Multiregional Age-Structured Populations with Changing Rates: Weak and Stochastic Ergodic Theorems

Cohen, J.E. (1981). Multiregional Age-Structured Populations with Changing Rates: Weak and Stochastic Ergodic Theorems. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-81-033

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Abstract

A biological population, human or nonhuman, may experience multiple states in two ways.

First, it may visit different states in the course of time, the whole population experiencing the same (possibly age-specific) vital rates at any one time. For example, a troop of baboons moves from one area to another of its range, with associated changes in food supply and risks of predation (Altmann and Altmann, 1970). A human population experiences fluctuating crop yields from one year to the next, with associated effects on childbearing and survival. There are serial changes of state of a homogeneous population.

Second, the population may be subdivided into inhomogeneous subpopulations that exercise different states in parallel. Individuals may migrate from one state to another in the course of time. The states may correspond to geographical regions, work status, marital status, health status, or other classifications (Rogers, 1980).

The purpose of this paper is to describe population models in which serial and parallel inhomogeneity are combined. In demography, theorems that describe long-run behavior that is independent of initial conditions are called ergodic theorems. Weak ergodic theorems assume that the rates that govern a population's evolution themselves follow some deterministic trajectory. Le Bras (1977) gave the first weak ergodic theorem for multiregional age-structured populations. We shall give four weak ergodic theorems that are more general than that of Le Bras. Stochastic ergodic theorems assume that the rates that govern a population's evolution are selected from a set of possible rates by some stochastic process. We shall state a stochastic ergodic theorem that assumes that the rates of birth, death, and migration or other transition are selected by a Markov chain.

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