Survival and aging in the wild via residual demography

Abstract

Information about the age distribution and survival of wild populations is of much interest in ecology and biodemography, but is hard to obtain. Established schemes such as capture–recapture often are not feasible. In the proposed residual demography paradigm, individuals are randomly sampled from the wild population at unknown ages and the resulting captive cohort is reared out in the laboratory until death. Under some basic assumptions one obtains a demographic convolution equation that involves the unknown age distribution of the wild population, the observed survival function of the captive cohort, and the observed survival function of a reference cohort that is independently raised in the laboratory from birth. We adopt a statistical penalized least squares method for the deconvolution of this equation, aiming at extracting the age distribution of the wild population under suitable constraints. Under stationarity of the population, the age density is proportional to the survival function of the wild population and can thus be inferred. Several extensions are discussed. Residual demography is demonstrated for data on fruit flies Bactrocera oleae.

Introduction

Understanding aging in the wild is one of the most important problems in biodemography, yet available methodology is limited. With appropriate demographic tools, empirical data derived from field studies are expected to be used to frame and test theories of aging, inform research concerned with aging mechanisms, and establish baselines for the natural history of aging (Finch, 1990, Finch, 2001, Promislow, 1991, Abrams, 1993, Gaillard et al., 1994, Reznick et al., 2001, Tatar and Yin, 2001). We introduce the new concept of residual demography with the aim to deduce the age distribution of individuals in the wild. Residual demography utilizes data on laboratory survival from both wild-caught individuals of unknown age and wild-type individuals of known age. We demonstrate that the age structure of the wild population can be deduced from the survival of these individuals in the laboratory, in the general situation where survival in the wild and in the laboratory might differ. This study is motivated by the need to develop a method to estimate age structure in field populations of insects.

Residual demography includes two experimental steps, namely (1) the capture and transfer of wild individuals from the field to the laboratory where they are reared out under defined conditions and their date of death is recorded (captive cohort); (2) creation of a reference cohort by obtaining individuals at birth from the wild and rearing them under the same defined laboratory conditions and recording death rates. The analysis of these data is then based on statistical deconvolution methodology. The aim is to construct an age distribution of the wild population that, when combined with the survival schedule of the reference cohort, yields the distribution of deaths observed in the captive cohort. If the population is stationary and a stable equilibrium of the age distribution has been reached, the density of the age distribution in the wild corresponds to the survival function of the population, a fact that is known from renewal theory (Feller, 1950).

A demographic identity to infer aging and survival in the wild has been discussed previously under the assumption that survival in the wild and after marking individuals is subject to the same force of mortality (Müller et al., 2004). This major restriction is dropped in the proposed residual demography approach, where the force of mortality is allowed to change upon transferring an individual of unknown age from the wild to the laboratory. This reflects underlying biology more closely and makes it possible to obtain inference for the age distribution in the wild in the face of differing mortality rates between field and laboratory. We also briefly discuss extensions of this new approach to the case of non-stationary birth rates and unequal sampling probabilities of capture, and to the closely related problems of estimating age-at-capture of individuals, and of estimating the force of mortality (hazard function) in the wild.

We show that the assumption that the force of mortality acting on an individual depends solely on the individual's age and on whether the individual is in the wild or in the laboratory implies a demographic convolution equation. Consequently, the recovery of the unknown age distribution requires to solve a statistical deconvolution problem (see Madden et al., 1996 for an overview). An implementation by penalized least squares is shown to provide a feasible solution to this inverse problem. Crucial for this specific deconvolution problem are functional constraints such as smoothness, non-negativity and monotonicity which can be conveniently incorporated via suitable penalty terms.

Assumptions made for the application of the proposed approach in biodemographic studies are that the survival of sampled individuals in captivity is determined by the corresponding age-specific mortality in captivity, irrespective of when the capture occurs, and that each individual present in the wild population has the same chance of being sampled, irrespective of age. If additional information is available, some of these assumptions can be relaxed. For inference about survival in the wild, going beyond inferring merely the age distribution in the wild, another needed assumption is stationarity of the population. This assumption also may be relaxed if additional information is available. We demonstrate the proposed deconvolution methodology with cohorts sampled from the fruit flies Bactrocera oleae.

In Section 2, we derive the novel demographic convolution equation that underlies the concept of residual demography and discuss various extensions. In Section 3, we discuss the deconvolution and its actual implementation. A biodemographic data set is used in Section 4 to demonstrate the methodology in action. Further discussion follows in Section 5.