In 1951, Skellem suggested that the spread of a population was similar to the spread of a chemical component via diffusion [13]. Let's consider the spread of Native Americans across North America. For simplicity we assume one-dimensional diffusion of the population across the plane of North America from west to east. The population spreads via diffusion with a constant value of diffusivity, \(D\). We account for births, \(b\), and deaths, \(d\), by assuming that they are akin to chemical reactions that are first order in the population density, \(P\), with some reaction constants, \(k_{b}^{\prime \prime}\) and \(k_{d}^{\prime \prime}\).

a. What is the differential equation describing the population density, \(P\) ?

b. Assuming an initial, exploratory, population, \(P_{o}\), at \(x=0 ; t=0\), and a slow spread of that population across the continent, what are the boundary conditions for this problem?

c. Show that the following solution satisfies the differential equation.
\[P(x, t)=\frac{P_{o}}{\sqrt{4 \pi D t}} \exp \left(k^{\prime \prime} t-\frac{x^{2}}{4 D t}\right) \quad k^{\prime \prime}=k_{b}^{\prime \prime}-k_{d}^{\prime \prime}\]

d. If we consider regions of equal population density, we can determine the speed at which the population spreads. Show that if \(P_{i}\) is a particular value of the population, such a contour is represented by:
\[\frac{x}{t}= \pm\left[4 k^{\prime \prime} D-\frac{2 D}{t} \ln (4 \pi D t)-\frac{4 D}{t} \ln \left(\frac{P_{i}}{P_{o}}\right)\right]^{1 / 2}\]

e. Demonstrate that for long times the solution in part

(d) simplifies to a constant speed given by:
\[\frac{x}{t}= \pm \sqrt{4 k^{\prime \prime} D}\]

f. If we assume it took 1000 years for humans to spread from Alaska to Tierra del Fuego, what was their speed and hence \(k^{\prime \prime} D\) ?