## Carey’s Equality and Fundamental Theorem on Stationary Population Models

### By Arni S.R. Srinivasa Rao (Georgia Regents University)

Arni S.R. Srinivasa Rao, Georgia Regents University, Augusta and James R. Carey, University of California, Davis, who met for the first time during an MBI workshop in June 2013, jointly formulated and proved a result dubbed by many "the fundamental theorem" of stationary population models. The event that inspired this collaboration was a three-day symposium held at MBI to honor the birth centenary of a renowned mathematical demographer Nathan Keyfitz.

As a result of their initial conversations at the symposium and the subsequent work, Rao and Carey provided a generalization of what is known as Carey’s equality or life table identity. The identity, first described in the early 2000's, states that age composition and the distribution of remaining lifespans are identical in a stationary population. This equivalence is important as it can be used to estimate age structure if information is available on time to death. Whereas Carey's equality was observed empirically, for instance, in the symmetric survival patterns of a captive cohort of Mediterranean fruit flies (data illustrated in the picture) and their follow-up cohort, up until now it was not clear under precisely what conditions was this phenomenon expected to be true in idealized wild populations.

As a consequence of their discussions at MBI, Rao and Carey were not only able to provide an elegant proof of a precise mathematical statement extending and explaining Carey's equality, but also obtained a series of new results which help understanding internal structures of aging process and age-structure of wild populations.

In their proofs Rao and Carey used arguments from algebra, combinatorics, logic and principles of stationary populations. Both their results as well as the mathematical techniques used have already received considerable attention among the mathematical demographers across the world. Several participants of the MBI Keyfitz event, including Ronald Lee of UC Berkeley have commented that the results of Rao and Carey are truly fundamental to further development of the mathematical theory of demography. The results of Rao and Carey are to soon appear in print, with a specific acknowledgement of the role MBI has played in initiating their collaboration.