Logistic Growth, Part 4

Logistics growth function. A biological population with plenty of food, space to grow, and no threat from predators, tends to grow at a rate that is proportional to the population -- that is, in each unit of time, a certain percentage of the individuals produce new individuals. If reproduction takes place more or less continuously, then this growth rate is represented by. where P is the population as a function of time t , and r is the proportionality constant. We know that all solutions of this natural-growth equation have the form. where P 0 is the population at time t = 0 . In short, unconstrained natural growth is exponential growth. Of course, most populations are constrained by limitations on resources -- even in the short run -- and none is unconstrained forever. The following figure shows two possible courses for growth of a population, the green curve following an exponential (unconstrained) pattern, the blue curve constrained so that the population is always less than some number K . When the population is small relative to K , the two patterns are virtually identical -- that is, the constraint doesn't make much difference. But, for the second population, as P becomes a significant fraction of K , the curves begin to diverge, and as P gets close to K , the growth rate drops to 0 . We may account for the growth rate declining to 0 by including in the model a factor of 1 - P/K -- which is close to 1 (i.e., has no effect) when P is much smaller than K , and which is close to 0 when P is close to K . The resulting model, is called the logistic growth model or the Verhulst model . The word "logistic" has no particular meaning in this context, except that it is commonly accepted. The second name honors P. F. Verhulst, a Belgian mathematician who studied this idea in the 19th century. Using data from the first five U.S. censuses, he made a prediction in 1840 of the U.S. population in 1940 -- and was off by less than 1%. This table shows the data available to Verhulst: