US Colonization Invasions Through Fractal River Networks

6 US Colonization Invasions Through Fractal River Networks 

An essential characteristic of the US transition westwards was the fact that settlers did not occupy all of the territory, as homogeneous models assume [137], but followed the course of the major rivers and lakes and settled near them to make use of their resources [113,133]. Therefore, landscape heterogeneities should have played an essential role in the process of migration. This situation is similar to the case of dispersion of biological species along the margins of rivers and streams [220,63]. 

Great efforts have been undertaken to describe the intricate geometry of river networks [363]. In Chap. 6 we illustrated some models, such as the Peano network, which resembles fractal river basins. Another well-known structure that was proposed to describe better the evolution and formation of river basins and that agrees with most of the observations is the Optimal Channels Network (OCN) model, based on some optimization principles that minimize the energy expenditure by the net- 

The jump distance distribution w(x) for an isotropic random walk across the backbone is w x) = jS(x + 1) + S(x — 1). Equation (5.30) yields the Hamilton-Jacobi equation for the CTRW model  

To estimate the velocity of human migration one can consider r to be the time between successive generations. A value of r = 25 yr is usually assumed [137]. The growth parameter r can be obtained directly from [256], where the population vs time plot for the United States in the 19th century was fitted to a logistic curve, obtaining r = 0.031 0.001 yr  

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