Caswell Hal

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Caswell

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Hal

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0000-0003-4394-6894

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Frequency-dependent two-sex models : a new approach to sex ratio evolution with multiple maternal conditions

(John Wiley & Sons, 2016-09-07) Shyu, Esther ; Caswell, Hal

Mothers that experience different individual or environmental conditions may produce different proportions of male to female offspring. The Trivers-Willard hypothesis, for instance, suggests that mothers with different qualities (size, health, etc.) will use different sex ratios if maternal quality differentially affects sex-specific reproductive success. Condition-dependent, or facultative, sex ratio strategies like these allow multiple sex ratios to coexist within a population. They also create complex population structure due to the presence of multiple maternal conditions. As a result, modeling facultative sex ratio evolution requires not only sex ratio strategies with multiple components, but also two-sex population models with explicit stage structure. To this end, we combine nonlinear, frequency-dependent matrix models and multidimensional adaptive dynamics to create a new framework for studying sex ratio evolution. We illustrate the applications of this framework with two case studies where the sex ratios depend one of two possible maternal conditions (age or quality). In these cases, we identify evolutionarily singular sex ratio strategies, find instances where one maternal condition produces exclusively male or female offspring, and show that sex ratio biases depend on the relative reproductive value ratios for each sex.
Selective harvest of sooty shearwater chicks : effects on population dynamics and sustainability

( 2004-09-17) Hunter, Christine M. ; Caswell, Hal

Selectivity of harvest influences harvest sustainability because individuals with different characteristics contribute differently to population growth. We investigate the effects of selection based on chick weight on a traditional harvest of the sooty shearwater Puffinus griseus by Rakiura Maori in New Zealand.
Climate change threatens polar bear populations : a stochastic demographic analysis

(Ecological Society of America, 2010-10) Hunter, Christine M. ; Caswell, Hal ; Runge, Michael C. ; Regehr, Eric V. ; Amstrup, Steve C. ; Stirling, Ian

The polar bear (Ursus maritimus) depends on sea ice for feeding, breeding, and movement. Significant reductions in Arctic sea ice are forecast to continue because of climate warming. We evaluated the impacts of climate change on polar bears in the southern Beaufort Sea by means of a demographic analysis, combining deterministic, stochastic, environment-dependent matrix population models with forecasts of future sea ice conditions from IPCC general circulation models (GCMs). The matrix population models classified individuals by age and breeding status; mothers and dependent cubs were treated as units. Parameter estimates were obtained from a capture–recapture study conducted from 2001 to 2006. Candidate statistical models allowed vital rates to vary with time and as functions of a sea ice covariate. Model averaging was used to produce the vital rate estimates, and a parametric bootstrap procedure was used to quantify model selection and parameter estimation uncertainty. Deterministic models projected population growth in years with more extensive ice coverage (2001–2003) and population decline in years with less ice coverage (2004–2005). LTRE (life table response experiment) analysis showed that the reduction in λ in years with low sea ice was due primarily to reduced adult female survival, and secondarily to reduced breeding. A stochastic model with two environmental states, good and poor sea ice conditions, projected a declining stochastic growth rate, log λs, as the frequency of poor ice years increased. The observed frequency of poor ice years since 1979 would imply log λs ≈ − 0.01, which agrees with available (albeit crude) observations of population size. The stochastic model was linked to a set of 10 GCMs compiled by the IPCC; the models were chosen for their ability to reproduce historical observations of sea ice and were forced with “business as usual” (A1B) greenhouse gas emissions. The resulting stochastic population projections showed drastic declines in the polar bear population by the end of the 21st century. These projections were instrumental in the decision to list the polar bear as a threatened species under the U.S. Endangered Species Act.
Sensitivity analysis of reactive ecological dynamics

( 2008-02) Verdy, Ariane ; Caswell, Hal

Ecological systems with asymptotically stable equilibria may exhibit significant transient dynamics following perturbations. In some cases, these transient dynamics include the possibility of excursions away from the equilibrium before the eventual return; systems that exhibit such amplification of perturbations are called reactive. Reactivity is a common property of ecological systems, and the amplification can be large and long-lasting. The transient response of a reactive ecosystem depends on the parameters of the underlying model. To investigate this dependence, we develop sensitivity analyses for indices of transient dynamics (reactivity, the amplification envelope, and the optimal perturbation) in both continuous- and discrete-time models written in matrix form. The sensitivity calculations require expressions, some of them new, for the derivatives of equilibria, eigenvalues, singular values, and singular vectors, obtained using matrix calculus. Sensitivity analysis provides a quantitative framework for investigating the mechanisms leading to transient growth. We apply the methodology to a predator-prey model and a size-structured food web model. The results suggest predator-driven and prey-driven mechanisms for transient amplification resulting from multispecies interactions.
Age, stage and senescence in plants

(John Wiley & Sons, 2013-04-24) Caswell, Hal ; Salguero-Gomez, Roberto

Senescence (an increase in the mortality rate or force of mortality, or a decrease in fertility, with increasing age) is a widespread phenomenon. Theories about the evolution of senescence have long focused on the age trajectories of the selection gradients on mortality and fertility. In purely age-classified models, these selection gradients are non-increasing with age, implying that traits expressed early in life have a greater impact on fitness than traits expressed later in life. This pattern leads inevitably to the evolution of senescence if there are trade-offs between early and late performance. It has long been suspected that the stage- or size-dependent demography typical of plants might change these conclusions. In this paper, we develop a model that includes both stage- and age-dependence and derive the age-dependent, stage-dependent and age×stage-dependent selection gradients on mortality and fertility. We applied this model to stage-classified population projection matrices for 36 species of plants, from a wide variety of growth forms (from mosses to trees) and habitats. We found that the age-specific selection gradients within a life cycle stage can exhibit increases with age (we call these contra-senescent selection gradients). In later stages, often large size classes in plant demography, the duration of these contra-senescent gradients can exceed the life expectancy by several fold. Synthesis. The interaction of age- and stage-dependence in plants leads to selection pressures on senescence fundamentally different from those found in previous, age-classified theories. This result may explain the observation that large plants seem less subject to senescence than most kinds of animals. The methods presented here can lead to improved analysis of both age-dependent and stage-dependent demographic properties of plant populations.
Mating, births, and transitions : a flexible two-sex matrix model for evolutionary demography

(Springer, 2018-06-01) Shyu, Esther ; Caswell, Hal

Models of sexually-reproducing populations that consider only a single sex cannot capture the effects of sex-specific demographic differences and mate availability. We present a new framework for two-sex demographic models that implements and extends the birth-matrix mating-rule approach of Pollak. The model is a continuous-time matrix model that explicitly includes the processes of mating (which is nonlinear but homogeneous), offspring production, and demographic transitions and survival. The resulting nonlinear model converges to exponential growth with an equilibrium population composition. The model can incorporate age- or stage-structured life histories and flexible mating functions. As an example, we apply the model to analyze the effects of mating strategies (polygamy or monogamy, and mated unions composed of males and females, of variable duration) on the response to sex-biased harvesting. The combination of demographic complexity with the interaction of the sexes can have major population dynamic effects and can change the outcome of evolution on sex-related characters.
COMADRE : a global data base of animal demography

(John Wiley & Sons, 2016-01-27) Salguero-Gomez, Roberto ; Jones, Owen R. ; Archer, C. Ruth ; Bein, Christoph ; de Buhr, Hendrik ; Farack, Claudia ; Gottschalk, France ; Hartmann, Alexander ; Henning, Anne ; Hoppe, Gabriel ; Romer, Gesa ; Ruoff, Tara ; Sommer, Veronika ; Wille, Julia ; Voigt, Jakob ; Zeh, Stefan ; Vieregg, Dirk ; Buckley, Yvonne M. ; Che-Castaldo, Judy ; Hodgson, David ; Scheuerlein, Alexander ; Caswell, Hal ; Vaupel, James W.

The open-data scientific philosophy is being widely adopted and proving to promote considerable progress in ecology and evolution. Open-data global data bases now exist on animal migration, species distribution, conservation status, etc. However, a gap exists for data on population dynamics spanning the rich diversity of the animal kingdom world-wide. This information is fundamental to our understanding of the conditions that have shaped variation in animal life histories and their relationships with the environment, as well as the determinants of invasion and extinction. Matrix population models (MPMs) are among the most widely used demographic tools by animal ecologists. MPMs project population dynamics based on the reproduction, survival and development of individuals in a population over their life cycle. The outputs from MPMs have direct biological interpretations, facilitating comparisons among animal species as different as Caenorhabditis elegans, Loxodonta africana and Homo sapiens. Thousands of animal demographic records exist in the form of MPMs, but they are dispersed throughout the literature, rendering comparative analyses difficult. Here, we introduce the COMADRE Animal Matrix Database, an open-data online repository, which in its version 1.0.0 contains data on 345 species world-wide, from 402 studies with a total of 1625 population projection matrices. COMADRE also contains ancillary information (e.g. ecoregion, taxonomy, biogeography, etc.) that facilitates interpretation of the numerous demographic metrics that can be derived from its MPMs. We provide R code to some of these examples. Synthesis: We introduce the COMADRE Animal Matrix Database, a resource for animal demography. Its open-data nature, together with its ancillary information, will facilitate comparative analysis, as will the growing availability of databases focusing on other aspects of the rich animal diversity, and tools to query and combine them. Through future frequent updates of COMADRE, and its integration with other online resources, we encourage animal ecologists to tackle global ecological and evolutionary questions with unprecedented sample size.
Interacting effects of unobserved heterogeneity and individual stochasticity in the life-history of the Southern fulmar

( 2017-09-19) Jenouvrier, Stephanie ; Aubry, Lise M. ; Barbraud, Christophe ; Weimerskirch, Henri ; Caswell, Hal

Individuals are heterogeneous in many ways. Some of these differences are incorporated as individual states (e.g., age, size, breeding status) in population models. However, substantial amounts of heterogeneity may remain unaccounted for, due to genetic, maternal, or environmental factors. Such unobserved heterogeneity (UH) affects the behavior of heterogeneous cohorts via intra-cohort selection and contributes to inter-individual variance in demographic outcomes such as longevity and lifetime reproduction. Variance is also produced by individual stochasticity, due to random events in the life cycle of wild organisms, yet no study thus far has attempted to decompose the variance in demographic outcomes into contributions from unobserved heterogeneity and individual stochasticity for an animal population in the wild. We developed a stage-classified matrix population model for the Southern fulmar breeding on Ile des Pétrels, Antarctica. We applied multi-event, multi-state markrecapture methods to estimate a finite mixture model accounting for UH in all vital rates and Markov chain methods to calculate demographic outcomes. Finally, we partitioned the variance in demographic outcomes into contributions from unobserved heterogeneity and individual stochasticity. We identify three UH groups, differing substantially in longevity, lifetime reproductive output, age at first reproduction, and in the proportion of the life spent in each reproductive state. 14% of individuals at fledging have a delayed but high probability of recruitment and extended reproductive lifespan. 67% of individuals are less likely to reach adulthood, recruit late and skip breeding often but have the highest adult survival rate. 19% of individuals recruit early and attempt to breed often. They are likely to raise their offspring successfully, but experience a relatively short lifespan. Unobserved heterogeneity only explains a small fraction of the variances in longevity (5.9%), age at first reproduction (3.7%) and lifetime reproduction (22%). UH can affect the entire life cycle, including survival, development, and reproductive rates, with consequences over the lifetime of individuals and impacts on cohort dynamics. The respective role of unobserved heterogeneity versus individual stochasticity varies greatly among demographic outcomes. We discuss the implication of our finding for the gradient of life-history strategies observed among species and argue that individual differences should always be accounted for in demographic studies of wild populations.
The Atlantic salmon (Salmo salar) population of the Matamek River, Quebec : 1967-1984 data report

(Woods Hole Oceanographic Institution, 1986-07) Naiman, Robert J. ; Morin, Roderick ; Caswell, Hal ; Montgomery, W. Linn ; Klopfer, Eileen ; Kana, Todd M.

From 1967 to 1984 the Matamek Research Station, located near Sept-Iles, Quebec, was the focus of a research program on salmonid production in boreal river ecosystems. Research was conducted under the auspices of Woods Hole Oceanographic Institution and in cooperation with the Ministere du Loisir, de la Chasse et de la Peche du Quebec and representatives of several universities in Canada and the United States. One of the central activities throughout the history of the Station was monitoring of Atlantic salmon (Salmo salar) in the Matamek River. All salmon life history stages were involved, although the greatest effort was spent in estimates of parr population size at select sites and in estimates of population size, age and sex ratios of sea-run adults entering the river to spawn. Effort and methods were not consistent from year to year due to changes in program focus and improvements in techniques. Nonetheless, we believe the data represent the only long-term record for an Atlantic salmon population on the North Shore of the St. Lawrence. Heretofore, information on Atlantic salmon from the Matamek River was available only in published works or in the Matamek Annual Reports published by Woods Hole Oceanographic Institution. Because of the potential value for analyses of salmon population dynamics and life history, we have assembled the raw data, with neither analysis nor interpretation, in this report. During the period of 1980-1984, considerable effort was exerted in collecting data from original sources (field notebooks, scale envelopes, etc.) and in correcting errors in the data. We cannot, of course, guarantee complete accuracy. Nonetheless, this collection of information is the most complete and accurate compilation possible at this time. The data are presented as records for individual fish, and are ordered by date and by life history stage. We include a key to the designations of columns and to conventions used in coding data. All entries are raw data as initially recorded and coded; no analyses are available beyond those used by various individual authors in their preparation of reports or publications. While these data are made available for general use through this compilation, we request that proper acknowledgment be given the Matamek Research Program of Woods Hole Oceanographic Institution, under whose directorship this compilation was accomplished.
The stage-structured epidemic : linking disease and demography with a multi-state matrix approach model

( 2010-05) Klepac, Petra ; Caswell, Hal

Stage-structured epidemic models provide a way to connect the interacting processes of infection and demography. Reproduction and development can replenish the pool of susceptible hosts, and demographic structure leads to heterogeneous transmission and disease risk. Epidemics, in turn, can increase mortality or reduce fertility of the host population. Here we present a framework that integrates both demography and epidemiology in models for stage-structured epidemics. We use the vec-permutation matrix approach to classify individuals jointly by their demographic stage and infection status. We describe demographic and epidemic processes as alternating in time with a periodic matrix models. The application of matrix calculus to this framework allows for the calculation of R0 and sensitivity analysis.
A demographic model for sex ratio evolution and the effects of sex-biased offspring costs

(John Wiley & Sons, 2016-02-09) Shyu, Esther ; Caswell, Hal

The evolution of the primary sex ratio, the proportion of male births in an individual's offspring production strategy, is a frequency-dependent process that selects against the more common sex. Because reproduction is shaped by the entire life cycle, sex ratio theory would benefit from explicitly two-sex models that include some form of life cycle structure. We present a demographic approach to sex ratio evolution that combines adaptive dynamics with nonlinear matrix population models. We also determine the evolutionary and convergence stability of singular strategies using matrix calculus. These methods allow the incorporation of any population structure, including multiple sexes and stages, into evolutionary projections. Using this framework, we compare how four different interpretations of sex-biased offspring costs affect sex ratio evolution. We find that demographic differences affect evolutionary outcomes and that, contrary to prior belief, sex-biased mortality after parental investment can bias the primary sex ratio (but not the corresponding reproductive value ratio). These results differ qualitatively from the widely held conclusions of previous models that neglect demographic structure.
Stochastic flood and precipitation regimes and the population dynamics of a threatened floodplain plant

(Ecological Society of America, 2005-06) Smith, Marian ; Caswell, Hal ; Mettler-Cherry, Paige

Boltonia decurrens is an endangered plant restricted to the Illinois River Valley. Its complex life cycle has evolved in response to the dynamics of the historic flood regime, which has changed dramatically in the last century due to the construction of navigation dams and agricultural levees. To explore the effects of these changes, we developed deterministic and stochastic matrix population models of the demography of Boltonia. We used periodic matrix models to incorporate intra-annual seasonal variation. We estimated parameters as a function of the timing of spring flood recession (early or late) and of growing season precipitation (high or low). Late floods and/or low precipitation reduce population growth (λ). Early floods and high precipitation lead to explosive population growth. Elasticity analysis shows that changes in floods and precipitation alter the life history pathways responsible for population growth, from annual to biennial and eventually clonal pathways. We constructed and analyzed a stochastic model in which flood timing and precipitation vary independently, and we computed the stochastic growth rate (log λs) and the variance growth rate (σ2) as functions of the frequency of late floods and low precipitation. Using historical data on floods and rainfall over the last 100 years, we found that log λs has declined as a result of hydrological changes accompanying the regulation of the river. Stochastic elasticity analysis showed that over that time the contribution of annual life history pathways to log λs has declined as the contributions of biennial and clonal pathways have increased. Over the same time period, σ2 has increased, in agreement with observations of large fluctuations in local B. decurrens populations. Undoubtedly, many plant and animal species evolved in concert with dynamic habitats and are now threatened by anthropogenic changes in those dynamics. The data and analyses used in this study can be applied to management and development strategies to preserve other dynamic systems.
Population growth rate of a common understory herb decreases non-linearly across a gradient of deer herbivory

( 2008-11-09) Knight, Tiffany M. ; Caswell, Hal ; Kalisz, Susan

Overabundant white-tailed deer (Odocoileus virginianus) are a significant management problem in North America that exert unprecedented herbivory pressure on native understory forest communities. Conserving understory plant populations requires quantifying a sustainable level of deer herbivory. To date, most population projection models consider only deer presence and absence. To estimate population growth rate along a gradient of herbivory, we focused on Trillium grandiflorum because it is a common understory species and a bellwether of deer effects and forest decline. We used matrix population models, and employed both prospective and retrospective analyses using a regression life table response experiment (LTRE). Deer affect size, stage and population dynamics of T. grandiflorum. Because deer target flowering and large non-flowering stages of T. grandiflorum, these individuals do not produce seed in the year they are browsed and are more likely to regress in stage and size in the following growing season relative to non-browsed plants. Importantly, sustained high browse levels result in populations dominated by small, non-flowering individuals. Our LTRE revealed a significant negative and decelerating relationship between herbivory and λ. This non-linearity occurs at the highest herbivory levels because highly browsed populations become dominated by stages that deer do not consume and are thus buffered from rapid decline. However, population extinction is expected when herbivory is greater than the pivotal value of ~15%. Our study demonstrates that levels of deer herbivory commonly experienced by forest understory perennials are sufficient to cause the loss of T. grandiflorum and likely other co-occurring palatable species.
Sensitivity analysis of periodic matrix population models

( 2012-04) Caswell, Hal ; Shyu, Esther

Periodic matrix models are frequently used to describe cyclic temporal variation (seasonal or interannual) and to account for the operation of multiple processes (e.g., demography and dispersal) within a single projection interval. In either case, the models take the form of peri- odic matrix products. The perturbation analysis of periodic models must trace the e ects of parameter changes, at each phase of the cycle, on output variables that are calculated over the entire cycle. Here, we apply matrix calculus to obtain the sensitivity and elasticity of scalar-, vector-, or matrix-valued output variables. We apply the method to linear models for periodic environments (including seasonal harvest models), to vec-permutation models in which individ- uals are classi ed by multiple criteria, and to nonlinear models including both immediate and delayed density dependence. The results can be used to evaluate management strategies and to study selection gradients in periodic environments.
Individual stochasticity in the life history strategies of animals and plants

(Public Library of Science, 2022-09-23) Varas Enríquez, Pablo José ; Van Daalen, Silke F. ; Caswell, Hal

The life histories of organisms are expressed as rates of development, reproduction, and survival. However, individuals may experience differential outcomes for the same set of rates. Such individual stochasticity generates variance around familiar mean measures of life history traits, such as life expectancy and the reproductive number R0. By writing life cycles as Markov chains, we calculate variance and other indices of variability for longevity, lifetime reproductive output (LRO), age at offspring production, and age at maturity for 83 animal and 332 plant populations from the COMADRE and COMPADRE matrix databases. We find that the magnitude within and variability between populations in variance indices in LRO, especially, are surprisingly high. We furthermore use principal components analysis to assess how the inclusion of variance indices of different demographic outcomes affects life history constraints. We find that these indices, to a similar or greater degree than the mean, explain the variation in life history strategies among plants and animals.
Detecting reactivity

(Ecological Society of America, 2009-10) Neubert, Michael G. ; Caswell, Hal ; Solow, Andrew R.

By definition, ecological systems at a stable equilibrium eventually return to the equilibrium point following a small perturbation. In the short term, however, perturbations can grow. Equilibria that exhibit transient growth following perturbation are said to be reactive. In this report, we present a statistical method for detecting reactivity from multivariate time series. The test is simple and computationally tractable, and it can be applied to short time series. Its main limitation is that it is based on a model of population dynamics that is linear on a logarithmic scale. Our results suggest that the test is robust when the dynamics are nonlinear on the log scale but that it may incorrectly classify an equilibrium as reactive when the reactivity is close to zero.
Calculating second derivatives of population growth rates for ecology and evolution

(John Wiley & Sons, 2014-05-19) Shyu, Esther ; Caswell, Hal

Second derivatives of the population growth rate measure the curvature of its response to demographic, physiological or environmental parameters. The second derivatives quantify the response of sensitivity results to perturbations, provide a classification of types of selection and provide one way to calculate sensitivities of the stochastic growth rate. Using matrix calculus, we derive the second derivatives of three population growth rate measures: the discrete-time growth rate λ, the continuous-time growth rate r = log λ and the net reproductive rate R0, which measures per-generation growth. We present a suite of formulae for the second derivatives of each growth rate and show how to compute these derivatives with respect to projection matrix entries and to lower-level parameters affecting those matrix entries. We also illustrate several ecological and evolutionary applications for these second derivative calculations with a case study for the tropical herb Calathea ovandensis.
The rarity of survival to old age does not drive the evolution of senescence

(Springer, 2016-05-04) Wensink, Maarten J. ; Caswell, Hal ; Baudisch, Annette

The evolution of senescence is often explained by arguing that, in nature, few individuals survive to be old and hence it is evolutionarily unimportant what happens to organisms when they are old. A corollary to this idea is that extrinsically imposed mortality, because it reduces the chance of surviving to be old, favors the evolution of senescence. We show that these ideas, although widespread, are incorrect. Selection leading to senescence does not depend directly on survival to old age, but on the shape of the stable age distribution, and we discuss the implications of this important distinction. We show that the selection gradient on mortality declines with age even in the hypothetical case of zero mortality, when survivorship does not decline. Changing the survivorship function by imposing age independent mortality has no affect on the selection gradients. A similar result exists for optimization models: age independent mortality does not change the optimal result. We propose an alternative, brief explanation for the decline of selection gradients, and hence the evolution of senescence.
Beyond R0 : demographic models for variability of lifetime reproductive output

(Public Library of Science, 2011-06-29) Caswell, Hal

The net reproductive rate measures the expected lifetime reproductive output of an individual, and plays an important role in demography, ecology, evolution, and epidemiology. Well-established methods exist to calculate it from age- or stage-classified demographic data. As an expectation, provides no information on variability; empirical measurements of lifetime reproduction universally show high levels of variability, and often positive skewness among individuals. This is often interpreted as evidence of heterogeneity, and thus of an opportunity for natural selection. However, variability provides evidence of heterogeneity only if it exceeds the level of variability to be expected in a cohort of identical individuals all experiencing the same vital rates. Such comparisons require a way to calculate the statistics of lifetime reproduction from demographic data. Here, a new approach is presented, using the theory of Markov chains with rewards, obtaining all the moments of the distribution of lifetime reproduction. The approach applies to age- or stage-classified models, to constant, periodic, or stochastic environments, and to any kind of reproductive schedule. As examples, I analyze data from six empirical studies, of a variety of animal and plant taxa (nematodes, polychaetes, humans, and several species of perennial plants).
Incorporating ‘recruitment’ in matrix projection models : estimation, parameters, and the influence of model structure

( 2010-07) Cooch, Evan G. ; Cam, Emmanuelle ; Caswell, Hal

Advances in the estimation of population parameters using encounter data from marked individuals have made it possible to include estimates of the probability of recruitment in population projection models. However, the projected growth rate of the population, and the sensitivity of projected growth to changes in recruitment, can vary significantly depending upon both the structural form of the model and how recruitment is parameterized. We show that the common practices of (1) collapsing some age classes into a single, terminal ‘aggregated’ age-class, and (2) parameterizing recruitment using the proportion of recruited individuals (breeders) in a given age-class may confound analysis of age-based (Leslie) matrix projection models in some instances, relative to state-based projection models where recruited and pre-recruited individuals are treated as separate states. Failing to account for these differences can lead to misinterpretation of the relative role of recruitment in the dynamics of an age-structured population.We show that such problems can be avoided, either by structural changes to the terminal aggregated age-class in age-based models, or by using using a state-based model instead. Since all the metrics of general interest from a classical age-based matrix models are readily derived from a state-based model equivalent, this suggests there may be little reason to use the classical age-based approach in situations where recruitment is a parameter of interest.

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