In Chap. 7, we mentioned the migration model presented by Harold Hotelling in an MA thesis 1921, which has been completely forgotten, and apparently only been commented by two economists, the present author and a very close senior collaborator, professor Martin J. Beckmann. As a non-printed thesis it was not even available until Sir Alan Wilson published this very original work in Environment and Planning in 1978, but that too is now long ago and seems to have made no difference to its attraction. The model was reinvented twice, in genetics, and in mathematical ecology. Especially the latter, due to Skellam, 30 years after Hotelling, became a great success. The model combines Verhulst population growth with diffusion in space, based on migration from crowded areas. To judge from its title “A Mathematical Theory of Migration” it would seem to be highly useful for modelling today’s population movements, but modelling follows fashion, and is, unfortunately, seldom inspired by issues in society. Anyhow, the model is, like the Hotelling duopoly discussed at length, set in one dimension, and Hotelling only provided two approximate solutions—close to zero population, and close to saturation population. We again want to take it to 2D geographical space. The original model is set in continuous space and time. In order to make it conform to the format of the present book, we suggest to discretize space to a square lattice, and time into periods. Growth alone, modelled by a logistic, might cause interesting new phenomena. As we know logistic growth in continuous time can be solved in closed form, whereas its discrete version became the most studied chaos model.
- Hotelling H (1921) A mathematical theory of migration. MA thesis, University of Washington, republished in Environment and Planning A10:1223–1239Google Scholar