Director, Stanford Center for Population Research (2010 - 2015)
Director, Demography Core, Center for Demography and Economics of Health and Aging (2009 - 2014)
Current Research and Scholarly Interests
Shripad Tuljapurkar is Professor of Biology and the Dean & Virginia Morrison Professor of Population Studies at Stanford University. His research areas include stochastic dynamics of human and natural populations; life history evolution, especially senescence; prehistoric societies; and probability forecasts including sex ratios, mortality, aging and fiscal balance.
Tuljapurkar directs Stanford’s Center for Population Research and the demography program at Stanford’s Center for the Demography and Economics of Health and Aging. He is a member of the Center for the Demography and Economics of Aging at the University of California, Berkeley. He has led a panel on aging for the International Union for the Scientific Study of Population and served on the Technical Advisory Panel to the US Social Security Administration. He received the 1996 Mindel Sheps Award from the Population Association of America, and a John Simon Guggenheim Fellowship in 1998.
- Essential Statistics for Human Biology
BIO 108, HUMBIO 85A (Spr)
- Matrix Methods for Dynamic Models and Data Analysis
BIO 329 (Aut)
- Stochastic Methods for Simulation, Dynamics and Data Analysis
BIO 330 (Win)
Independent Studies (10)
- Advanced Research Laboratory in Experimental Biology
BIO 199 (Aut, Win, Spr, Sum)
- Curricular Practical Training
BIO 292 (Sum)
- Directed Reading in Biology
BIO 198 (Aut, Win, Spr, Sum)
- Directed Reading in Environment and Resources
ENVRES 398 (Aut, Win, Spr, Sum)
- Directed Research in Environment and Resources
ENVRES 399 (Aut, Win, Spr, Sum)
- Graduate Research
BIO 300 (Aut, Win, Spr, Sum)
- Out-of-Department Advanced Research Laboratory in Experimental Biology
BIO 199X (Sum)
- Out-of-Department Directed Reading
BIO 198X (Aut, Win, Spr, Sum)
- Out-of-Department Graduate Research
BIO 300X (Aut, Win, Spr, Sum)
- Teaching of Biology
BIO 290 (Aut, Win, Spr)
- Advanced Research Laboratory in Experimental Biology
Prior Year Courses
- Essential Statistics for Human Biology
BIO 108, HUMBIO 85A (Spr)
- Essential Mathematics for Research in Life and Social Sciences
BIO 220 (Win)
- Essential Statistics for Human Biology
HUMBIO 85A (Spr)
- Essential Statistics for Human Biology
Graduate and Fellowship Programs
Biology (School of Humanities and Sciences) (Phd Program)
Des differences, pourquoi? Transmission, maintenance and effects of phenotypic variance
JOURNAL OF ANIMAL ECOLOGY
2016; 85 (2): 356-370
Despite the observed distribution of variable individual phenotypes, survival and reproductive performance in wild populations, models of population dynamics often focus on mean demographic rates. Populations are constituted by individuals with different phenotypes and thus different performances. However, many models of population dynamics provide no understanding of the influence of this phenotypic variation on population dynamics. In this paper, we investigate how the relationships between demographic rates and phenotype distribution influence the transmission and the upholding of phenotypic variation, and population dynamics. We used integral projection models to measure associations between differences of phenotypic trait (size or mass) among individuals and demographic rates, growth and inheritance, and then quantify the influence of phenotypic variation on population dynamics. We build an analytical and general model resulting from simplifications assuming small phenotypic variance. We illustrate our model with two case studies: a short- and a long-lived life history. Population growth rate r is determined by a Lotka style equation in which survival and fertility are averaged over a phenotypic distribution that changes with age. Here, we further decomposed r to show how much it is affected by shifts in phenotypic average as well as variance. We derived the elasticities of r to the first and second derivative of each demographic rate. In particular, we show that the nonlinearity of change in selective pressure with phenotype matters more to population dynamics than the strength of this selection. In other words, the variance of a given trait will be most important when the strength of selection increases (or decreases) nonlinearly with that trait. Inheritance shapes the distribution of newborn phenotypes. Even if newborns have a fixed average phenotype, the variance among newborns increases with phenotypic variance among mothers, strength of inheritance and developmental variation. We explain how the components of inheritance can influence phenotypic variance and thus the demographic rates and population dynamics. In particular, when mothers of different ages produce offspring of different mean phenotype, the inheritance function can have a large influence on both the mean and variance of the trait at different ages and thus on the population growth rate. We provide new tools to understand how phenotypic variation influences population dynamics and discuss in which life histories we expect this influence to be large. For instance, in our short-lived life history, individual variability has larger effect than in our long-lived life history. We conclude by indicating future directions of analysis.
View details for DOI 10.1111/1365-2656.12477
View details for Web of Science ID 000370959300005
View details for PubMedID 26899422
Deciphering life history transcriptomes in different environments
2015; 24 (1): 151-179
We compared whole transcriptome variation in six pre-adult stages and seven adult female ages in two populations of cactophilic Drosophila mojavensis reared on two host plants to understand how differences in gene expression influence standing life history variation. We used singular value decomposition (SVD) to identify dominant trajectories of life cycle gene expression variation, performed pairwise comparisons of stage and age differences in gene expression across the life cycle, identified when genes exhibited maximum levels of life cycle gene expression, and assessed population and host cactus effects on gene expression. Life cycle SVD analysis returned four significant components of transcriptional variation, revealing functional enrichment of genes responsible for growth, metabolic function, sensory perception, neural function, translation and ageing. Host cactus effects on female gene expression revealed population- and stage-specific differences, including significant host plant effects on larval metabolism and development, as well as adult neurotransmitter binding and courtship behaviour gene expression levels. In 3- to 6-day-old virgin females, significant upregulation of genes associated with meiosis and oogenesis was accompanied by downregulation of genes associated with somatic maintenance, evidence for a life history trade-off. The transcriptome of D. mojavensis reared in natural environments throughout its life cycle revealed core developmental transitions and genome-wide influences on life history variation in natural populations.
View details for DOI 10.1111/mec.13017
View details for Web of Science ID 000347446300012
View details for PubMedID 25442828
Generation Time, Net Reproductive Rate, and Growth in Stage-Age-Structured Populations
2014; 183 (6): 771-783
Abstract Major insights into the relationship between life-history features and fitness have come from Lotka's proof that population growth rate is determined by the level (expected amount) of reproduction and the average timing of reproduction of an individual. But this classical result is limited to age-structured populations. Here we generalize this result to populations structured by stage and age by providing a new, unique measure of reproductive timing (Tc) that, along with net reproductive rate (R0), has a direct mathematical relationship to and approximates growth rate (r). We use simple examples to show how reproductive timing Tc and level R0 are shaped by stage dynamics (individual trait changes), selection on the trait, and parent-offspring phenotypic correlation. We also show how population structure can affect dispersion in reproduction among ages and stages. These macroscopic features of the life history determine population growth rate r and reveal a complex interplay of trait dynamics, timing, and level of reproduction. Our results contribute to a new framework of population and evolutionary dynamics in stage-and-age-structured populations.
View details for DOI 10.1086/675894
View details for Web of Science ID 000335721700005
View details for PubMedID 24823821
Divergence in Age Patterns of Mortality Change Drives International Divergence in Lifespan Inequality
2014; 51 (3): 1003-1017
In the past six decades, lifespan inequality has varied greatly within and among countries even while life expectancy has continued to increase. How and why does mortality change generate this diversity? We derive a precise link between changes in age-specific mortality and lifespan inequality, measured as the variance of age at death. Key to this relationship is a young-old threshold age, below and above which mortality decline respectively decreases and increases lifespan inequality. First, we show for Sweden that shifts in the threshold's location have modified the correlation between changes in life expectancy and lifespan inequality over the last two centuries. Second, we analyze the post-World War II (WWII) trajectories of lifespan inequality in a set of developed countries-Japan, Canada, and the United States-where thresholds centered on retirement age. Our method reveals how divergence in the age pattern of mortality change drives international divergence in lifespan inequality. Most strikingly, early in the 1980s, mortality increases in young U.S. males led to a continuation of high lifespan inequality in the United States; in Canada, however, the decline of inequality continued. In general, our wider international comparisons show that mortality change varied most at young working ages after WWII, particularly for males. We conclude that if mortality continues to stagnate at young ages yet declines steadily at old ages, increases in lifespan inequality will become a common feature of future demographic change.
View details for DOI 10.1007/s13524-014-0287-8
View details for Web of Science ID 000337029400013
View details for PubMedID 24756909
Neutral theory for life histories and individual variability in fitness components
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
2012; 109 (12): 4684-4689
Individuals within populations can differ substantially in their life span and their lifetime reproductive success but such realized individual variation in fitness components need not reflect underlying heritable fitness differences visible to natural selection. Even so, biologists commonly argue that large differences in fitness components are likely adaptive, resulting from and driving evolution by natural selection. To examine this argument we use unique formulas to compute exactly the variance in life span and in lifetime reproductive success among individuals with identical (genotypic) vital rates (assuming a common genotype for all individuals). Such individuals have identical fitness but vary substantially in their realized individual fitness components. We show by example that our computed variances and corresponding simulated distribution of fitness components match those observed in real populations. Of course, (genotypic) vital rates in real populations are expected to differ by small but evolutionarily important amounts among genotypes, but we show that such differences only modestly increase variances in fitness components. We conclude that observed differences in fitness components may likely be evolutionarily neutral, at least to the extent that they are indistinguishable from distributions generated by neutral processes. Important consequences of large neutral variation are the following: Heritabilities for fitness components are likely to be small (which is in fact the case), small selective differences in life histories will be hard to measure, and the effects of random drift will be amplified in natural populations by the large variances among individuals.
View details for DOI 10.1073/pnas.1018096109
View details for Web of Science ID 000301712600067
View details for PubMedID 22392997
Dynamic heterogeneity and life histories
WILEY-BLACKWELL. 2010: 65-72
Biodemography is increasingly focused on the large and persistent differences between individuals within populations in fitness components (age at death, reproductive success) and fitness-related components (health, biomarkers) in humans and other species. To study such variation we propose the use of dynamic models of observable phenotypes of individuals. Phenotypic change in turn determines variation among individuals in their fitness components over the life course. We refer to this dynamic accumulation of fitness differences as dynamic heterogeneity and illustrate it for an animal population in which longitudinal data are studied using multistate capture-mark-recapture models. Although our approach can be applied to any characteristic, for our empirical example we use reproduction as the phenotypic character to define stages. We indicate how our stage-structured model describes the nature of the variation among individual characteristics that is generated by dynamic heterogeneity. We conclude by discussing our ongoing and planned work on animals and humans. We also discuss the connections between our work and recent work on human mortality, disability and health, and life course theory.
View details for DOI 10.1111/j.1749-6632.2010.05519.x
View details for Web of Science ID 000287380800007
View details for PubMedID 20738276
Inequality in life spans and a new perspective on mortality convergence across industrialized countries
POPULATION AND DEVELOPMENT REVIEW
2005; 31 (4): 645-?
View details for Web of Science ID 000234490200002
Population forecasting for fiscal planning: Issues and innovations
DEMOGRAPHIC CHANGE AND FISCAL POLICY
View details for Web of Science ID 000179946200002
STOCHASTIC POPULATION FORECASTS FOR THE UNITED-STATES - BEYOND HIGH, MEDIUM, AND LOW
JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
1994; 89 (428): 1175-1189
View details for Web of Science ID A1994PU33000003
Racial and Socioeconomic Variation in Genetic Markers of Telomere Length: A Cross-Sectional Study of U.S. Older Adults.
2016; 11: 296-301
Shorter telomere length (TL) has been associated with stress and adverse socioeconomic conditions, yet U.S. blacks have longer TL than whites. The role of genetic versus environmental factors in explaining TL by race and socioeconomic position (SEP) remains unclear.We used data from the U.S. Health and Retirement Study (N=11,934) to test the hypothesis that there are differences in TL-associated SNPs by race and SEP. We constructed a TL polygenic risk score (PRS) and examined its association with race/ethnicity, educational attainment, assets, gender, and age.U.S. blacks were more likely to have a lower PRS for TL, as were older individuals and men. Racial differences in TL were statistically accounted for when controlling for population structure using genetic principal components. The GWAS-derived SNPs for TL, however, may not have consistent associations with TL across different racial/ethnic groups.This study showed that associations of race/ethnicity with TL differed when accounting for population stratification. The role of race/ethnicity for TL remains uncertain, however, as the genetic determinants of TL may differ by race/ethnicity. Future GWAS samples should include racially diverse participants to allow for better characterization of the determinants of TL in human populations.
View details for DOI 10.1016/j.ebiom.2016.08.015
View details for PubMedID 27566956
- Reply to Yang et al.: GCTA produces unreliable heritability estimates. Proceedings of the National Academy of Sciences of the United States of America 2016; 113 (32): E4581-?
The effects of asymmetric competition on the life history of Trinidadian guppies
2016; 19 (3): 268-278
The effects of asymmetric interactions on population dynamics has been widely investigated, but there has been little work aimed at understanding how life history parameters like generation time, life expectancy and the variance in lifetime reproductive success are impacted by different types of competition. We develop a new framework for incorporating trait-mediated density-dependence into size-structured models and use Trinidadian guppies to show how different types of competitive interactions impact life history parameters. Our results show the degree of symmetry in competitive interactions can have dramatic effects on the speed of the life history. For some vital rates, shifting the competitive superiority from small to large individuals resulted in a doubling of the generation time. Such large influences of competitive symmetry on the timescale of demographic processes, and hence evolution, highlights the interwoven nature of ecological and evolutionary processes and the importance of density-dependence in understanding eco-evolutionary dynamics.
View details for DOI 10.1111/ele.12563
View details for Web of Science ID 000369986000006
View details for PubMedID 26843397
- Linking demographic responses and life history tactics from longitudinal data in mammals OIKOS 2016; 125 (3): 395-404
- Quantifying the influence of measured and unmeasured individual differences on demography JOURNAL OF ANIMAL ECOLOGY 2015; 84 (5): 1434-1445
Measuring selective constraint on fertility in human life histories
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
2015; 112 (29): 8982-8986
Human life histories combine late age at first reproduction, long reproductive span, relatively high fertility, and substantial postreproductive survival. However, even among the most fecund populations, human fertility falls far below its theoretical maximum. The extent of parental care required for successful offspring recruitment and widespread fertility decline under proper economic conditions suggest that selection on fertility is constrained by trade-offs with recruitment. Here we measure the trade-offs between life history traits under selection by approximating the slope of the selective constraint curve on two traits at the observed values. Using a selection of populations that span human demographic space, we find that the substitution elasticity of fertility for infant survival shows age-related patterns, with minimum substitution elasticities ranging from 14 to 22 for the four populations. The age of this minimum occurs earlier in the high-mortality populations relative to generation time than it does in the low-mortality populations. The human curves are qualitatively similar to one of two comparable nonhuman primate age-specific substitution elasticity curves. The curve for rhesus macaques has a similar shape but is shifted down, meaning that the threshold for switching from investing in survival to fertility is lower at all ages. The magnitude of the substitution elasticities is similar between chimpanzees and humans but the shape is quite different, rising more slowly for a longer fraction of the chimpanzee life cycle. The steeply rising substitution elasticities with age in humans has clear implications for the evolution of reproductive senescence.
View details for DOI 10.1073/pnas.1422037112
View details for Web of Science ID 000358225100062
View details for PubMedID 26150499
- Influence of Life-History Tactics on Transient Dynamics: A Comparative Analysis across Mammalian Populations AMERICAN NATURALIST 2014; 184 (5): 673-683
Influence of life-history tactics on transient dynamics: a comparative analysis across mammalian populations.
2014; 184 (5): 673-683
Most mammalian populations suffer from natural or human-induced disturbances; populations are no longer at the equilibrium (i.e., at stable [st]age distribution) and exhibit transient dynamics. From a literature survey, we studied patterns of transient dynamics for mammalian species spanning a large range of life-history tactics and population growth rates. For each population, we built an age-structured matrix and calculated six metrics of transient dynamics. After controlling for possible confounding effects of the phylogenetic relatedness among species using a phylogenetic principal component analysis and phylogenetic generalized least squares models, we found that short-term demographic responses of mammalian populations to disturbance are shaped by generation time and growth rate. Species with a slow pace of life (i.e., species with a late maturity, a low fecundity, and a long life span) displayed decreases in population size after a disturbance, whereas fast-living species increased in population size. The magnitude of short-term variation in population size increased with asymptotic population growth, being buffered in slow-growing species (i.e., species with a low population growth rate) but large in fast-growing species. By demonstrating direct links between transient dynamics, life history (generation time), and ecology (demographic regime), our comparative analysis of transient dynamics clearly improves our understanding of population dynamics in variable environments and has clear implications for future studies of the interplay between evolutionary and ecological dynamics. As most populations in the wild are not at equilibrium, we recommend that analyses of transient dynamics be performed when studying population dynamics in variable environments.
View details for DOI 10.1086/677929
View details for PubMedID 25325750
BIRTH-ORDER DIFFERENCES CAN DRIVE NATURAL SELECTION ON AGING
2014; 68 (3): 886-892
Senescence-the deterioration of survival and reproductive capacity with increasing age-is generally held to be an evolutionary consequence of the declining strength of natural selection with increasing age. The diversity in rates of aging observed in nature suggests that the rate at which age-specific selection weakens is determined by species-specific ecological factors. We propose that, in iteroparous species, relationships between parental age, offspring birth order, and environment may affect selection on senescence. Later-born siblings have, on average, older parents than do first borns. Offspring born to older parents may experience different environments in terms of family support or inherited resources, factors often mediated by competition from siblings. Thus, age-specific selection on parents may change if the environment produces birth-order related gradients in reproductive success. We use an age-and-stage structured population model to investigate the impact of sibling environmental inequality on the expected evolution of senescence. We show that accelerated senescence evolves when later-born siblings are likely to experience an environment detrimental to lifetime reproduction. In general, sibling inequality is likely to be of particular importance for the evolution of senescence in species such as humans, where family interactions and resource inheritance have important roles in determining lifetime reproduction.
View details for DOI 10.1111/evo.12319
View details for Web of Science ID 000332046700022
The Invisible Cliff: Abrupt Imposition of Malthusian Equilibrium in a Natural-Fertility, Agrarian Society
2014; 9 (1)
Analysis of a natural fertility agrarian society with a multi-variate model of population ecology isolates three distinct phases of population growth following settlement of a new habitat: (1) a sometimes lengthy copial phase of surplus food production and constant vital rates; (2) a brief transition phase in which food shortages rapidly cause increased mortality and lessened fertility; and (3) a Malthusian phase of indefinite length in which vital rates and quality of life are depressed, sometimes strikingly so. Copial phase duration declines with increases in the size of the founding group, maximum life expectancy and fertility; it increases with habitat area and yield per hectare; and, it is unaffected by the sensitivity of vital rates to hunger. Transition phase duration is unaffected by size of founding population and area of settlement; it declines with yield, life expectancy, fertility and the sensitivity of vital rates to hunger. We characterize the transition phase as the Malthusian transition interval (MTI), in order to highlight how little time populations generally have to adjust. Under food-limited density dependence, the copial phase passes quickly to an equilibrium of grim Malthusian constraints, in the manner of a runner dashing over an invisible cliff. The three-phase pattern diverges from widely held intuitions based on standard Lotka-Verhulst approaches to population regulation, with implications for the analysis of socio-cultural evolution, agricultural intensification, bioarchaeological interpretation of food stress in prehistoric societies, and state-level collapse.
View details for DOI 10.1371/journal.pone.0087541
View details for Web of Science ID 000330621900130
View details for PubMedID 24498131
- Defoliation and bark harvesting affect life-history traits of a tropical tree JOURNAL OF ECOLOGY 2013; 101 (6): 1563-1571
Contributions of Covariance: Decomposing the Components of Stochastic Population Growth in Cypripedium calceolus
2013; 181 (3): 410-420
Although correlations between vital rates can have important effects on evolution and demography, few studies have investigated their effects on population dynamics. Here, we extend life-table response experiments (LTREs) to variable environments, showing how to quantify contributions made by (1) mean vital rates, (2) variability driven by environmental fluctuations, (3) correlations implying demographic trade-offs and reflecting stage transition synchrony, and (4) elasticities reflecting local selection pressures. Applying our methods to the lady's slipper orchid Cypripedium calceolus, we found that mean rates accounted for 77.1% of all effects on the stochastic growth rate, variability accounted for 12.6%, elasticities accounted for 6.6%, and correlations accounted for 3.7%. Stochastic effects accounted for 17.6%, 15.3%, and 35.9% of the total in our three populations. Larger elasticities to transitions between dormancy states and stronger correlations between emergence and survival suggest that one population was under greater pressure to remain active while the other two showed survival payoffs for dormancy in poor years. Strong negative correlations between dormancy, emergence, and stasis balanced opposing contributions, resulting in near stationarity in two populations. These new methods provide an additional tool for researchers investigating stochastic population dynamics and should be useful for a broad range of applications in basic ecology and conservation biology.
View details for DOI 10.1086/669155
View details for Web of Science ID 000315927900012
View details for PubMedID 23448889
- Beyond the mean: sensitivities of the variance of population growth METHODS IN ECOLOGY AND EVOLUTION 2013; 4 (3): 290-298
- Structured Population Models: Introduction THEORETICAL POPULATION BIOLOGY 2012; 82 (4): 241-243
Linking the population growth rate and the age-at-death distribution
THEORETICAL POPULATION BIOLOGY
2012; 82 (4): 244-252
The population growth rate is linked to the distribution of age at death. We demonstrate that this link arises because both the birth and death rates depend on the variance of age-at-death. This bears the prospect to separate the influences of the age patterns of fertility and mortality on population growth rate. Here, we show how the age pattern of death affects population growth. Using this insight we derive a new approximation of the population growth rate that uses the first and second moments of the age-at-death distribution. We apply our new approximation to 46 mammalian life tables (including humans) and show that it is on par with the most prominent other approximations.
View details for DOI 10.1016/j.tpb.2012.09.003
View details for Web of Science ID 000311981400002
View details for PubMedID 23103877
- Editorial for the Special Issue: Biodemographic determinants of lifespan EXPERIMENTAL GERONTOLOGY 2012; 47 (10): 755-758
- Building and testing models of long-term agricultural intensification and population dynamics: A case study from the Leeward Kohala Field System, Hawai'i ECOLOGICAL MODELLING 2012; 227: 18-28
- Stochastic LTRE analysis of the effects of herbivory on the population dynamics of a perennial grassland herb OIKOS 2012; 121 (2): 211-218
Derivatives of the stochastic growth rate
THEORETICAL POPULATION BIOLOGY
2011; 80 (1): 1-15
We consider stochastic matrix models for population driven by random environments which form a Markov chain. The top Lyapunov exponent a, which describes the long-term growth rate, depends smoothly on the demographic parameters (represented as matrix entries) and on the parameters that define the stochastic matrix of the driving Markov chain. The derivatives of a-the "stochastic elasticities"-with respect to changes in the demographic parameters were derived by Tuljapurkar (1990). These results are here extended to a formula for the derivatives with respect to changes in the Markov chain driving the environments. We supplement these formulas with rigorous bounds on computational estimation errors, and with rigorous derivations of both the new and old formulas.
View details for DOI 10.1016/j.tpb.2011.03.004
View details for Web of Science ID 000292353700001
View details for PubMedID 21463645
- Variance in death and its implications for modeling and forecasting mortality DEMOGRAPHIC RESEARCH 2011; 24: 497-525
- Demography as the Human Story POPULATION AND DEVELOPMENT REVIEW 2011; 37 (1): 166-171
- Static and dynamic expression of life history traits in the northern fulmar Fulmarus glacialis OIKOS 2011; 120 (3): 369-380
Using evolutionary demography to link life history theory, quantitative genetics and population ecology
JOURNAL OF ANIMAL ECOLOGY
2010; 79 (6): 1226-1240
1. There is a growing number of empirical reports of environmental change simultaneously influencing population dynamics, life history and quantitative characters. We do not have a well-developed understanding of links between the dynamics of these quantities. 2. Insight into the joint dynamics of populations, quantitative characters and life history can be gained by deriving a model that allows the calculation of fundamental quantities that underpin population ecology, evolutionary biology and life history. The parameterization and analysis of such a model for a specific system can be used to predict how a population will respond to environmental change. 3. Age-stage-structured models can be constructed from character-demography associations that describe age-specific relationships between the character and: (i) survival; (ii) fertility; (iii) ontogenetic development of the character among survivors; and (iv) the distribution of reproductive allocation. 4. These models can be used to calculate a wide range of useful biological quantities including population growth and structure; terms in the Price equation including selection differentials; estimates of biometric heritabilities; and life history descriptors including generation time. We showcase the method through parameterization of a model using data from a well-studied population of Soay sheep Ovis aries. 5. Perturbation analysis is used to investigate how the quantities listed in summary point 4 change as each parameter in each character-demography function is altered. 6. A wide range of joint dynamics of life history, quantitative characters and population growth can be generated in response to changes in different character-demography associations; we argue this explains the diversity of observations on the consequences of environmental change from studies of free-living populations. 7. The approach we describe has the potential to explain within and between species patterns in quantitative characters, life history and population dynamics.
View details for DOI 10.1111/j.1365-2656.2010.01734.x
View details for Web of Science ID 000283074000010
View details for PubMedID 20704627
- A NEW WAY TO INTEGRATE SELECTION WHEN BOTH DEMOGRAPHY AND SELECTION GRADIENTS VARY OVER TIME INTERNATIONAL JOURNAL OF PLANT SCIENCES 2010; 171 (9): 945-959
Coupled dynamics of body mass and population growth in response to environmental change
2010; 466 (7305): 482-U5
Environmental change has altered the phenology, morphological traits and population dynamics of many species. However, the links underlying these joint responses remain largely unknown owing to a paucity of long-term data and the lack of an appropriate analytical framework. Here we investigate the link between phenotypic and demographic responses to environmental change using a new methodology and a long-term (1976-2008) data set from a hibernating mammal (the yellow-bellied marmot) inhabiting a dynamic subalpine habitat. We demonstrate how earlier emergence from hibernation and earlier weaning of young has led to a longer growing season and larger body masses before hibernation. The resulting shift in both the phenotype and the relationship between phenotype and fitness components led to a decline in adult mortality, which in turn triggered an abrupt increase in population size in recent years. Direct and trait-mediated effects of environmental change made comparable contributions to the observed marked increase in population growth. Our results help explain how a shift in phenology can cause simultaneous phenotypic and demographic changes, and highlight the need for a theory integrating ecological and evolutionary dynamics in stochastic environments.
View details for DOI 10.1038/nature09210
View details for Web of Science ID 000280141200034
View details for PubMedID 20651690
Plant populations track rather than buffer climate fluctuations
2010; 13 (6): 736-743
Climate change not only affects mean temperature and precipitation but also exacerbates temporal fluctuations in these conditions. However, we know relatively little about how species respond to such climate fluctuations, with respect to variation in vital rates (i.e. survival, growth and reproduction of individuals) and population fluctuations. We examine whether populations display evidence of buffering against environmental variation in one of two ways: (1) through negative covariances among vital rates, or (2) reduction of variation in those vital rates to which population growth is most sensitive. We analyse time series of demographic data for 40 plant species and show that there is no evidence for either of these mechanisms. In species in which there is evidence for vital rate covariation, positive covariances between reproduction and survival rates predominate, and tend to magnify the effect of variability. Increasing climate variability is therefore expected to increase population fluctuations and extinction risks.
View details for DOI 10.1111/j.1461-0248.2010.01470.x
View details for Web of Science ID 000277867100008
View details for PubMedID 20426793
Dynamic heterogeneity and life history variability in the kittiwake
JOURNAL OF ANIMAL ECOLOGY
2010; 79 (2): 436-444
1. Understanding the evolution of life histories requires an assessment of the process that generates variation in life histories. Within-population heterogeneity of life histories can be dynamically generated by stochastic variation of reproduction and survival or be generated by individual differences that are fixed at birth. 2. We show for the kittiwake that dynamic heterogeneity is a sufficient explanation of observed variation of life histories. 3. The total heterogeneity in life histories has a small contribution from reproductive stage dynamics and a large contribution from survival differences. We quantify the diversity in life histories by metrics computed from the generating stochastic process. 4. We show how dynamic heterogeneity can be used as a null model and also how it can lead to positive associations between reproduction and survival across the life span. 5. We believe our approach to identifying the nature of among-individual heterogeneity yields important insights into the forces that generate within-population variation of life-history traits. It provides an alternative to claims that fixed individual differences are a major determinant of heterogeneity in life histories.
View details for DOI 10.1111/j.1365-2656.2009.01653.x
View details for Web of Science ID 000274321200016
View details for PubMedID 20102422
- Demographic effects of extreme weather events on a short-lived calcareous grassland species: stochastic life table response experiments JOURNAL OF ECOLOGY 2010; 98 (2): 255-267
Environmental variance, population growth and evolution
JOURNAL OF ANIMAL ECOLOGY
2010; 79 (1): 1-3
Environmental fluctuations on time scales of one to tens of generations are increasingly recognized as important determinants of population dynamics and microevolution. Jonzén et al. in this issue analyse how the vital rates of red kangaroos depend on annual rainfall, and estimate the elasticities of stochastic growth rate to the means and variances of the vital rates, as well as to the mean and variance of rainfall. Their results demonstrate how ecological and evolutionary studies can benefit from including explicit environmental drivers when modelling populations, and from the use of mean and variance elasticities.
View details for DOI 10.1111/j.1365-2656.2009.01619.x
View details for Web of Science ID 000272656600001
View details for PubMedID 20409157
Population and prehistory III: Food-dependent demography in variable environments
THEORETICAL POPULATION BIOLOGY
2009; 76 (3): 179-188
The population dynamics of preindustrial societies depend intimately on their surroundings, and food is a primary means through which environment influences population size and individual well-being. Food production requires labor; thus, dependence of survival and fertility on food involves dependence of a population's future on its current state. We use a perturbation approach to analyze the effects of random environmental variation on this nonlinear, age-structured system. We show that in expanding populations, direct environmental effects dominate induced population fluctuations, so environmental variability has little effect on mean hunger levels, although it does decrease population growth. The growth rate determines the time until population is limited by space. This limitation introduces a tradeoff between population density and well-being, so population effects become more important than the direct effects of the environment: environmental fluctuation increases mortality, releasing density dependence and raising average well-being for survivors. We discuss the social implications of these findings for the long-term fate of populations as they transition from expansion into limitation, given that conditions leading to high well-being during growth depress well-being during limitation.
View details for DOI 10.1016/j.tpd.2009.06.003
View details for Web of Science ID 000271283500003
View details for PubMedID 19540865
A time to grow and a time to die: a new way to analyze the dynamics of size, light, age, and death of tropical trees
2009; 90 (10): 2766-2778
In tropical rain forests, rates of forest turnover and tree species' life-history differences are shaped by the life expectancy of trees and the time taken by seedlings to reach the canopy. These measures are therefore of both theoretical and applied interest. However, the relationship between size, age, and life expectancy is poorly understood. In this paper, we show how to obtain, in a dynamic environment, age-related population parameters from data on size and light transitions and survival of individuals over single time steps. We accomplish this goal by combining two types of analysis (integral projection modeling and age-from-stage analysis for variable environments) in a new way. The method uses an index of crown illumination (CI) to capture the key tree life-history axis of movement through the light environment. We use this method to analyze data on nine tropical tree species, chosen to sample two main gradients, juvenile recruitment niche (gap/nongap) and adult crown position niche (subcanopy, canopy-emergent). We validate the method using independent estimates of age and size from growth rings and 14C from some of the same species at the same site and use our results to examine correlations among age-related population parameters. Finally, we discuss the implications of these new results for life histories of tropical trees.
View details for Web of Science ID 000270274200013
View details for PubMedID 19886486
- DEMOGRAPHY Babies make a comeback NATURE 2009; 460 (7256): 693-694
Estimating stochastic elasticities directly from longitudinal data
2009; 12 (8): 806-812
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.
View details for DOI 10.1111/j.1461-0248.2009.01330.x
View details for Web of Science ID 000267660600009
View details for PubMedID 19552649
The Dynamics of Phenotypic Change and the Shrinking Sheep of St. Kilda
2009; 325 (5939): 464-467
Environmental change, including climate change, can cause rapid phenotypic change via both ecological and evolutionary processes. Because ecological and evolutionary dynamics are intimately linked, a major challenge is to identify their relative roles. We exactly decomposed the change in mean body weight in a free-living population of Soay sheep into all the processes that contribute to change. Ecological processes contribute most, with selection--the underpinning of adaptive evolution--explaining little of the observed phenotypic trend. Our results enable us to explain why selection has so little effect even though weight is heritable, and why environmental change has caused a decline in the body size of Soay sheep.
View details for DOI 10.1126/science.1173668
View details for Web of Science ID 000268255100053
View details for PubMedID 19574350
From stochastic environments to life histories and back
PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY B-BIOLOGICAL SCIENCES
2009; 364 (1523): 1499-1509
Environmental stochasticity is known to play an important role in life-history evolution, but most general theory assumes a constant environment. In this paper, we examine life-history evolution in a variable environment, by decomposing average individual fitness (measured by the long-run stochastic growth rate) into contributions from average vital rates and their temporal variation. We examine how generation time, demographic dispersion (measured by the dispersion of reproductive events across the lifespan), demographic resilience (measured by damping time), within-year variances in vital rates, within-year correlations between vital rates and between-year correlations in vital rates combine to determine average individual fitness of stylized life histories. In a fluctuating environment, we show that there is often a range of cohort generation times at which the fitness is at a maximum. Thus, we expect 'optimal' phenotypes in fluctuating environments to differ from optimal phenotypes in constant environments. We show that stochastic growth rates are strongly affected by demographic dispersion, even when deterministic growth rates are not, and that demographic dispersion also determines the response of life-history-specific average fitness to within- and between-year correlations. Serial correlations can have a strong effect on fitness, and, depending on the structure of the life history, may act to increase or decrease fitness. The approach we outline takes a useful first step in developing general life-history theory for non-constant environments.
View details for DOI 10.1098/rstb.2009.0021
View details for Web of Science ID 000265732200003
View details for PubMedID 19414465
Hierarchical demography: a general approach with an application to honey bees
2009; 90 (2): 556-566
Hierarchical population structure, where individuals are aggregated into colonies or similar groups that themselves grow, survive or perish, and potentially produce offspring groups, is an important feature of many biological systems, most notably eusocial organisms such as the honey bee, Apis mellifera. Despite this hierarchical structure, there is a paucity of analytical models and theory linking the dynamics of individuals within colonies to the dynamics of a population of colonies. We present an analytical framework that provides a simple, robust, and predictive theory for the population dynamics of hierarchical organisms. Our framework explicitly describes and links demographic dynamics for the different levels in the hierarchy (individuals, groups, population). We illustrate the application of the framework by developing a model for honey bees and analyzing the effects of life history traits such as worker life span and size at swarming on the growth rate of populations. We conclude by discussing possible extensions of the model that increase its realism and expand its usefulness beyond swarm-founding, monogynous, eusocial insects.
View details for Web of Science ID 000263570800030
View details for PubMedID 19323239
Dynamic heterogeneity in life histories
2009; 12 (1): 93-106
Longitudinal data on natural populations have been analysed using multistage models in which survival depends on reproductive stage, and individuals change stages according to a Markov chain. These models are special cases of stage-structured population models. We show that stage-structured models generate dynamic heterogeneity: life-history differences produced by stochastic stratum dynamics. We characterize dynamic heterogeneity in a range of species across taxa by properties of the Markov chain: the entropy, which describes the extent of heterogeneity, and the subdominant eigenvalue, which describes the persistence of reproductive success during the life of an individual. Trajectories of reproductive stage determine survivorship, and we analyse the variance in lifespan within and between trajectories of reproductive stage. We show how stage-structured models can be used to predict realized distributions of lifetime reproductive success. Dynamic heterogeneity contrasts with fixed heterogeneity: unobserved differences that generate variation between life histories. We show by an example that observed distributions of lifetime reproductive success are often consistent with the claim that little or no fixed heterogeneity influences this trait. We propose that dynamic heterogeneity provides a 'neutral' model for assessing the possible role of unobserved 'quality' differences between individuals. We discuss fitness for dynamic life histories, and the implications of dynamic heterogeneity for the evolution of life histories and senescence.
View details for DOI 10.1111/j.1461-0248.2008.01262.x
View details for Web of Science ID 000261625500011
View details for PubMedID 19016825
Evolution of Delayed Reproduction in Uncertain Environments: A Life-History Perspective
2008; 172 (6): 797-805
Environmental uncertainty alone can select for delayed reproduction; however, its relative role in the evolution of delayed reproduction across life histories is not known. Along a life-history spectrum from low-survival/high-fertility species to high-survival/low-fertility species, we show that the latter are more likely to evolve delayed reproduction if fertility varies over time. By contrast, if survival varies over time, low-survival life histories are more likely to evolve delays. If there is variation in both survival and fertility, and if this variation is positively associated, the evolutionarily stable reproductive delay is decreased (relative to independent variation in survival and fertility). Conversely, if variation in survival and fertility is negatively associated, the evolutionarily stable reproductive delay is increased. We further show that environmental uncertainty can drive the evolution of delayed reproduction in an iteroparous organism but only in the special case where juvenile survival is greater than adult survival. For common iteroparous life histories (adult survival > juvenile survival), environmental uncertainty does not select for delayed reproduction. Thus, any benefits that delayed reproduction might have on reproduction or survival could be especially important in explaining the common observation of delayed reproduction in many vertebrates and perennial plants.
View details for DOI 10.1086/592867
View details for Web of Science ID 000261235500009
View details for PubMedID 18959491
The Dynamics of a Quantitative Trait in an Age-Structured Population Living in a Variable Environment
2008; 172 (5): 599-612
Time series of rapid phenotypic change have been documented in age-structured populations living in the wild. Researchers are often interested in identifying the processes responsible for such change. We derive an equation to exactly decompose change in the mean value of a phenotypic trait into contributions from fluctuations in the demographic structure and age-specific viability selection, fertility selection, phenotypic plasticity, and differences between offspring and parental trait values. We treat fitness as a sum of its components rather than as a scalar and explicitly consider age structure by focusing on short time steps, which are appropriate for describing phenotypic change in species with overlapping generations. We apply the method to examine stasis in birth weight in a well-characterized population of red deer. Stasis is achieved because positive viability selection for an increase in birth weight is countered by parents producing offspring that are, on average, smaller than they were at birth. This is one of many ways in which equilibria in the mean value of a phenotypic trait can be maintained. The age-structured Price equation we derive has the potential to provide considerable insight into the processes generating now frequently reported cases of rapid phenotypic change.
View details for DOI 10.1086/591693
View details for Web of Science ID 000260186000001
View details for PubMedID 18840061
How can economic schemes curtail the increasing sex ratio at birth in China?
2008; 19: 1831-1850
Fertility decline, driven by the one-child policy, and son preference have contributed to an alarming difference in the number of live male and female births in China. We present a quantitative model where people choose to sex-select because they perceive that married sons are more valuable than married daughters. Due to the predominant patrilocal kinship system in China, daughters-in-law provide valuable emotional and financial support, enhancing the perceived present value of married sons. We argue that inter-generational transfer data will help ascertain the extent to which economic schemes (such as pension plans for families with no sons) can curtail the increasing sex ratio at birth.
View details for Web of Science ID 000259977900001
View details for PubMedID 21113272
Population and prehistory II: Space-limited human populations in constant environments
THEORETICAL POPULATION BIOLOGY
2008; 74 (2): 147-160
We present a population model to examine the forces that determined the quality and quantity of human life in early agricultural societies where cultivable area is limited. The model is driven by the non-linear and interdependent relationships between the age distribution of a population, its behavior and technology, and the nature of its environment. The common currency in the model is the production of food, on which age-specific rates of birth and death depend. There is a single non-trivial equilibrium population at which productivity balances caloric needs. One of the most powerful controls on equilibrium hunger level is fertility control. Gains against hunger are accompanied by decreases in population size. Increasing worker productivity does increase equilibrium population size but does not improve welfare at equilibrium. As a case study we apply the model to the population of a Polynesian valley before European contact.
View details for DOI 10.1016/j.tpb.2008.05.007
View details for Web of Science ID 000259165700001
View details for PubMedID 18598711
Stage dynamics, period survival, and mortality plateaus
2008; 172 (2): 203-215
Mortality plateaus at advanced ages have been found in many species, but their biological causes remain unclear. Here, we exploit age-from-stage methods for organisms with stage-structured demography to study cohort dynamics, obtaining age patterns of mortality by weighting one-period stage-specific survivals by expected age-specific stage structure. Cohort dynamics behave as a killed Markov process. Using as examples two African grasses, one pine tree, a temperate forest perennial herb, and a subtropical shrub in a hurricane-driven forest, we illustrate diverse patterns that may emerge. Age-specific mortality always reaches a plateau at advanced ages, but the plateau may be reached rapidly or slowly, and the trajectory may follow positive or negative senescence along the way. In variable environments, birth state influences mortality at early but not late ages, although its effect on the level of survivorship persists. A new parameter micro omega summarizes the risk of mortality averaged over the entire lifetime in a variable environment. Recent aging models for humans that employ nonobservable abstract states of "vitality" are also known to produce diverse trajectories and similar asymptotic behavior. We discuss connections, contrasts, and implications of our results to these models for the study of aging.
View details for DOI 10.1086/589453
View details for Web of Science ID 000257986200008
View details for PubMedID 18616387
Senescence rates are determined by ranking on the fast-slow life-history continuum
2008; 11 (7): 664-673
Comparative analyses of survival senescence by using life tables have identified generalizations including the observation that mammals senesce faster than similar-sized birds. These generalizations have been challenged because of limitations of life-table approaches and the growing appreciation that senescence is more than an increasing probability of death. Without using life tables, we examine senescence rates in annual individual fitness using 20 individual-based data sets of terrestrial vertebrates with contrasting life histories and body size. We find that senescence is widespread in the wild and equally likely to occur in survival and reproduction. Additionally, mammals senesce faster than birds because they have a faster life history for a given body size. By allowing us to disentangle the effects of two major fitness components our methods allow an assessment of the robustness of the prevalent life-table approach. Focusing on one aspect of life history - survival or recruitment - can provide reliable information on overall senescence.
View details for DOI 10.1111/j.1461-0248.2008.01187.x
View details for Web of Science ID 000256376900002
View details for PubMedID 18445028
Population and prehistory I: Food-dependent population growth in constant environments
THEORETICAL POPULATION BIOLOGY
2008; 73 (4): 473-482
We present a demographic model that describes the feedbacks between food supply, human mortality and fertility rates, and labor availability in expanding populations, where arable land area is not limiting. This model provides a quantitative framework to describe how environment, technology, and culture interact to influence the fates of preindustrial agricultural populations. We present equilibrium conditions and derive approximations for the equilibrium population growth rate, food availability, and other food-dependent measures of population well-being. We examine how the approximations respond to environmental changes and to human choices, and find that the impact of environmental quality depends upon whether it manifests through agricultural yield or maximum (food-independent) survival rates. Human choices can complement or offset environmental effects: greater labor investments increase both population growth and well-being, and therefore can counteract lower agricultural yield, while fertility control decreases the growth rate but can increase or decrease well-being. Finally we establish equilibrium stability criteria, and argue that the potential for loss of local stability at low population growth rates could have important consequences for populations that suffer significant environmental or demographic shocks.
View details for DOI 10.1016/j.tpb.2008.03.001
View details for Web of Science ID 000256996000002
View details for PubMedID 18439637
Longevity can buffer plant and animal populations against changing climatic variability
2008; 89 (1): 19-25
Both means and year-to-year variances of climate variables such as temperature and precipitation are predicted to change. However, the potential impact of changing climatic variability on the fate of populations has been largely unexamined. We analyzed multiyear demographic data for 36 plant and animal species with a broad range of life histories and types of environment to ask how sensitive their long-term stochastic population growth rates are likely to be to changes in the means and standard deviations of vital rates (survival, reproduction, growth) in response to changing climate. We quantified responsiveness using elasticities of the long-term population growth rate predicted by stochastic projection matrix models. Short-lived species (insects and annual plants and algae) are predicted to be more strongly (and negatively) affected by increasing vital rate variability relative to longer-lived species (perennial plants, birds, ungulates). Taxonomic affiliation has little power to explain sensitivity to increasing variability once longevity has been taken into account. Our results highlight the potential vulnerability of short-lived species to an increasingly variable climate, but also suggest that problems associated with short-lived undesirable species (agricultural pests, disease vectors, invasive weedy plants) may be exacerbated in regions where climate variability decreases.
View details for Web of Science ID 000253717200003
View details for PubMedID 18376542
Time, transients and elasticity
2007; 10 (12): 1143-1153
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.
View details for DOI 10.1111/j.1461-0248.2007.01108.x
View details for Web of Science ID 000250700800004
View details for PubMedID 17883410
- Detecting variability in demographic rates: randomization with the Kullback-Leibler distance JOURNAL OF ECOLOGY 2007; 95 (6): 1370-1380
Why Men Matter: Mating Patterns Drive Evolution of Human Lifespan
2007; 2 (8)
Evolutionary theory predicts that senescence, a decline in survival rates with age, is the consequence of stronger selection on alleles that affect fertility or mortality earlier rather than later in life. Hamilton quantified this argument by showing that a rare mutation reducing survival is opposed by a selective force that declines with age over reproductive life. He used a female-only demographic model, predicting that female menopause at age ca. 50 yrs should be followed by a sharp increase in mortality, a "wall of death." Human lives obviously do not display such a wall. Explanations of the evolution of lifespan beyond the age of female menopause have proven difficult to describe as explicit genetic models. Here we argue that the inclusion of males and mating patterns extends Hamilton's theory and predicts the pattern of human senescence. We analyze a general two-sex model to show that selection favors survival for as long as men reproduce. Male fertility can only result from matings with fertile females, and we present a range of data showing that males much older than 50 yrs have substantial realized fertility through matings with younger females, a pattern that was likely typical among early humans. Thus old-age male fertility provides a selective force against autosomal deleterious mutations at ages far past female menopause with no sharp upper age limit, eliminating the wall of death. Our findings illustrate the evolutionary importance of males and mating preferences, and show that one-sex demographic models are insufficient to describe the forces that shape human senescence.
View details for DOI 10.1371/journal.pone.0000785
View details for Web of Science ID 000207455400002
View details for PubMedID 17726515
The evolutionary demography of ecological change: Linking trait variation and population growth
2007; 315 (5818): 1571-1574
Population dynamics and evolutionary change are linked by the fundamental biological processes of birth and death. This means that population growth may correlate with the strength of selection, whereas evolutionary change can leave an ecological signature. We decompose population growth in an age-structured population into contributions from variation in a quantitative trait. We report that the distribution of body sizes within a population of Soay sheep can markedly influence population dynamics, accounting for up to one-fifth of observed population growth. Our results suggest that there is substantial opportunity for evolutionary dynamics to leave an ecological signature and visa versa.
View details for DOI 10.1126/science.1139024
View details for Web of Science ID 000244934800051
View details for PubMedID 17363672
Sensitivity of the population growth rate to demographic variability within and between phases of the disturbance cycle
2006; 9 (12): 1331-1341
For species in disturbance-prone ecosystems, vital rates (survival, growth and reproduction) often vary both between and within phases of the cycle of disturbance and recovery; some of this variation is imposed by the environment, but some may represent adaptation of the life history to disturbance. Anthropogenic changes may amplify or impede these patterns of variation, and may have positive or negative effects on population growth. Using stochastic population projection matrix models, we develop stochastic elasticities (proportional derivatives of the long-run population growth rate) to gauge the population effects of three types of change in demographic variability (changes in within- and between-disturbance-phase variability and phase-specific changes). Computing these elasticities for five species of disturbance-influenced perennial plants, we pinpoint demographic rates that may reveal adaptation to disturbance, and we demonstrate that species may differ in their responses to different types of changes in demographic variability driven by climate change.
View details for DOI 10.1111/j.1461-0248.2006.00988.x
View details for Web of Science ID 000242196900007
View details for PubMedID 17118007
- Risky business: Temporal and spatial variation in preindustrial dryland agriculture HUMAN ECOLOGY 2006; 34 (6): 739-763
From stage to age in variable environments: Life expectancy and survivorship
2006; 87 (6): 1497-1509
Stage-based demographic data are now available on many species of plants and some animals, and they often display temporal and spatial variability. We provide exact formulas to compute age-specific life expectancy and survivorship from stage-based data for three models of temporal variability: cycles, serially independent random variation, and a Markov chain. These models provide a comprehensive description of patterns of temporal variation. Our formulas describe the effects of cohort (birth) environmental condition on mortality at all ages, and of the effects on survivorship of environmental variability experienced over the course of life. This paper complements existing methods for time-invariant stage-based data, and adds to the information on population growth and dynamics available from stochastic demography.
View details for Web of Science ID 000238658400019
View details for PubMedID 16869426
- Temporal autocorrelation and stochastic population growth ECOLOGY LETTERS 2006; 9 (3): 324-334
Temporal autocorrelation and stochastic population growth.
2006; 9 (3): 327-337
How much does environmental autocorrelation matter to the growth of structured populations in real life contexts? Interannual variances in vital rates certainly do, but it has been suggested that between-year correlations may not. We present an analytical approximation to stochastic growth rate for multistate Markovian environments and show that it is accurate by testing it in two empirically based examples. We find that temporal autocorrelation has sizeable effect on growth rates of structured populations, larger in many cases than the effect of interannual variability. Our approximation defines a sensitivity to autocorrelated variability, showing how demographic damping and environmental pattern interact to determine a population's stochastic growth rate.
View details for PubMedID 16958899
Plant-animal interactions in random environments: Habitat-stage elasticity, seed predators, and hurricanes
2005; 86 (12): 3312-3322
View details for Web of Science ID 000234066700023
Elasticities in variable environments: Properties and implications
2005; 166 (4): 481-495
Elasticities in stochastic matrix models are used to understand both population and evolutionary dynamics. We examine three such elasticities: stochastic elasticity E(ij)(S) with respect to the (i, j) matrix element, the elasticity E(ij)(S mu) with respect to the mean mu(ij) of the matrix element, and the elasticity E(ij)(S sigma) with respect to the variability sigma(ij) of the matrix element. We show that the stochastic elasticity E(S) does not accurately describe the effect of variability; one should use E(S sigma) and E(S mu). We establish two general properties of these elasticities: a sum rule that connects them and a limit on the sum of the E(S sigma). We discuss the implications of these properties for the analysis of buffering and selection on the average rates versus the variability of rates.
View details for Web of Science ID 000232270600008
View details for PubMedID 16224704
Future mortality: a bumpy road to Shangri-La?
Science of aging knowledge environment : SAGE KE
2005; 2005 (14): pe9-?
Americans are getting fatter, and it is known that increased obesity may increase the risk of death. Olshansky et al. have argued that this increase in obesity will likely slow, or even reverse, increases in life expectancy in the United States and perhaps save U.S. Social Security as a result. We discuss historical changes in the mortality rate and the reasons why other analyses argue that life expectancies will continue to increase. We also discuss the limitations of using single risk factors such as obesity as predictors of mortality risk. Finally, we explore the relation between risk factors and the long-term historical increase in human life expectancy.
View details for PubMedID 15814821
A FORMAL MODEL OF AGE-STRUCTURAL TRANSITIONS
POPULATION, RESOURCES AND DEVELOPMENT
2005; 1: 91-105
View details for Web of Science ID 000271189300005
Random scenario forecasts versus stochastic forecasts
INTERNATIONAL STATISTICAL REVIEW
2004; 72 (2): 185-199
View details for Web of Science ID 000222970200003
Soils, agriculture, and society in precontact Hawai
2004; 304 (5677): 1665-1669
Before European contact, Hawai'i supported large human populations in complex societies that were based on multiple pathways of intensive agriculture. We show that soils within a long-abandoned 60-square-kilometer dryland agricultural complex are substantially richer in bases and phosphorus than are those just outside it, and that this enrichment predated the establishment of intensive agriculture. Climate and soil fertility combined to constrain large dryland agricultural systems and the societies they supported to well-defined portions of just the younger islands within the Hawaiian archipelago; societies on the older islands were based on irrigated wetland agriculture. Similar processes may have influenced the dynamics of agricultural intensification across the tropics.
View details for Web of Science ID 000221934300049
View details for PubMedID 15192228
- Demography in the 21st century: Introduction THEORETICAL POPULATION BIOLOGY 2004; 65 (4): 317-317
Using the Lee-Carter method to forecast mortality for populations with limited data
INTERNATIONAL STATISTICAL REVIEW
2004; 72 (1): 19-36
View details for Web of Science ID 000222159200003
The many growth rates and elasticities of populations in random environments
2003; 162 (4): 489-502
Despite considerable interest in the dynamics of populations subject to temporally varying environments, alternate population growth rates and their sensitivities remain incompletely understood. For a Markovian environment, we compare and contrast the meanings of the stochastic growth rate (lambdaS), the growth rate of average population (lambdaM), the growth rate for average transition rates (lambdaA), and the growth rate of an aggregate represented by a megamatrix (shown here to equal lambdaM). We distinguish these growth rates by the averages that define them. We illustrate our results using data on an understory shrub in a hurricane-disturbed landscape, employing a range of hurricane frequencies. We demonstrate important differences among growth rates: lambdaS
lambdaM. We show that stochastic elasticity, ESij, and megamatrix elasticity, EMij, describe a complex perturbation of both means and variances of rates by the same proportion. Megamatrix elasticities respond slightly and stochastic elasticities respond strongly to changing the frequency of disturbance in the habitat (in our example, the frequency of hurricanes). The elasticity EAij of lambdaA does not predict changes in the other elasticities. Because ES, although commonly utilized, is difficult to interpret, we introduce elasticities with a more direct interpretation: ESmu for perturbations of means and ESsigma for variances. We argue that a fundamental tool for studying selection pressures in varying environments is the response of growth rate to vital rates in all habitat states.
View details for Web of Science ID 000186172300009
View details for PubMedID 14582010
Reproductive effort in variable environments, or environmental variation is for the birds
2001; 82 (9): 2659-2665
View details for Web of Science ID 000171049100023
Escape in time: stay young or age gracefully?
2000; 133 (1-2): 143-159
View details for Web of Science ID 000089318800014
A universal pattern of mortality decline in the G7 countries
2000; 405 (6788): 789-792
Human lifespan has increased enormously this century. But we remain uncertain about the forces that reduce mortality, and about the cost implications of ageing populations and their associated social burden. The poor understanding of the factors driving mortality decline, and the difficulty of forecasting mortality are due in part to the pronounced irregularity of annual to decadal mortality change. Here we examine mortality over five decades in the G7 countries (Canada, France, Germany, Italy, Japan, UK, US). In every country over this period, mortality at each age has declined exponentially at a roughly constant rate. This trend places a constraint on any theory of society-driven mortality decline, and provides a basis for stochastic mortality forecasting. We find that median forecasts of life expectancy are substantially larger than in existing official forecasts. In terms of the costs of ageing, we forecast values of the dependency ratio (that is, the ratio of people over 65 to working people) in 2050 that are between 6% (UK) and 40% (Japan) higher than official forecasts.
View details for Web of Science ID 000087620600050
View details for PubMedID 10866199
Validation, probability-weighted priors, and information in stochastic forecasts
INTERNATIONAL JOURNAL OF FORECASTING
1999; 15 (3): 259-271
View details for Web of Science ID 000081062600003
Population momentum for gradual demographic transitions
POPULATION STUDIES-A JOURNAL OF DEMOGRAPHY
1999; 53 (2): 255-262
View details for Web of Science ID 000081475100010
Uncertain demographic futures and social security finances
AMERICAN ECONOMIC REVIEW
1998; 88 (2): 237-241
View details for Web of Science ID 000073658200045
Stochastic forecasts for social security
FRONTIERS IN THE ECONOMICS OF AGING
View details for Web of Science ID 000078443200018
Demographic uncertainty and the stable equivalent population
MATHEMATICAL AND COMPUTER MODELLING
1997; 26 (6): 39-56
View details for Web of Science ID A1997YG11800006
- Demography - Taking the measure of uncertainty NATURE 1997; 387 (6635): 760-761
Death and taxes: Longer life, consumption, and social security
1997; 34 (1): 67-81
We analyze in three steps the influence of the projected mortality decline on the long run finances of the Social Security System. First, on a theoretical level, mortality decline adds person years of life which are distributed across the life cycle. The interaction of this distribution with the age distribution of labor earnings minus consumption, or of taxes minus benefits, partially determines the corresponding steady state financial consequences of mortality decline. The effect of mortality decline on population growth rates also matters, but is negligible in low mortality populations. Second, examination of past mortality trends in the United States and of international trends in low mortality populations, suggests that mortality will decline faster than foreseen by the Social Security Administration's forecasts. Third, we combine the work of the first two parts in dynamic simulations to examine the implications of mortality decline and of alternative forecasts of mortality for the finances of the social security system. Also, we use stochastic population forecasts to assess the influence of uncertainty about mortality decline on uncertainty about finances; we find that uncertainty about fertility still has more important implications than uncertainty about mortality, contrary to sensitivity tests in the official forecasts.
View details for Web of Science ID A1997WP31400005
View details for PubMedID 9074832
1995; 269 (5221): 148-?
View details for PubMedID 17789831
Disaggregation in population forecasting: do we need it? And how to do it simply.
Mathematical population studies
1995; 5 (3): 217-?
"We have described a method for reducing the dimensionality of the forecasting problem by parsimoniously modeling the evolution over time of the age schedules of vital rates. This method steers a middle course between forecasting aggregates and forecasting individual age specific rates: we reduce the problem to forecasting a single parameter for fertility and another one for mortality. We have described a number of refinements and extensions of those basic methods, which preserve their underlying structure and simplicity. In particular, we show how one can fit the model more simply, incorporate lower bounds to the forecasts of rates, disaggregate by sex or race, and prepare integrated forecasts of rates for a collection of regions. We also discuss alternate approaches to forecasting the estimated indices of fertility and mortality, including state-space methods. These many versions of the basic method have yielded remarkably similar results." (SUMMARY IN FRE)
View details for PubMedID 12290947
HIGH SEX-RATIOS IN CHINA FUTURE
1995; 267 (5199): 874-876
In China in recent years, male live births have exceeded those of females by amounts far greater than those that occur naturally in human populations, a trend with significant demographic consequences. The resulting imbalance in the first-marriage market is estimated to be about 1 million males per year after 2010. These "excess" males were not easily accommodated in models with substantial changes in first-marriage patterns. The current sex ratio at birth has little effect on a couple's probability of having at least one son, so future increases in the sex ratio may well occur, especially given increasing access to sex-selective abortion.
View details for Web of Science ID A1995QG20700055
View details for PubMedID 7846529
High sex ratio at birth and its consequences.
Chinese journal of population science
1995; 7 (3): 213-221
View details for PubMedID 12290857
Population growth changes targets for immunization.
1995; 23 (1): 5-?
View details for PubMedID 12319060
LOOP ANALYSIS - EVALUATING LIFE-HISTORY PATHWAYS IN POPULATION PROJECTION MATRICES
1994; 75 (8): 2410-2415
View details for Web of Science ID A1994PW55400021
Stochastic population forecasts for the United States: beyond high, medium, and low.
Journal of the American Statistical Association
1994; 89 (428): 1-?
"This article presents and implements a new method for making stochastic population forecasts that provide consistent probability intervals. We blend mathematical demography and statistical time series methods to estimate stochastic models of fertility and mortality based on U.S. data back to 1900 and then use the theory of random-matrix products to forecast various demographic measures and their associated probability intervals to the year 2065. Our expected total population sizes agree quite closely with the Census medium projections, and our 95 percent probability intervals are close to the Census high and low scenarios. But Census intervals in 2065 for ages 65+ are nearly three times as broad as ours, and for 85+ are nearly twice as broad. In contrast, our intervals for the total dependency and youth dependency ratios are more than twice as broad as theirs, and our ratio for the elderly dependency ratio is 12 times as great as theirs. These items have major implications for policy, and these contrasting indications of uncertainty clearly show the limitations of the conventional scenario-based methods."
View details for PubMedID 12155397
NONLINEAR FEEDBACK DYNAMICS IN FISHERIES - ANALYSIS OF THE DERISO-SCHNUTE MODEL
CANADIAN JOURNAL OF FISHERIES AND AQUATIC SCIENCES
1994; 51 (7): 1462-1473
View details for Web of Science ID A1994PP18900001
MIGRATION IN VARIABLE ENVIRONMENTS - EXPLORING LIFE-HISTORY EVOLUTION USING STRUCTURED POPULATION-MODELS
JOURNAL OF THEORETICAL BIOLOGY
1994; 166 (1): 75-90
In this paper, we explore the hypothesis that environmental variability favors the evolution of migration. Using the single-locus invasion condition for a novel allele in a variable environment, we derive conditions where increased migration rates between two sites are favored. We find that while there is a strong advantage to migrants entering a resident population with no migration, there is little advantage to migrants entering a population where the residents migrate at a different rate. Instead of an optimal rate of migration, there is a range of favored migration rates. Negative spatial correlation and a population structure including more than two sites accentuate the advantage of migration. Extending this model to include the effects of developmental delay (e.g. seed dormancy or diapause) on the evolution of migration, we find that higher levels of such delay reduce the advantage to migrants.
View details for Web of Science ID A1994MV13000006
View details for PubMedID 8145562
ENVIRONMENTAL UNCERTAINTY AND VARIABLE DIAPAUSE
THEORETICAL POPULATION BIOLOGY
1993; 43 (3): 251-280
We analyze a stage-structured model of a population that displays variable diapause in a randomly varying environment. The ruggedness of the environment is measured by the extent of random variation in per-capita reproductive success. We show how variable diapause and environmental characteristics affect the population's stochastic growth rate. In rugged unpredictable environments, phenotypes that show some tendency to diapause are found to have a higher growth rate than nondiapausing phenotypes. In harsh rugged environments, some tendency to diapause may be all that permits population persistence. Positive serial autocorrelation causes the optimal diapause fraction to decrease, while negative autocorrelation causes that fraction to increase. The structured model behaves very differently from a scalar model for large diapause fractions even in uncorrelated environments, and in many cases predicts a broad optimum. The difference between models is due to the extreme variability of stage structure in populations subject to even small variability when diapause tendency is high.
View details for Web of Science ID A1993LF75300001
View details for PubMedID 8327984
ENTROPY AND CONVERGENCE IN DYNAMICS AND DEMOGRAPHY
JOURNAL OF MATHEMATICAL BIOLOGY
1993; 31 (3): 253-271
Demographic dynamics is formally equivalent to the dynamics of a Markov chain, as is true of some nonlinear dynamical systems. Convergence to demographic equilibrium can be studied in terms of convergence in the Markov chain. Tuljapurkar (1982) showed that population entropy (Kolmogorov-Sinai entropy) provides information on the rate of this convergence. This paper begins by considering finite state Markov chains, providing elementary proofs of the relationship between convergence rate and entropy, and discusses in detail the uses and limitations of entropy as a convergence measure; these results also apply to Markovian dynamical systems. Next, new qualitative and quantitative arguments are used to discuss the demographic meaning of entropy. An exact relationship is established giving population entropy in terms of the eigenvalues of the Leslie matrix characteristic equation. Finally, the significance of imprimitive and periodic limits is discussed in relation to population entropy.
View details for Web of Science ID A1993KN16300003
View details for PubMedID 8468536
STOCHASTIC POPULATION FORECASTS AND THEIR USES
INTERNATIONAL JOURNAL OF FORECASTING
1992; 8 (3): 385-391
"The properties and uses of stochastic forecasts are discussed here. For linear stochastic projections, we show how the computation of forecast moments and the statistical distribution of forecasts depend on the multiplicative and autoregressive structure of the dynamics. Both scalar and vector projection methods are discussed, and their similarities are explored. Next we discuss the uses of stochastic forecasts, arguing that it is important to relate forecasts to the specific decision-making criteria of particular forecast users. The example of [the U.S. system of] Social Security is used to show how a dynamic programming approach may be used to explore alternative decisions in a probabilistic context."
View details for Web of Science ID A1992KE72100007
View details for PubMedID 12157865
DISEASE IN CHANGING POPULATIONS - GROWTH AND DISEQUILIBRIUM
THEORETICAL POPULATION BIOLOGY
1991; 40 (3): 322-353
This paper examines simple age-structured models of childhood disease epidemiology, focusing on nonstationary populations which characterize LDCs. An age-structured model of childhood disease epidemiology for nonstationary populations is formulated which incorporates explicit scaling assumptions with respect both to time and to population density. The static equilibrium properties and the dynamic local stability of the model are analyzed, as are the effects of random variability due to fluctuations in demographic structure. We determine the consequences of population growth rate for: the critical level of immunization needed to eradicate an endemic disease, the transient epidemic period, the return time which measures the stability of departures from epidemiological equilibrium, and the power spectrum of epidemiological fluctuations and combined demographic-epidemiological fluctuations. Growing populations are found to be significantly different from stationary ones in each of these characteristics. The policy implications of these findings are discussed.
View details for Web of Science ID A1991GW78400003
View details for PubMedID 1808755
The mathematics of infection.
1991; 254 (5031): 591-592
View details for PubMedID 17806979
DELAYED REPRODUCTION AND FITNESS IN VARIABLE ENVIRONMENTS
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
1990; 87 (3): 1139-1143
Many organisms delay the initiation of reproduction even though such delay is not adaptive in a constant environment. Theoretical arguments in this paper show that delaying reproduction can increase fitness in a sufficiently variable environment. This paper uses stochastic demography to analyze the fluctuating population structure produced by environmental uncertainty. The results explain previously puzzling features of life cycle delays observed in nature, predicting that populations of the same species living in environments of differing harshness can display different life history phenotypes, a number of distinct life history phenotypes can coexist neutrally within a single population, and genetic polymorphisms are easily maintained if heterozygotes have intermediate life history phenotypes.
View details for Web of Science ID A1990CM07700060
View details for PubMedID 2300574
AGE STRUCTURE, ENVIRONMENTAL FLUCTUATIONS, AND HERMAPHRODITIC SEX ALLOCATION
1990; 64: 1-7
This paper studies sex allocation in an age-structured population of hermaphrodites living in a temporally fluctuating environment. The general condition for the evolutionary stable state (ESS) of allocation is derived for density-independent dynamics. This condition is used to determine the effect on the deterministic ESS of a dependence of survival rates on allocation. It is also used to identify the special conditions under which a stochastic ESS is given by a product rule and show how demographic structure and the correlation structure of vital rates determines the stochastic ESS.
View details for Web of Science ID A1990CU86900001
View details for PubMedID 2312342
POPULATION-DYNAMICS IN VARIABLE ENVIRONMENTS .7. THE DEMOGRAPHY AND EVOLUTION OF ITEROPARITY
1989; 133 (6): 901-923
View details for Web of Science ID A1989AG01500009
AN UNCERTAIN LIFE - DEMOGRAPHY IN RANDOM-ENVIRONMENTS
THEORETICAL POPULATION BIOLOGY
1989; 35 (3): 227-294
This paper concisely reviews the demography of populations with random vital rates, highlights examples and techniques which yield insight into population dynamics, summarizes the state of significant applications of the theory, and points to open problems. The central picture in this theory is of a time-varying but statistically stationary equilibrium for population, sharply distinct from the notions of classical demography. The deepest biological insights from the theory reveal the temporal structure of life histories to be a rich arena for natural selection.
View details for Web of Science ID A1989AE97100001
View details for PubMedID 2756495
CYCLES IN NONLINEAR AGE-STRUCTURED MODELS .1. RENEWAL-EQUATIONS
THEORETICAL POPULATION BIOLOGY
1987; 32 (1): 26-41
A variety of density-dependent population models can be described by nonlinear renewal equations. This paper develops analytical tools for such models to study the sustained population cycles which arise by bifurcation. The results obtained describe explicitly the direction of bifurcation, and the period, form, and dynamic stability of sustained cycles. The results are illustrated by application to a cohort-controlled model of human populations which has been proposed as a formalization of the Easterlin effect.
View details for Web of Science ID A1987J665700003
View details for PubMedID 3660273
- DEMOGRAPHY IN STOCHASTIC ENVIRONMENTS .2. GROWTH AND CONVERGENCE-RATES JOURNAL OF MATHEMATICAL BIOLOGY 1986; 24 (5): 569-581
POPULATION-DYNAMICS IN VARIABLE ENVIRONMENTS .6. CYCLICAL ENVIRONMENTS
THEORETICAL POPULATION BIOLOGY
1985; 28 (1): 1-17
This paper studies the dynamics of an age-structured population which experiences cyclical variation in vital rates. The principal features of population behavior are found to be contained in an explicitly calculable response function. Three distinct regimes of qualitative behavior are described when cycle period is respectively much less than, of the order of, and much greater than the average generation length. These results make explicit the way in which transient properties corresponding to average vital rates determine population response to cycles.
View details for Web of Science ID A1985ARQ7900001
View details for PubMedID 4060082
DEMOGRAPHY IN STOCHASTIC ENVIRONMENTS .1. EXACT DISTRIBUTIONS OF AGE STRUCTURE
JOURNAL OF MATHEMATICAL BIOLOGY
1984; 19 (3): 335-350
The steady state distribution of age structure is studied for populations with two age classes and stochastic vital rates. For a serially uncorrelated dichotomic vital rate the distribution of age structure is found analytically to be a singular steplike function; outside a specific region of vital rate values the singular function crosses a threshold to a smooth function. For a vital rate following a correlated two state Markov process the joint distributions of age structure and environment are found analytically to be singular steplike functions; again a threshold marks a transition to a smooth function. For fecundities which are serially uncorrelated but continuously distributed the age structure distribution is obtained as a smooth analytic function for all parameter values. These explicit results have applications to studies of age structure and average growth rate.
View details for Web of Science ID A1984TC88000007
View details for PubMedID 6470584
POPULATION-DYNAMICS IN VARIABLE ENVIRONMENTS .4. WEAK ERGODICITY IN THE LOTKA EQUATION
JOURNAL OF MATHEMATICAL BIOLOGY
1982; 14 (2): 221-230
The Hilbert projective metric is applied to the continuous-time Lotka equation in demography to establish weak ergodicity: populations with the same time-varying fecundity and mortality schedules ultimately have the same age composition. The analysis displays clearly the dynamic content of Lotka's equation and identifies a contraction operator which forces convergence of birth sequences over time. The relationship between primitivity in the discrete (Leslie) and continuous (Lotka) demographic models is made clear.
View details for Web of Science ID A1982NY63500007
View details for PubMedID 7119584
POPULATION-DYNAMICS IN VARIABLE ENVIRONMENTS .1. LONG-RUN GROWTH-RATES AND EXTINCTION
THEORETICAL POPULATION BIOLOGY
1980; 18 (3): 314-342
View details for Web of Science ID A1980LF39200002
STOCHASTIC INSTABILITY AND LIAPUNOV STABILITY
JOURNAL OF MATHEMATICAL BIOLOGY
1979; 8 (2): 133-145
View details for Web of Science ID A1979HR22100003
LIAPUNOV-FUNCTIONS - GEOMETRY AND STABILITY
JOURNAL OF MATHEMATICAL BIOLOGY
1979; 8 (1): 25-32
View details for Web of Science ID A1979HF35700003
HETEROSIS AS AN EXPLANATION FOR LARGE AMOUNTS OF GENIC POLYMORPHISM
1978; 88 (1): 149-169
By using both numerical and analytical approaches, we have shown that heterosis alone is not a mechanism for maintaining many alleles segregating at a locus. Even when all heterozygous are more fit than all homozygotes, the proportion of fitness arrays that will lead to a stable, feasible equilibrium of more than 6 or 7 alleles is vanishingly small. More alleles can be maintained if, in addition to heterosis, it is assumed that there is very little variation in fitness from heterozygote to heterozygote, with the ratio of mean heterosis to standard deviation of fitness among heterozygotes in the neighborhood of 10. When such conditions hold, the allelic frequency distribution and equilibrium will be very uniform, with all alleles very close to equal frequency (see PDF). It is much more likely that stable equilibria for multiple alleles will be best explained by multiple niche selection.
View details for Web of Science ID A1978ER32000011
View details for PubMedID 17248790
- DYNAMIC EQUILIBRIUM UNDER PERIODIC PERTURBATIONS IN SIMPLE ECOSYSTEM MODELS JOURNAL OF THEORETICAL BIOLOGY 1977; 66 (2): 327-343
STABILITY OF LOTKA-VOLTERRA SYSTEMS
1975; 257 (5525): 388-389
View details for Web of Science ID A1975AS00300032