This paper presents an approach to modelling population density as a function of space and time, both of which are treated over a continuous domain. An equilibrium (or alternatively, an optimal) distribution of urban population density is specified exogenously. The rate of change of the actual population density is then considered to be a function of the deviations between the actual and the equilibrium (or optimal) density distributions, over time. This adjustment process is interpreted as being realized by internal migration, immigration/emigration, and/or natural growth. A formal specification of the models results in integral—differential equations. A solution procedure that uses double Laplace transforms and partial Laplace transforms with respect to space and time is outlined. Properties relating to the limiting distributions are discussed. An example that makes use of exponential functions is presented; and procedures for computing explicit, and possibly heuristic, solutions are noted.
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