On growth and dispersal of populations

In his unpublished MA thesis of 1921, the young Hotelling invented an ingenious model of population growth and diffusion. The contribution remained widely unknown until Waldo Tobler and Alan Wilson edited it as an article in 1978. Not even then did it trigger off any outburst of contributions. Meanwhile, in 1951, Skellam invented exactly the same model for non-human populations. Unlike in economics, it was a great success in ecology. In a recent contribution Skellam is named Father of Ecological Diffusion and more than a thousand articles on the subject are listed. There were many attempts at solving the equation, but none were quite successful. The Hotelling and Skellam models both assumed a given saturation population that nature could support. In 1985 the present author suggested that an explicit production function be introduced in the Hotelling model, as man produces his own means of subsistence. Moreover it was proposed that diffusion be related to spatial differences in per capita productivity, rather than to those in population density. The present contribution is a rejoinder. It is shown that the stationary solutions to the original Hotelling model are periodic and dip into negative populations, whereas this is avoided by the modified model suggested. It can thus be used to show how agglomerative structures may evolve.