Regulated growth: growth curve, rate of increase, and capacity limit
A population can never grow indefinitely, because no indefinite amounts of nutriments, energy, and habitat exist. Each habitat has a capacity (K), that can in the most simple case be regarded as a constant. K equals accordingly the highest possible number of individuals in the respective habitat.
As long as the number N is small if compared to K, the population can grow unrestricted. As soon as N begins to equal K, the growth rate decreases until N and K have reached an equilibrium. Such a growth function can mathematically be expressed as
dN / dt = rN(K - N) / K
or
(dN / N) + (dN / K - N) = r dt
The growth rate is the first differentiation of the growth curve. It shows that r has a maximum value. Integration leads to
t = [1/ r] ln [Nt (K - N0)] / [(K - Nt)N0]
An example will explain things a little: The time it takes a population to grow from N0 = 1,000 to Nt = 2,000 individuals takes 98 years under an assumed population increase of 1 percent per year (r = 0.01) and a habitat capacity of K = 5,000 individuals:
t = 1 / 0,01 ln (2000 x 4000) / (3000 x 1000) = 98 years
Neglecting the limit of the capacity, the time is
t = 1 / r ln Nt / N0
leading to a result of 69 years in our example.
During the course of the evolution of organisms, two fitness-increasing strategies developed: the r- and the K-strategy. The r-strategy is characterized by a high rate of propagation. It occurs especially with species specialized on colonizing new habitats with variable conditions or with species with strongly fluctuating population sizes. The K-strategy, in contrast, describes a regulated, density-dependent propagation in view of the capacity limit of the habitat K. It occurs in species living in stable habitats, where a high rate of propagation is of no advantage. It is regarded as more progressive than the r-strategy in an evolutionary sense. In nature, all conceivable transitions between these two extremes occur. A given species will therefore mainly adopt one strategy, even though shares of the other strategy cannot be overlooked. Sometimes, extern circumstances like unpredicted changes of the living conditions trigger a change from one strategy to the other.
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