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Of Elephants and Men: A Discrete, Stochastic Model for Long-Lived Species with Complex Life Histories

Large relatively long-lived species comprise a considerable portion of the biomass of many ecosystems and usually occupy key positions in their food chains, but their population dynamics are poorly understood. The deterministic Lotka-Volterra (Lotka 1925; Volterra 1926) and Verhulst-Pearl (Verhulst 1838; Pearl and Reed 1920) logistic growth models, which have formed the basis for most modern population theory, assume that population growth is only a function of population size and therefore that individuals are indistinguishable; an assumption that seems particularly inappropriate for long-lived species, which are characterized by deferred and intermittent breeding, relatively high adult survivorship, and correspondingly long maximum longevities. A further problem with long-lived species concerns overlapping biological states: Females give birth more than once and although breeding is generally synchronized among females in a population, their calving intervals (i.e., time interval between two successive conceptions) overlap. Typically, reproductive history also differs among females of the same cohort. The purpose of this paper is to present a model and the basis for a body of theory for populations of individuals with complex life histories, such as deferred and intermittent breeding, and overlapping calving intervals. We will describe this model of female life histories, give properties of the model, and discuss some biological implications of these properties of analytical results derived from the model. The first part of the paper focuses on the mathematical aspects. First we present the mathematical model for individual females and derive properties for populations characterized by time-invarying processes. The specific results we report are: a method for calculating the relationship between the change in population size and cohort-dependent characteristics such as age-specific fecundity and mortality; a method for calculating the average number of female offspring produced in the lifetime of one female, the probability of extinction of such a population and, if the population does not become extinct, its composition after