The Calculus Diaries - Jennifer Ouellette [57]
Maybe it had something to do with being one of eight children, but Malthus’s proposed solution to overpopulation included restricting the family size of the lower classes to ensure that parents did not produce more children than they could support. It sounds more elitist than he perhaps intended: Malthus thought having too many children doomed the lower classes to poverty, making it impossible for them to rise above those conditions and improve their lot in life. Then again, he also flirted with the notion of eugenics—a term not coined until 1883—by pondering whether the techniques of animal husbandry might be applied to breed out undesirable qualities in people, although he didn’t feel this was a realistic goal: “As the human race, however, could not be improved in this way without condemning all the bad specimens to celibacy, it is not probable that an attention to breed should ever become general.” (Yes, even in the 1800s, people realized that abstinence alone is not a viable solution for family planning.)
This sort of pessimistic thinking did not win Malthus any popularity contests at a time when fervent social reformers preached the gospel of erasing all the ills of man if only one could implement the proper social structures. The model is not without merit, but the Malthusian growth equation is only applicable under specific conditions, such as scientists growing bacteria in a lab in a perfectly controlled environment. Even then, while growth occurs exponentially for a time, it does not continue forever.
It fell to his contemporary, the Brussels-born Pierre Verhulst, to improve upon the basic idea by devising a more sophisticated model that more accurately reflected real-world population dynamics. Verhulst said there are forces at work to prevent exponential growth in the population, and these forces increase in direct proportion to the ratio of the excess population to the total population. In other words, population growth depends not just on the size of the population, but also on how far that size is from its upper limit.
The crux is something called carrying capacity (which we can denote by K): the maximum population size that any given habitat can support. If the population of ants starts to double every year—grows exponentially—there will be two thousand ants the first year, and the next year there will twice as many. But there is a limited supply of food and other resources, so if that exponential growth rate continues unchecked, the population of ants will rapidly consume all the available resources. Exponential growth simply cannot be sustained indefinitely. Once the food runs out, the ants will begin to die out too. Verhulst’s model34 shows that if the population is less than the maximum, the population will increase rather steeply because people have lots of food. But then when it gets closer to the maximum sustainable population, the rate at which the population increases slows down.
Look closely and you will recognize the telltale sign of a derivative. In the Verhulst model, the derivative of the population with respect to time—that is, the rate of change in population ( p) or the number of additional people over time—would be proportional to the number of people at time 0 (now) multiplied by the maximum sustainable population minus the current population. If the population actually followed that equation, it would start out low and show exponential growth at first. But then the rate of growth would begin to slow as it approached the maximal population, eventually leveling off to become stable