The Systematic Formulation of Tractable Single-Species Population Models Incorporating Age Structure

Abstract

A mathematically rigorous approach to modeling the effects of age structure, in which the life-history of a species is divided into age classes of arbitrary duration was developed with special attention to the sub-populations of the various classes. By assuming that all individuals in a particular age class have the same birth and death rates (which may be time- and density-dependent), the normal integro-differential equations describing an age-structured population with overlapping generations was reduced to a set of coupled ordinary delay-differential equations which are readily integrated numerically. The use of the formalism in the construction and analysis of 2 models of laboratory insect populations is illustrated: a detailed model of Nicholson''s blowflies and a strategic model of larval competition intended to refine the design of experiments in progress on the dynamics of the Indian meal-moth Plodia interpunctella (Huebner). The model of Nicholson''s blowflies is dynamically identical to a model previously derived heuristically. The parameters were fitted using a more comprehensive range of data, and the passage of large, quasi-cyclic fluctuations through the age structure was illustrated. The larval competition model was used to distinguish the effects of uniform competition (all larvae competing) and cohort competition (larvae of a given age competing). The former can produce quasi-cycles of a type characteristic of delayed regulation, while the latter causes quasi-cycles consisting of bursts of population propagating through the age structure. The larval competition model was also used to demonstrate that apparently minor changes in the description of adult survival can induce dramatic alterations in model behavior. Rigor in the formation of the model is emphasized to distinguish genuinely interesting dynamics from mathematical artifacts.