## Info

The word logistic is derived from the French word logistique, which means to compute. The scientist and mathematician Verhulst wanted to be able to compute the population trajectory of France. He knew that using the exponential growth equation dN/dt = rN would not work because the population grows without bound. This happens because with exponential growth the per capita growth rate is a constant (r). We don't know what Verhulst was thinking, but it might have gone something like this: "I know that a constant per capita growth rate will not be a good representation, and it must be true that per capita growth rate declines as population size increases. Suppose that per capita growth rate falls to zero when the population size is K. What is the simplest way to connect the points (0, r) and (K, 0)? Of course - a line. C'est bon.'' Furthermore, there is only one line that connects the maximum per capita growth rate r when N= 0 and per capita growth rate = 0 when N= K. There are an infinite number of nonlinear ways that we could do it. For example, a per capita growth rate of the form r(1 — (N/K)a), for any value of a > 0, works equally well to achieve the goal of connecting the maximum and zero per capita growth rates. So, the logistic is not a law of nature, but is a simple and somewhat unique representation of nature. In Figure 2.7b, I show the per capita growth rate for the same parameters as in Figure 2.7a.

Let us now think about the dynamics of a population starting at size N(0) and following logistic growth. If N(0) > K, then the growth rate of the population is negative and the population will decline towards K. If N(0) > 0 but small, the population will grow, albeit slowly at first, but then as population size increases, the growth rate increases too (even though per capita growth rate is always declining, the product of per capita growth rate and population size increases until N = K/2). Once the population size exceeds K/2, growth rate begins to slow, ultimately reaching 0 as the population approaches K. We thus expect the picture of population size versus time to be S-shaped or sigmoidal and it is (Figure 2.7c).

Although Eq. (2.26) is a nonlinear equation, it can be solved exactly (that is how I generated the trajectories in Figure 2.7c) and everyone should do it at least once in his or her career. The exercise is to show that the solution of Eq. (2.26) is N(t) = [N(0)Kert] / [K + N(0)(ert — 1)]. To help you along, I offer two hints (the method of partial fractions, if you want to check your calculus text). First, separate the differential equation so that Eq. (2.26) becomes dN

Second, recognize that the left hand side of this expression looks like a common denominator, so write

0 0