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Using these techniques, Malthus' population principle of growth was later transformed into a mathematical model known as the logistic equation: = (), where N is the population size, r is the intrinsic rate of natural increase, and K is the carrying capacity of the population.
Original image of a logistic curve, contrasted with what Verhulst called a "logarithmic curve" (in modern terms, "exponential curve") The logistic function was introduced in a series of three papers by Pierre François Verhulst between 1838 and 1847, who devised it as a model of population growth by adjusting the exponential growth model, under the guidance of Adolphe Quetelet. [5]
Thomas Malthus was one of the first to note that populations grew with a geometric pattern while contemplating the fate of humankind. [3] One of the most basic and milestone models of population growth was the logistic model of population growth formulated by Pierre François Verhulst in 1838.
As mentioned above, the logistic map can be used as a model to consider the fluctuation of population size. In this case, the variable x of the logistic map is the number of individuals of an organism divided by the maximum population size, so the possible values of x are limited to 0 ≤ x ≤ 1.
P 0 = P(0) is the initial population size, r = the population growth rate, which Ronald Fisher called the Malthusian parameter of population growth in The Genetical Theory of Natural Selection, [2] and Alfred J. Lotka called the intrinsic rate of increase, [3] [4] t = time. The model can also be written in the form of a differential equation:
Under the logistic model, population growth rate between these two limits is most often assumed to be sigmoidal (Figure 1). There is scientific evidence that some populations do grow in a logistic fashion towards a stable equilibrium – a commonly cited example is the logistic growth of yeast. The equation describing logistic growth is: [13]
For the competition equations, the logistic equation is the basis. The logistic population model, when used by ecologists often takes the following form: = (). Here x is the size of the population at a given time, r is inherent per-capita growth rate, and K is the carrying capacity.
Thomas Robert Malthus, after whom Malthusianism is named. Malthusianism is a theory that population growth is potentially exponential, according to the Malthusian growth model, while the growth of the food supply or other resources is linear, which eventually reduces living standards to the point of triggering a population decline.