## Diffusion and logistic population growth invasions the Fisher equation and traveling waves

We conclude this chapter with a short introduction to a complicated topic, and one that comes the closest to pure mathematics yet - we are going to show that a solution to a question exists, but we are not going to actually find the solution. By way of motivation, we begin with the empirical phenomenon.

In Figure 2.21a, I show the spatial distribution of the variegated leafhopper (VLH, Erythronewra varzabz'/zs) which is a pest of grapes in California (Settle and Wilson 1990), during an invasion in which E. varzaMz's more or less replaced a congener, the grape leafhopper E. e/egantw/a. Note that in 1985, the proportion of VLH was 1 for distances less than about 3 km and dropped to 50% at about 5 km. However, in 1986 these respective distances are about 7 km Distance (km)

Figure 2.21. (a) The invasion pattern of a leafhopper in California (after Settle and Wilson 1990; with permission). Note that 1986 pattern is similar to that of 1985, only shifted to the right. We call this a traveling wave (of invasion). (b) A caricature of the traveling wave of invasion for three different times. Our goal is to understand how diffusion and logistic population growth combine to move the initial profile to the right.

Distance (km)

Figure 2.21. (a) The invasion pattern of a leafhopper in California (after Settle and Wilson 1990; with permission). Note that 1986 pattern is similar to that of 1985, only shifted to the right. We call this a traveling wave (of invasion). (b) A caricature of the traveling wave of invasion for three different times. Our goal is to understand how diffusion and logistic population growth combine to move the initial profile to the right.

and 20 km; it is as if the entire 1985 graph had shifted to the right. Figure 2.21b is an idealized example of this phenomenon. The abscissa is space and the ordinate is a function u(x, t), which will become explicit in a moment, shown at three different times - the subsequent times have the same shape, but translated to the right. This kind of spatial-temporal behavior is called a traveling wave.

R. A. Fisher thought a lot about this question in the context of the spread of an advantageous allele. That is, imagine a single locus with two alleles, a and A. Assume that A is more fit, but that the population is initially mainly a; thus the fitness of the genotype AA is greater than that of the genotype aa and heterozygotes are somewhere in between. Suppose that we denote the frequency of A by u(t). Then, in the absence of spatial effects, the dynamics of u(t) are du

where s > 0 is the selection coefficient, a function of the fitnesses of the different genotypes AA, Aa and aa. As long as u(0) > 0, so that some of the advantageous allele is present, we see that u(t) will grow logistically towards 1.

Fisher modified the dynamics given by Eq. (2.60) to include space by assuming that there was undirected diffusion in space that accompanied the logistic growth in time. Hence the resulting equation would be u = y uxx + su(1 — u) (2.61)

We will consider an infinite spatial domain, but defer for a bit the discussion of boundary conditions. For the initial condition, we assume u(x, 0) similar to the profile in Figure 2.21b.

To simplify Eq. (2.61) (and to show how exactly the same equation arises in the discussion of invading organisms, rather than invading genes), we will begin by scaling variables. First, divide both sides of the equation by s. The left hand side is now (1/s)(Su/St), so that if we defined a new time variable by t' = st, the left hand side would be Su/St'. After division by s, the first term on the right hand side of Eq. (2.61) will be (o-2/2s) (S2u/Sx2), so that if we define a new space like variable by y = \J(2s/^2)x the entire equation will become u/ = uyy + u(1 — u). Understanding that we are using scaled variables, we can thus just as easily consider the equation ut = uxx + u(1 — u) which is called the Fisher equation.

Exercise 2.16 (E)

The model for logistic population growth and non-directed diffusion of an invading organism would be Nt = (a2/2)Nxx + rN[1 — (N/K)]. What scalings are needed to convert this to the same form as the Fisher equation (2.62)?

Now a traveling wave, such as shown in Figure 2.21b, keeps its shape as time changes but is displaced. Thus, at some time t, if we want to know the value of u(x, t), we ask for the corresponding value of u at the initial time, but at a spatial point that is moved backwards from x. If the wave is traveling at speed c > 0, then to reach the point x at time t, it had to start at x — ct at time 0. Thus u(x, t) is only a function, let's call it U, of the combination x — ct, which we will call r. In symbols, we write that u(x, t) = U(r), where r = x — ct. Then the chain rule tells us Su/St = (dU/dr)(Sr/St) = —c(dU/dr) and 82u/8x2 = d2U/dr2. We thus are able to convert Eq. (2.62) from a partial differential equation for u(x, t) to an ordinary differential equation for U(t):