Regression Approximations for Continuous-Time, Stochastic Compartmental Models of Discrete Interacting Populations

Canadian lynx and snowshoe hare interact in a predator-prey relationship in which lynx abundance is more strongly dependent upon hare abundance than vice versa. A four-compartment stochastic compartmental model is specified that represents net balances of live individuals in each population, together with cumulative, nondecreasing balances of dead individuals of both populations. Reproduction is modeled as time-variant Poisson arrival processes. Residence times in live compartments, equivalent to ages of individuals, are modeled as conditional distributions, dependent upon abundances of the other population in each case. Models of residence-time distributions are hypothesized for each population and fitted to available data on abundance in live compartments over an 85-year period. Multiple regression techniques are used to fit the models, from which conditional distributions of population abundances in live compartments are used to compute numerical estimates of net balances from year 1850 to year 1935 and beyond. Interaction terms in the model give rise to marginal distributions of population abundances that are non-Poisson even though reproduction of both populations is assumed to be Poisson. The basic assumption about interactions is that any such prey-predator effect over a period of time of residence of two individuals of opposite species can be summarized by a function of current time alone.