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In the present article Hotelling's model of population growth and migration of 1921 is 'revisited'. After a discussion of the stationary solutions and their stability the main point is made. The model itself is structurally unstable, but can be easily stabilized by adding a simple autonomous migration component. By this, the solution curves, in the shape of constant amplitude population waves over space for the original model, either become damped in one direction and explosive in another or are replaced by just one single spatial limit cycle.

The paper is a development of the Hotelling model of population growth and migration. In the original model, a logistic growth term is combined with linear diffusion. Since man produces his own means of subsistance and himself decides which standard of living per capita is desirable, the 'saturation' population cannot be regarded as a constant, and production technology and technological development are introduced. At the same time it is noted that the reach of migration has increased with developments in transportation. Basic nonlinearities are introduced by use of a production technology with increasing-decreasing returns to scale. It is demonstrated how industrial takeoffs, population transitions, and agglomerative spatial patterns can emerge by changing the model parameters.

A spatially extended production and exchange economy is considered. It is first established that specialization and trade arise even in the absence of increasing returns, comparative advantages, or immobile resources. Next, it is demonstrated that all trade in a multicommodity economy is transported along the same trajectories, which are determined by the land-rent gradient. Finally, applying the concept of structural stability to the land-rent gradient field, it is shown that the configuration of central places and their surrounding hinterlands must form a quadratic, not a hexagonal, pattern, where the specialization zones around each central place locally have von Thünen character.

Structural stability and structural change in a continuous two-dimensional model of a space economy are considered, and it is suggested that structural stability of flows or parametric families of flows is a reasonable assumption and secures the robustness of the models constructed. Within the format of continuous flows it is possible to characterize globally the structurally stable flows, by use of the generic theory of differential equations, and to characterize locally, the structural change of flows up to topological equivalence, by use of two-dimensional catastrophe theory.

This paper uses the concept of structural stability for dynamic systems to find out which flow patterns for goods, and hence which location patterns, are structurally stable, that is, qualitatively unaffected by perturbations caused by partial relocations of economic activity. The model used is the Beckmann continuous-space market model, generalized to a production economy and interdependent labour and goods markets. The basic stable flow pattern is quadratic, with one node surrounded by four saddles and one saddle surrounded by four nodes as the basic graph.

In continuous transportation models, minimum-cost paths are obtained by variational methods, in analogy with geometrical optics. Previous contributions on this subject have stated the appropriate Euler equations and discussed the possibility of defining a transportation-cost surface on which optimal paths are geodesics. The present paper discusses the sufficiency conditions for optimal paths and the existence conditions for cost surfaces, in terms of Gaussian theory. It is also demonstrated that, when the transportation model is isotropic, the cost surface is conformal to the plane.

In a continuous transportation model with a scalar velocity field, dependent on the ratio of traffic to transportation capacity, and with trips generated by a gravity or entropy hypothesis, the equilibrium traffic distribution is derived along with the optimal capacity distribution. Possible improvements by introducing congestion tolls are discussed. It is seen that capacity should be distributed in proportion to traffic, so that velocity of transfer is constant, trips go along straight lines, and traffic is minimized. This implies that capacity is concentrated to the central parts. Congestion tolls are superfluous when capacity is optimally allocated.

The paper deals with the problem of how to define traffic flow in a continuous transportation model. A hydrodynamic analogy with continuously distributed sources and sinks, where a continuum of vector fields represents the communication system, is used. Traffic is defined as an integral of the moduli of all these vector fields. The definition is seen to provide for conservation of traffic. The measure is compared to those obtained by counting 'cordon crossings'. It is seen that a measure obtained in this way agrees with the one proposed if the cordon crossings are weighted by the reciprocals of the cosines of the incidence angle.