Diffusion and exponential population growth

We now consider how diffusion and population growth interact. That is, instead of simply exponential population growth in time or diffusion in space, we consider population size N(x, t) depending upon both spatial point x and time t. This population will be characterized by the equation where I have suppressed the dependence of N on x and t. As before, we will require an initial condition and boundary conditions. We will assume that N(x, 0) is specified and that the population is confined to a region [0, L]. In that case the appropriate boundary conditions are

Before doing any mathematics, let us spend time thinking about Eq. (2.57). We begin with a profile of population size in time, N(x, 0). One such a profile (made up by me) is shown in Figure 2.19a. If we were to describe this profile, we might say that there is a cline of increasing population size, with some small deviations from what looks to be a straight line. It is those deviations that we are interested in learning about, so to focus on them we define the average population size by N = (1/L) J^ N(x, 0)dx and the deviation n(x, 0) by n(x, 0) = N(x, 0)— N In Figure 2.19b, I show the scaled value of n(x, 0), scaled by the average (that is, I am plotting n(x, 0)/N).

We know that if the population started out completely homogeneous in space with initial value NV, then its size at any later time would be Nert, so let us define n(x, t) = N(x, t)— Nert. We already know n(x, 0) and since the boundary conditions for N(x, t) involve derivatives, we have the same boundary conditions for n(x, t). Regarding the equation that n(x, t) satisfies, see Exercise 2.15.

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