Logistic Growth

The starting point for describing the evolution of a renewable resource stock is the logistic growth function. Using t to denote time, a simple logistic growth function has the form G(t) = rS(1—S/K). The variable r is the intrinsic growth rate and K is the environmental carrying capacity, or maximum pos­sible size of the resource stock. G(t) is the growth rate defined in biomass units and G/S is the proportional growth rate (i. e., a number such as 0.1 or 10%). If the forest stock S reaches a level equal to carrying capacity K, then G = 0 and no further growth occurs. Similarly, if there is no forest stock, then S = 0 and no growth occurs. If the forest stock is small relative to carrying capacity (but positive), then S/K is negligible and G = rS. In this case, proportional growth G/S approaches r. Thus, we can think of r as the proportional growth rate in the absence of congestion effects. As the forest stock increases, there is congestion (competition for space, rainfall, sunlight, etc.), with the result that the proportional growth rate tends to decline. However, the base to which that growth rate is applied increases as S increases. Therefore, the absolute growth rate in biomass units increases with S up to a critical point beyond which the congestion effect dominates the increasing base effect and the absolute growth rate declines. A logistic growth function is illustrated in Fig. 1. It is clear from Fig. 1 that the maximum sustainable yield occurs at the maximum

Logistic Growth

Stock

FIGURE 1 Logistic growth.

growth rate. In other words, it would be possible to maintain a stock of K/2 and harvest at this maximum growth rate in steady state.

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