrete or continuous, and allow the classification to change during the life cycle. These features are present in plant populations with size and age as important predictors of individual fate, populations with a persistent bank of dormant seeds or eggs, and animal species with complex life cycles. We also present a case-study based on a 6-year field study of the Illyrian thistle, Onopordum illyricum, to demonstrate how easily a stochastic integral model can be parameterized from field data and then applied using familiar matrix software and methods. Thistle demography is affected by multiple traits (size, age and a latent "quality" variable), which would be difficult to accommodate in a classical matrix model. We use the model to explore the evolution of size- and age-dependent flowering using an evolutionarily stable strategy (ESS) approach. We find close agreement between the observed flowering behavior and the predicted ESS from the stochastic model, whereas the ESS predicted from a deterministic version of the model is very different from observed flowering behavior. These results strongly suggest that the flowering strategy in O. illyricum is an adaptation to random between-year variation in vital rates.     

Time, transients and elasticity

How does life history affects the short‐term elasticities of population growth rate? We decompose short‐term elasticity as a sum of (i) the effect of the perturbation in rates on the unperturbed population structure and (ii) the effect of the original vital rates on the difference in structure between the original and the perturbed population. We provide exact analytical formulas for these components. In a population at its stable stage distribution (SSD), short‐term elasticity is determined mainly by the SSD and reproductive value. In a non‐stable population, short‐term elasticity depends also on the projection of initial structure on the SSD, equal to population momentum. Non‐stable stage structures matter most to elasticity if stages are missing that take time to fill in. We show how the demographic damping rate of the original population determines the rate at which short‐term elasticity converges to its limiting values.     

Estimating stochastic elasticities directly from longitudinal data

The elasticities of long-run population growth rate with respect to vital rates are useful in studying selection on vital rates, and in evaluating management policy that aims to control vital rates. In temporally varying environments, elasticity is often calculated from simulations that assume a probability distribution for the environmental states. Here we develop a method to estimate elasticities directly from demographic data. Using a time-series of demographic matrices and age-structure we construct a consistent statistical estimator of elasticity that converges to the correct limiting value as the sample length increases. We also construct confidence intervals for elasticities from temporal data and suggest tools for testing hypotheses about the strength of selection. We use data on a natural population to show that our method can indeed accurately estimate elasticities using relatively short time series.     

A cautionary note on elasticity analyses in a ternary plot using randomly generated population matrices

Matrix population models are one of the most common mathematical models in ecology, which describe the dynamics of stage-structured populations and provide us many population statistics. One of the statistics, elasticity onto population growth rate, is frequently used and represents the degree of the relative impact of life history parameters to the population growth rate. Due to the utility of elasticities for cross-taxonomic comparisons, Silvertown and his coauthors have published multiple papers and reported the relationship between elasticities and life forms (or life history) in multiple plant species, using a triangle map (called “ternary plot”). To understand why their elasticities are located in specific regions of the ternary plot, we constructed four archetypes of population matrices, from which we simulated 24,000 randomly generated population matrices and obtained the consequent elasticities. We found a large discrepancy when comparing our results to those in Silvertown et al.'s study (Conserv Biol 10:591–597, 1996): for our simulated matrices where rapid transitions were not allowed (e.g., trees), the elasticity distribution resulted in a line across the ternary plot. We provided the mathematical proof for this result, and found that its slope depends on matrix dimension. We also used 1230 matrices from the COMPADRE Plant Matrix Database and calculated the elasticities. Our simulated results were validated with field data from COMPADRE: two straight lines appeared in the ternary plot. Furthermore, we answered several addressed questions, such as, “Is there any special elasticity distribution in matrices with high population growth rates?” and “Why are the elasticities of natural populations concentrated in the upper half of the ternary plot?”.     

Dynamics of populations with changing rates: generalization of the stable population theory

A general and complete exposition of the dynamics of populations with changing vital rates is given in the discrete time formulation. Results obtained are stronger than Lopez's or Hajnal's. In addition to the proof of existence of limits, an explicit expression for the age distribution is obtained by considering forward products of population projection matrices, while an explicit expression for the generalized reproductive values is obtained by considering backward products. The forward and backward characteristic equations respectively determine the forward and backward growth rates. The relative age distribution is compared to the alternative expression of Y.J. Kim (1986, Demography 23 (3), 451-461), which is the discrete version of S.H. Preston and A.J. Coale (1982, Pop. Index 48 (2), 217-259).     

Sequence structure of Lowary/Widom clones forming strong nucleosomes

Lowary and Widom selected from random sequences those which form exceptionally stable nucleosomes, including clone 601, the current champion of strong nucleosome (SN) sequences. This unique sequence database (LW sequences) carries sequence elements which confer stability on the nucleosomes formed on the sequences, and, thus, may serve as source of information on the structure of “ideal” or close to ideal nucleosome DNA sequence. An important clue is also provided by crystallographic study of Vasudevan and coauthors on clone 601 nucleosomes. It demonstrated that YR·YR dinucleotide stacks (primarily TA·TA) follow one another at distances 10 or 11 bases or multiples thereof, such that they all are located on the interface between DNA and histone octamer. Combining this important information with alignment of the YR-containing 10-mers and 11-mers from LW sequences, the bendability matrices of the stable nucleosome DNA are derived. The matrices suggest that the periodically repeated TA (YR), RR, and YY dinucleotides are the main sequence features of the SNs. This consensus coincides with the one for recently discovered SNs with visibly periodic DNA sequences. Thus, the experimentally observed stable LW nucleosomes and SNs derived computationally appear to represent the same entity – exceptionally stable SNs.     

Modeling and Analysis of a Density-Dependent Stochastic Integral Projection Model for a Disturbance Specialist Plant and Its Seed Bank

In many plant species dormant seeds can persist in the soil for one to several years. The formation of these seed banks is especially important for disturbance specialist plants, as seeds of these species germinate only in disturbed soil. Seed movement caused by disturbances affects the survival and germination probability of seeds in the seed bank, which subsequently affect population dynamics. In this paper, we develop a stochastic integral projection model for a general disturbance specialist plant-seed bank population that takes into account both the frequency and intensity of random disturbances, as well as vertical seed movement and density-dependent seedling establishment. We show that the probability measures associated with the plant-seed bank population converge weakly to a unique measure, independent of initial population. We also show that the population either persists with probability one or goes extinct with probability one, and provides a sharp criteria for this dichotomy. We apply our results to an example motivated by wild sunflower (Helianthus annuus) populations, and explore how the presence or absence of a “storage effect” impacts how a population responds to different disturbance scenarios.     

Backwrd population projection by a generalized inverse

The Leslie population projection matrix may be used to project forward in time the age distribution or age-sex distribution of a population. As it is a singular matrix, it does not have an inverse, and so it is not clear that there is a corresponding procedure for backward projection. In terms of the eigenvalues and eigenvectors of the Leslie matrix, certain generalized inverses are constructed that can sometimes be used advantageously for backward projection.     

Relationships in a simple harmonic mean two-sex fertility model

A continuous time two-sex stable population model that does not recognize age is examined under the “harmonic mean” consistency condition of equation (2). Solutions for the stable population intrinsic growth rate r and the sex composition 5 are presented in equations (5) and (6). The process of stabilization is examined, and it is shown that, given two basic constraints, any initial population sex composition will eventually converge to the stable population value. An algebraic solution for the discrete case where the sex ratio at birth is unity is presented and used to describe the trajectory to stability of several hypothetical populations. A closed form algebraic expression for the trajectory to stability is presented for the continuous model in the special case of no mortality.     

New method of three-dimensional analysis of bipedal locomotion for the study of displacements of the body and body-parts centers of mass in man and non-human primates: Evolutionary framework

The current biomechanical interpretation of the chimpanzee's bipedal walking argues that larger lateral and vertical displacements of the body center of mass occur in the chimpanzee's “side-to-side” gait than in the human striding gait. The evolutionary hypothesis underlying this study is the following: during the evolution of human bipedalism one of the necessary changes could have been the progressive reduction of these displacements of the body center of mass. In order to quantitatively test this hypothesis, it is necessary to obtain simultaneously the trajectories of the centers of mass of the whole body and of the different body parts. To solve this problem, a new method of three-dimensional analysis of walking, associated with a volumetric modelling of the body, has been developed based on finite-element modelling. An orthogonal synchrophotographic device yielding four synchronous pictures of the walking subject allows a qualitative analysis of the photographic sequences together with the results of their quantitative analysis. This method was applied to an adult man, a 3-year-old girl and a 9-year-old male chimpanzee. Our results suggest that the trajectory of the body center of mass of the human is distinguished from that of the chimpanzee not by a lower movement amplitude but by the synchronization of the transverse and vertical displacements into two periodic curves in phase with one another. The non-human primate uses its repertoire of arboreal movements in its bipedal terrestrial gait, provisionally referred to as a “rope-walker” gait. We show that the interpretation of a “side-to-side” gait is not applicable to the chimpanzee. We argue that similarly this interpretation and the initial hypothesis presuppose a basic symmetric structure of the gait, in relation to the sagittal plane of progression, similar to the human one. This lateral symmetry of the right and left displacements of the center of gravity, in phase with the right and left single supports of walking, is probably a very derived feature of the human gait. We suggest that low lateral and vertical displacements of the body center of mass are not indicative of a progressive bipedal gait and we discuss the new evolutionary implications of our results. © 1993 Wiley-Liss, Inc.     

Convergence rate estimation for the TKF91 model of biological sequence length evolution

The TKF91 model of biological sequence evolution describes changes in the sequence length via an infinite state-space birth-death process, which we term the TKF91-BD process. The TKF91 model assumes that, for any pair of modern sequences, the ancestral sequence has equilibrium length distribution, an assumption whose validity has not been rigorously investigated. We obtain explicit upper and lower bounds on the rate of convergence to equilibrium for the distribution of the TKF91-BD process. We show that the rate of convergence of the TKF91-BD process for protein sequences with parameter values inferred from sequence data on alpha and beta globins is too low to guarantee convergence to equilibrium on a reasonable timescale. For the analyzed nucleotide sequences, the convergence is faster, but the equilibrium sequence length is unrealistically small. The Jukes-Cantor model of nucleotide substitutions can converge considerably faster than the length evolution model for both amino acid and nucleotide sequences, while the speed of convergence for the Kimura model is close to that for the TKF91-BD process describing nucleotide sequences.     

Conformational Changes in DNA and Soft Modes

Abstract

In this paper we explore the applicability of the soft mode approach to study the conformational transitions of DNA. It is believed that the A-B conformation change is a first order transition. Soft mode theories only apply to the initial stages of a first order transition. However the mode softening in such a case can be the initiating factor which ultimately leads to the transition. The first order transition is, then, a breakdown of what otherwise would have been a true second order transition. The mode softening is causally connected to onset of the transition. We use the eigenvectors obtained from lattice dynamics calculations to identify the softmode. We use the eigenvector projections to form a force constimi matrix that is required to drive a mode soft. We explore the methods by which this force constant matrix can be formed. We suggest that the breaking of specific “water bridges” between phosphate groups in the two single strands can drive the conformation change.     

Developmental implications of dichotomous ossification sequences in the wrist region

As shown in radiographs of 3,764 children of European ancestry under 11 years of age, there are 55 dichotomous (present-absent/absent-present) ossification sequences for nine wrist region centers in boys and 48 such sequences for girls, with statistically significant sexual dimorphism in more than half of the “alternative” sequences. Substantial samples of Meso-American origin (“Chicanos”) and largely-African origin (“Blacks”) evidence additional dichotomous ossification sequences, with clear population differences, while Down's syndrome children (47,G +) show a major excess of the numerically rarer sequences of the wrist region.     

Stochastic LTRE analysis of the effects of herbivory on the population dynamics of a perennial grassland herb

Herbivores can have strong deleterious effects on vital rates (growth, reproduction, and survival) and thus negatively impact the population dynamics of plant species. In practice, however, these effects might be strongly correlated, for example as a result of tradeoffs between vital rates. To get better insights into the effects of herbivory on the population dynamics of the long‐lived grassland plant Primula veris population projection matrices were constructed from demographic data collected between 1999 and 2008 (nine annual transitions). Data were collected in two large grassland populations, each of which was subjected to two treatments (grazing by cattle versus a mowing treatment), yielding a total of 36 matrices. We applied a lower‐level vital rate life table response experiment (LTRE) using the small noise approximation (SNA) of the stochastic population growth rate to disentangle the contributions of changes in mean vital rates, variability in vital rates, correlations between vital rates and vital rate elasticities to the difference in the stochastic growth rate. Stochastic growth rates (a= log λ_S) were significantly lower in grazed than in mown plots (a= 0.0185 and 0.1019, respectively). SNA LTRE analysis showed that contributions of mean vital rates by far made the largest contribution to the observed difference in a between grazed and control plots. In particular, changes in sexual reproduction rates made the largest contributions to lower the stochastic growth rate in grazed plots: both adult flowering probabilities and flower and seed production were importantly lower in grazed populations, but these negative effects were largely buffered by increased establishment and seedling survival rates. Among the stochastic terms of the SNA decomposition, contributions of covariance and correlations between vital rates had the largest impact, whereas contributions of elasticities were smaller. The strongest correlation driver was the association between adult survival and seedling establishment, suggesting that environmental conditions favouring adult survival also are beneficial for seedling establishment. Overall, our results show that herbivory had a strong negative effect on the long‐term population growth rate of P. veris that was primarily mediated by differences in fecundity (flower and seed production) and germination.     

Computational analysis of chain flexibility and fluctuations in Rhizomucor miehei lipase.

We have performed molecular dynamics simulation of Rhizomucor miehei lipase (Rml) with explicit water molecules present. The simulation was carried out in periodic boundary conditions and conducted for 1. 2 ns in order to determine the concerted protein dynamics and to examine how well the essential motions are preserved along the trajectory. Protein motions are extracted by means of the essential dynamics analysis method for different lengths of the trajectory. Motions described by eigenvector 1 converge after approximately 200 ps and only small changes are observed with increasing simulation time. Protein dynamics along eigenvectors with larger indices, however, change with simulation time and generally, with increasing eigenvector index, longer simulation times are required for observing similar protein motions (along a particular eigenvector). Several regions in the protein show relatively large fluctuations and in particular motions in the active site lid and the segments Thr57-Asn63 and the active site hinge region Pro101-Gly104 are seen along several eigenvectors. These motions are generally associated with glycine residues, while no direct correlations are observed between these fluctuations and the positioning of prolines in the protein structure. The partial opening/closing of the lid is an example of induced fit mechanisms seen in other enzymes and could be a general mechanism for the activation of Rml.     

Population growth and structure in a variable environment

When a population experiences temporal changes in the vital rates due to environmental or biotic variation, change is not only expected in the rate of population growth but also in the structure of the population. In this study we present a method for transforming observed patterns (notably how vital rates change with temperature) into functions that can be used in population growth models and analysis of population structure. The method is exemplified by applying it to cohort studies in different constant temperatures of four species of aphids, Lipaphis erysimi (K.), Metopolophium dirhodum (Wlk.), Rhopaliosiphum padi and Macrosiphum avenae (F.). We use piece-wise linear functions to transform the vital rates of the cohort studies. The lifespans are divided into phases, each phase having linear rates. A projection matrix is formulated, where the elements are temperature dependent fecundities, survivorships and developmental rates. The major result is, contrary to what theory predicts as reasonable (Caswell 1989), that population structure of these aphid species will become almost fixed although the temperature varies. This result is consistent with findings of earlier field studies (Wiktelius 1982). A fixed population structure implies that it is possible to calculate the population growth rate on the basis of intrinsic rates of increase. By simulating different temperature regimes we also show that initial oscillations in the population structure dampen out after a few days. After initial oscillations, calculations of population growth using intrinsic rates of increase are consistent with calculations made by a matrix model.     

How phenotypic convergence arises in experimental evolution

Evolutionary convergence is a core issue in the study of adaptive evolution, as well as a highly debated topic at present. Few studies have analyzed this issue using a “real‐time” or evolutionary trajectory approach. Do populations that are initially differentiated converge to a similar adaptive state when experiencing a common novel environment? Drosophila subobscura populations founded from different locations and years showed initial differences and variation in evolutionary rates in several traits during short‐term (～20 generations) laboratory adaptation. Here, we extend that analysis to 40 more generations to analyze (1) how differences in evolutionary dynamics among populations change between shorter and longer time spans, and (2) whether evolutionary convergence occurs after 60 generations of evolution in a common environment. We found substantial variation in longer term evolutionary trajectories and differences between short‐ and longer term evolutionary dynamics. Although we observed pervasive patterns of convergence toward the character values of long‐established populations, populations still remain differentiated for several traits at the final generations analyzed. This pattern might involve transient divergence, as we report in some cases, indicating that more generations should lead to final convergence. These findings highlight the importance of longer term studies for understanding convergent evolution.