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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.
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.
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 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:
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.
Verhulst developed the logistic function in a series of three papers between 1838 and 1847, based on research on modeling population growth that he conducted in the mid 1830s, under the guidance of Adolphe Quetelet; see Logistic function § History for details. [1] Verhulst published in Verhulst (1838) the equation:
Because of this structure, the model can be considered as the discrete-time analogue of the continuous-time logistic equation for population growth introduced by Verhulst; for comparison, the logistic equation is = (), and its solution is
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.