Extending the reach of structured demographic models in ecology: Stochasticity and heterogeneity as sources of demographic variability

Background/Question/Methods

The fates of individuals differ. The difference may arise from heterogeneity in individual properties (e.g., age, size, developmental stage) or from randomly differing outcomes of stochastic demographic events (living or dying, reproducing or not).  As demographic analysis has developed from its age-classified roots in the works of Lotka, Pearl, and Leslie, it has extended its reach to include stochasticity, heterogeneity, the resulting variability among individuals, and sensitivity analysis of all these.  We are confronted by two important questions: how to deduce the individual stochasticity implied by a set of demographic rates, and how to incorporate hidden individual heterogeneity into demographic models.

Results/Conclusions

A model for individual stochasticity is obtained by describing the life cycle as a Markov chain and using this formulation to calculate the statistical properties of individual lives, represented as trajectories through the state space of the Markov chain. By associating "rewards" with the trajectories, we obtain a very general analytical solution for the mean, variance, and skewness of lifetime reproductive output, as well as other demographic outcomes.

 A model for studying heterogeneity among individuals is obtained by including extra dimensions in the individual state space.  Measurable heterogeneity is regularly incorporated into the i-state space. When this is done, differences due to size are incorporated into the size structure, and so on.  However, demographic analysis also confronts the issue of cryptic heterogeneity, differences that are unmeasurable, or at least unmeasured. Such heterogeneity goes by names such as frailty, quality, vitality, or fecundability.  When cryptic heterogeneity is incorporated into demographic models, as an extra i-state dimension, it becomes possible to separate the effects of individual stochasticity and heterogeneity. The resulting models are multistate matrix population models and the multistate Markov chains derived from them.

 I will present the demographic models for individual stochasticity and cryptic heterogeneity, and use these to compare the variance in longevity between human populations and laboratory population of invertebrate animals; in the former, individual stochasticity accounts for about 95% of the variance; in the latter, only about 50%. I will also compare observed and calculated standardized variance in lifetime reproductive output in vertebrate species and show that, in most cases, stochasticity can account for most or all of the observed variance.