Two-State Spatial Dynamics in the Absence of Age

This paper investigates the simplest multistate population model, a one-age-group, two-living-state model with constant rates of birth, death, and interstate movement. A general solution for the model is presented, and special attention is given to the process of convergence to stability and its relationship to spatial population momentum. The constant ultimate level of the speed of convergence, or the intrinsic force of convergence, is found to be twice the difference between the two eigenvalues of the model′s transition matrix. Spatial population momentum is shown to be directly proportional to the initial growth rate of the population under the stationary rates and inversely proportional to the intrinsic force of convergence. A population increases in size during the transition to zero growth only when the faster growing state is a larger fraction of the initial population than of the stationary population. Hypothetical calculations with an urban/rural population model illustrate the importance of the speed of convergence, and show the great growth potential inherent in spatial momentum.