Since the rate goes to zero as N goes to K, the population cannot increase beyond K (except by immigration). Thus, K, called the carrying capacity, reflects the inherent ability of the ecosystem to support the population. Both K and r0 can vary with environmental conditions; for example, they might decrease in time of drought. When the rate equation (14.24) is substituted into the growth equation (14.17), we get the important model of population growth in the presence of a limiting factor, called the logistic equation:

Figure 14.15 Logistic equation solution (14.26) with several parameter values, compared to exponential growth equation (14.18). The dashed line is N — 100. (a) logistic equation with ro — 1.0, K — 100, N(0) — 5.0; (b) logistic equation with ro — 0.75, K — 100, N(0) — 5.0; (c) logistic equation with r0 — 1.0, K — 70, N(0) — 5.0; (d) logistic equation with r0 — 1.0, K — 100, N(0) — 150; (e) exponential equation with r0 — 0.7, N(0) — 5.0.

Figure 14.15 Logistic equation solution (14.26) with several parameter values, compared to exponential growth equation (14.18). The dashed line is N — 100. (a) logistic equation with ro — 1.0, K — 100, N(0) — 5.0; (b) logistic equation with ro — 0.75, K — 100, N(0) — 5.0; (c) logistic equation with r0 — 1.0, K — 70, N(0) — 5.0; (d) logistic equation with r0 — 1.0, K — 100, N(0) — 150; (e) exponential equation with r0 — 0.7, N(0) — 5.0.

This equation has the following analytical solution for N at time t, N(t), based on the initial population N(0):

The solution has a sigmoid shape if N(0) < K/2 (Figure 14.15), sometimes referred to by ecologists as the S-shaped growth curve. This distinguishes the logistic growth curve from exponential growth, which is called the J-shaped growth curve (Figure 14.15e). Exponential growth is essentially like logistic growth but with an infinite carrying capacity. Curves (b) and (c) show the effect of varying the parameters, and curve (d) shows what happens if the initial population is greater than the carrying capacity.

Compare the logistic equation with the Monod model for microorganism growth [equation (11.6), Section 11.7.2]. The Monod model specifically includes the limiting resource needed for growth (substrate, S). In the Monod model, exponential growth is possible at any resource level as long as the resource concentration is held constant. Growth limitation will occur only if the disappearance of the resource is modeled separately. In the logistic equation, the organisms in a population are assumed to have to compete with each other for the resources, so as population increases, the growth rate decreases. The total amount of resource available to the population is assumed to be held constant.

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