16. Ageing in the Wild, Residual Demography and Discovery of a Stationary Population Equality

A feature of many interdisciplinary fields such as biodemography is that questions are often asked that had previously not been considered by scientists in either of the “parent” disciplines (i.e. demography and biology). Presumably this is because the questions were not thought of in the first place, or, alternatively, because the answers were perceived in the respective fields to be of no general interest, to serve no conceptual purpose or to solve no pertinent problem.

In this chapter I describe a personal experience in which a question emerged during discussions with several of my biodemography colleagues in the late 1990s that appeared not to have been asked before (at least in the same way): What can information gathered on field-captured Mediterranean fruit flies (medflies) monitored through death in the laboratory tell us about population ageing in the wild? This question was the precursor that, in the late 1990s, led to my discovery of a mathematical identity unique to stationary populations stated as “the fraction of individuals x days old in a stationary population equals the fraction of individuals that die x days later.” This identity was published in a jointly-authored paper with Hans Müller, Jane-Ling Wang and other colleagues (Müller et al., 2004), followed six years later by its proof the Carey’s Equality eponym (Vaupel, 2009a), which, in turn, was followed by a paper with original analytical concepts and new theoretical insights into the identity (Rao and Carey, 2015).

Arguably the greatest difference between population studies of humans and population studies of non-human species is the gulf in the availability of age-specific data. Whereas it is nearly ubiquitous in the former, it is mostly absent in the latter.

The situation is the near-exact opposite in the vast majority of population studies concerned with non-human species. For example, the accuracy is extremely low and the costs generally extremely high for virtually all of the methods used to estimate insect age (Lehane, 1985) including wear-and-tear (Tyndale-Biscoe, 1984), cuticular hydrocarbon layering (Gerade et al., 2004), accumulation of bio-compounds (Lehane, 1985), and transcriptional profiling (Cook, McMeniman, and O’Neil, 2008; Cook and Sinkins, 2010). No ageing method has ever been routinized as part of a standardized surveillance program in applied insect ecology such as for monitoring insect disease vectors (mosquitoes, tsetse flies) where insect age is an extremely important component in disease transmission (Cook, McMeniman, and O’Neil, 2008). For vertebrates there are some exceptions including (1) long-term mark-recapture studies on selected species of birds and large mammals (Nussey et al., 2006; Ozgul et al., 2009) and (2) ecological studies spanning many taxa that use post-mortem techniques to estimate age including otolith layering in fish (Campana and Thorrold, 2001; Limburg et al., 2013) and tooth wear in wildlife (Dinsmore and Johnson, 2012). Although there are a number of relatively recent papers for estimating age- and stage-specific life history parameters (Cochran and Ellner, 1992; Metcalf et al., 2009; Davison, 2011; Horvitz and Tuljapurkar, 2008), this is a different concept than that for estimating the age of individuals.

The profundity of not having information on individual and population age in studies of non-human species is not recognized by the majority of mainstream demographers because of their exclusive focus on humans. But this lack is deeply frustrating to the majority of population biologists and applied ecologists. This is because the absence of information on age and age structure in populations of non-human species severely limits the scope and depth of demographic analysis and modelling in several important respects. Firstly, the majority of the most sophisticated demographic models in the literature are developed for and concerned with human populations. These methods both assume and require information on individual-age and population-age structure. Therefore, without age data on non-human species, many of the classical demographic models including cohort life tables and age-structured population models apply in theoretical and laboratory contexts rather than in the wild settings where they are the most relevant.

Thirdly, the results of demographic studies in the laboratory are of marginal value without the availability of age data for cohorts and populations in the field. These limitations frequently preclude opportunities to refine, adapt and expand powerful demographic tools for use in analysis of populations of non-human species. They also restrict the range of possibilities for creating new demographic concepts and building new models based on the treasure-trove of life history (and thus demographic) characteristics observed across the Tree of Life.

In light of the near-absence of methods for estimating age structure in fruit fly populations and the importance of finding a method, several of my biodemography colleagues and I set about trying to develop new concepts for studying ageing in the wild. We believed that it might be possible to achieve a better outcome for estimating age structure in insect populations using demographic models than with costly and mostly inaccurate high-tech methods used to estimate the age of individual insects.

A retrospective examination of the process that took me from an idea to the discovery of the life table identity occurred in four stages. Remove any of the first three stages or even one of the within-stage details, and I would likely not have discovered the identity. The last stage was model formulation after having identified the key equivalency. Key parts of the following sections are taken from a paper by myself and co-workers written at approximately the same time as this one (Carey, Silverman, and Rao, 2019).

Stage I: Framing the Concept

The germ of the idea that ultimately led to the discovery of the population identity was motivated by my view, and that of many insect ecologists, that the conventional methods for estimating individual age and age structure in wild populations described earlier are sorely lacking. Thus, the question that arose in 1998 on a research retreat in Crete (Greece) involving myself, mathematical demographer Anatoli Yashin and geneticists Lawrence Harshman and Linda Partridge was: “What can be learned about aging in the wild from information gathered on field-captured fruit flies of unknown age monitored in the laboratory?” This initial question, the concept of which is illustrated in Figure 1, was framed around the potential use of biological information in what Partridge referred to as “residual demography”, e.g. the post-capture levels and patterns of egg laying; challenge assays such as starvation and desiccation resistance; health status and remaining longevity. None of us had an inkling that this biological idea would lay the groundwork for the discovery of a population identity.

Stage II: Simulation Studies

Inasmuch as I am trained as an insect ecologist and not as a mathematical modeler per se, I asked several of my mathematical demography colleagues what statistical concepts would be required to estimate the age structure of a fruit fly population using the post-capture survival information of wild-caught individuals of unknown age.

Stage III: Construction of a Heuristic Life Table

Shown in Table 1, the life-table-based model consisted of sub-components framed as separate but interconnected sub-tables. The first of these sub-tables (columns 1–4 in Table 1) contains the basic metrics of the stationary population with age x in column 1, and the number in the population at each age (Nx) in column 2 (i.e. 40, 30, 25, 5 and 0 individuals at ages 0, 1, 2, 3, and 4, respectively). The corresponding survival lx within this population and its age structure cx are given in columns 3 and 4, respectively.

The concept for the third subcomponent of Table 1 (columns 10–11) was to bring life table methods to bear on the captive cohort. Inasmuch as the sum of the fractions surviving in each of the sample sub-cohorts at each captive age (y) represents the total of the original surviving, these sums represent the survival schedule of the captive cohort, ly*. Although I found the survival column (column 10) interesting, it revealed nothing about the deeper mathematical connection between the population and the captive cohort. It was only when I computed the dy*column did I experience a Eureka moment — the values for the death distribution (column 11) were exactly equal to the values in the age distributions (column 4).

In his fascinating paper on the role of serendipity in science, Yaqub (2018) would likely classify my discovery of this stationary population equality as a “Mertonian serendipity” — discovery as the outcome of a targeted search that solved a problem in hand (age-structure estimation) via an unexpected route. The expected route would have been through the use of Bayesian models.

Stage IV: Model Formulation

In a section titled “A key demographic identity” my statistical colleagues Hans Müller and Jane-Ling Wang formulated a mathematical model based on the framework and concepts that I developed as given in Table 1 (Müller et al., 2004). Assuming population stationarity in this hypothetical case, the fraction of individuals age x in the population is given by cx = lxΣly = c₀lx. The death rates in the marked sample life table at captive age xʹ are by definition d^*y = l^*y − d^*y₊₁. These death rates are generated by subjects that enter the marked sample life table at various unknown ages, and survive to captive age y. For all individuals that enter the marked sample cohort at age z, the contribution to d^*y is therefore

cz … (1 − ) = cz( − ) = c₀(lz₊y − lz₊y₊₁)

where lz refers to the survival of the captured individuals at age z.

The contributions of individuals entering the marked sample life table at various ages are additive. Therefore, adding the contributions over all ages of entry z:

d^*y = Σ c₀(lz₊y − lz₊y₊₁) = c₀/ly

and therefore

d^*y = Σ cx

This formula states that the fraction of a population age x days equals the fraction of the population that die x days later. The shorthand for this equality is age structure equals death distribution. In his proof of this equality, Vaupel expressed the relationship differently — the proportion of the population with x years of life remaining is equal to the proportion dying x years in the future (Vaupel, 2009a; p. 8). The shorthand for this expression of the same equality is life lived equals life left.

Table 1 Illustration of the relationship between the age structure in a stationary population (columns 1–4), the captive (sample) cohort life (columns 5–9), and the captive cohort life table (columns 10–11). Framework from Table 1 in Hans Müller and others (Müller et al. 2004). The identity illustrated in this table as well as one for medfly stationary populations are visualized in Figs. 3–5 of the paper by Carey et al. (2019).

Given the depth and scope of the demography literature in general (Poston and Micklin, 2005) and of the mathematical demography (Land, Yang, and Yi, 2005) and life table literature in particular (Guillot, 2005), I was interested to know whether this identity had already been published in some form. My online searches in the demography, statistics and mathematical biology literature revealed that there were no previous papers on this identity in either the primary literature or in the searchable secondary and grey literature. Later perusals of demography texts (Preston, Heuveline, and Guillot, 2001; Keyfitz and Caswell, 2010), handbooks (Poston and Micklin, 2005), methods books (Siegel and Swanson, 2004; Caswell, 2001), dictionaries (Peterson and Peterson, 1986), encyclopaedias (Demeny and McNicoll 2003a, 2003b) and treatises (Caselli, Vallin, and Wunsch, 2006f) also indicated that this result was not present in the earlier literature.

To my astonishment, three years ago I learned from Tim Riffe that he had discovered a 1986 paper describing an equality in stationary populations that was expressed differently than mine but similarly to Vaupel’s life-lived and left version (Riffe, 2015). In his article, titled “Structure et dynamique des populations. La pyramide des années à vivre, aspects nationaux et exemples régionaux”, French demographer Nicolas Brouard (1986; pp. 160–61) wrote:

La population stationnaire a un intérêt pour le sujet qui nous préoccupe, car dans une population rigoureusement stationnaire la pyramide des âges est aussi égale à la pyramide des années à vivre. On montre [2] en effet que dans une population stationnaire, il y a autant d’individus ayant vécu x années que d’individus ayant x années à vivre.

[The stationary population has an interest in the subject that concerns us, because in a strictly stationary population the age pyramid is also equal to the pyramid of the years to live. It is shown [2] that in a stationary population, there are as many individuals having lived x years as individuals having x years to live.]

Brouard’s story reveals how his limited access to typesetting technology, the restricted distribution of the publications, the necessity of publishing in French rather than in English, the perceived absence of any utility of the identity and the lack of interest by his more senior colleagues conspired to keep his discovery (Brouard’s Theorem) hidden from mainstream demography for thirty years.

Papers related either to this identity, similar concepts or stationary population theory more generally, which were published many years prior to the paper that described this identity (Müller et al., 2004), include the two papers by Brouard (Brouard, 1986, 1989), the classic papers by Ryder (1965, 1973, 1975) on replacement populations, the article by Kim and Aron (1989) showing the equivalency of the average age and average expectation of remaining life in a stationary population, the book section by Keyfitz (1985; p. 74) containing a general formula for the average expectation of life in a stationary population, and the demography text by Preston and his co-authors (Preston, Heuveline, and Guillot, 2001; pp. 53–58) outlining the basic properties of a stationary population. More recent papers connecting life lived to life left (or age structure to death distributions) include one by Goldstein (2009) proving the earlier finding of Kim and Aron (1989) but in a different way; a paper by Rao and Carey (2015) with original approach and new conceptual insights into Carey’s Equality; symmetries between life lived and left in finite stationary populations (Villavicencio and Riffe 2016a); the relationship of random age and remaining lifetimes by Finkelstein and Vaupel (2014); and the paper by Vaupel and Villavicencio (Vaupel and Villavicencio, 2018) in which they extend stationary population identity to the stable, non-stationary case.

Although Brouard’s Theorem and Carey’s Equality both describe the same demographic identity, the contexts in which they were discovered and the concepts upon which they are based are fundamentally different. From my experience working with scientists in the professional worlds of both human (classical) demography and biodemography, the conceptual differences between the two flow naturally from these two worlds — the “life lived equals life left” perspective from demography, and the “age structure equals death distribution” perspective from biodemograpy. Indeed, until James Vaupel published the proof of the stationary population identity (Vaupel, 2009), I had never before thought of age structure (denoted c_x) as a “life lived” concept, or of the death distribution (denoted d_x) as a “life left” concept. Even though they are literally identical, they are neither semantically nor conceptually identical.

As a biodemographer working through each of the stages in the discovery process that I described earlier, my focus was on the potential use of life table theory as a basis for estimating the age structure of a life table (stationary) population. I am reasonably certain that I would never have conceived of measuring ‘life left’ as a way to estimate ‘life lived’ instead of the use of the conventional and more transparent life table concepts in which the death distribution provides a means to estimate age structure.

I believe the stationary population identity that I discovered in the context of medfly research has a number of important implications in both basic and applied contexts for human demography. I briefly describe several of these next.

Human Evolution

It is virtually certain that the population growth rates of various species of early hominids (e.g. Australopithicus spp.) in general and of prehistoric Homo sapiens in particular were stationary, or nearly so, the vast majority of the time (Johnson and Brook, 2011; Lee and Tuljapurkar, 2008; Boone, 2002). Indeed, it is estimated that during most of the Holocene, human populations worldwide grew at a long-term annual rate of 0.04% (Zahid, Robinson, and Kelly, 2016) which, for practical purposes, have age structures that are nearly indistinguishable from stationary populations.

Using the concept of the stationary population identity, life-table rates of prehistoric populations (Gage, 1998; Kaplan et al., 2000) imply that up to half of the population is 20 years old or greater, and at least a quarter are over 30 years old. This implies (from the identity) that 50% and 25% of the prehistoric populations had respectively 20 and 30 years remaining to live together. These basic demographic metrics provide important perspectives on the extent to which individuals in prehistoric societies shared lives (and thus their skills, language, stories, art, music and culture).

Demographic Principles

Inasmuch as stationary population theory and basic concepts are inextricably linked to population stationarity (Preston, Heuveline, and Guillot, 2001), the stationary population identity contributes to a number of fundamental principles upon which the field rests. For example, demographic transition theory is based on the concept of offsetting birth and death rates at the beginning and end of the transition (Mesle and Vallin, 2006); population momentum is based on the concept of stationarity as its end point (Caselli, Vallin, and Wunsch, 2006a); and zero population growth (ZPG) is, by definition, based on the concept of the cessation of population growth (Caselli, Vallin, and Wunsch, 2006a), and thus implies stationary populations. The stationary population identity can now be integrated into each of these basic demographic principles.

Replacement-level Populations in the Twenty-first Century

Future World

I believe there are several lessons that can be learned from this story. The first is that the proverb “necessity is the mother of invention” can also be reframed as “… the mother of discovery”. My quest to find a practical solution to the problem of population age structure estimation led to the discovery of the basic stationary population identity. A second lesson pertains to the virtues of pencil-and-paper modelling in science — it is both immediate and simple (Wong and Kjaergaard, 2012). This step forced me to think about the essence of the problem rather than becoming buried in its complexities. Simple is not the same as simplistic. A third lesson pertains to the discovery of new demographic principles. If the stationary population identity remained hidden for most of the 350-year history of the life table, there are almost certainly more undiscovered ones. I believe the most fertile ground for new demographic discoveries will be within the interdisciplinary paradigm of biodemography.

I will close with what I consider to be the wonderment of the discovery. How could this particular mathematical property of (arguably) the most studied of all demographic models — the life table — have remained hidden for so long? And once discovered, still not have found its way into the mainstream demographic literature for thirty more years? Why do all deaths that occur in a stationary fruit-fly population at twenty-five days post-capture, for example, sum exactly to the number that were precisely twenty-five days old in the original population? How is it possible to compute precisely the age structure of a stationary population without age data on even a single member? Think about that.

The concept appears not to make sense on its face. It is a discovery that is not obviously true. But once proven (Brouard, 1989; Vaupel, 2009b) becomes one that obviously is true.

Many of the findings that inform this chapter were supported in part by the National Institute on Aging grants P01 AG08761-10 and P01 AG022500-01, and through UC Berkeley CEDA grant P30AG0128. I thank Lawrence Harshman, Linda Partridge and Anatoli Yashin for lively discussions on residual demography in the late 1990s, and my UC Davis colleagues Hans Müller and Jane-Ling Wang for their formulations and insights. I am especially indebted to James Vaupel for recognizing the importance of this identity in the first place, for his mathematical proof, and for crediting me with its (re-) discovery. I am grateful to Arni Rao for his energy and enthusiasm for doing collaborative work with me on this identity as well as for his theoretical insights and innovative ideas after having first met him at the Nathan Keyfitz Memorial Symposium at Ohio University in 2013. I thank Tim Riffe for bringing Brouard’s Theorem to my (indeed the world’s) attention and, along with Francisco Villavicencio, for their insightful and constructive comments on an earlier version of this chapter. I am grateful to Nicolas Brouard for background information he provided me about the circumstances of his original discovery. Finally, I thank Raziel Joseph Davison for her constructive review of an earlier version of this paper, and Rebecca Sear, Ronald Lee and Oskar Burger for their encouragement, feedback and patience.