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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]
reproduction, where the population will increase at a rate proportional to the current population when the population size is small, starvation (density-dependent mortality), where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population.
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]
In logistic populations however, the intrinsic growth rate, also known as intrinsic rate of increase (r) is the relevant growth constant. Since generations of reproduction in a geometric population do not overlap (e.g. reproduce once a year) but do in an exponential population, geometric and exponential populations are usually considered to be ...
A Malthusian growth model, sometimes called a simple exponential growth model, is essentially exponential growth based on the idea of the function being proportional to the speed to which the function grows. The model is named after Thomas Robert Malthus, who wrote An Essay on the Principle of Population (1798), one of the earliest and most ...
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 .
When describing growth models, there are two main types of models that are most commonly used: exponential and logistic growth. When the per capita rate of increase takes the same positive value regardless of population size, the graph shows exponential growth.
K is the logistic growth rate or steepness of the curve [19] and = / The logistic growth curve depicts how population growth rate and carrying capacity are inter-connected. As illustrated in the logistic growth curve model, when the population size is small, the population increases exponentially.