Discrete demographic models with density-dependent vital rates

Abstract

The standard Leslie model of population growth in an age structured population is modified so as to incorporate density-dependent feedback control on each parameter of the standard projection matrix. Under fairly general conditions, the population converges to a stable age-distribution and a constant population size. This steady-state solution is uniquely determined by the parameters of the model. In general, fertility damping results in a flatter age-distribution than yielded by the undamped Leslie model. General survival damping results in the Leslie age-distribution. Post-infant survival damping results in a very steep age-distribution. For populations with high intrinsic growth rates, these differences in stable age-distributions are pronounced. For populations of low intrinsic growth rate, the patterns are the same, but the differences in stable age-distribution are more subtle. The age-distribution usually converges rapidly to the steady-state array, although population size generally takes longer to approach a stable value. Convergence properties are described for a series of cases which show periodicity. Such cases arise from “periodic” behavior of certain fertility-damping strategies, and ultimately approach a stable steady-state, although convergence may be very slow. Although the model is very general, it can be considerably simplified in practice. Special cases, which can be constructed, are the Malthusian (Leslie) model and the Logistic model. As a generality, the model is approximately Logistic, once the age-distribution approaches the steady-state array. One may use this fact for purposes of population projection.