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A logistic function or logistic curve is a common S-shaped curve ( sigmoid curve) with the equation. where. is the carrying capacity, the supremum of the values of the function; is the logistic growth rate, the steepness of the curve; and. is the value of the function's midpoint.
The generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves.
After dividing both sides of the equation by the population size N, in the logistic growth the left hand side of the equation represents the per capita population growth rate, which is dependent on the population size N, and decreases with increasing N throughout the entire range of population sizes.
The von Bertalanffy growth function ( VBGF ), or von Bertalanffy curve, is a type of growth curve for a time series and is named after Ludwig von Bertalanffy. It is a special case of the generalised logistic function.
The logistic growth equation is an effective tool for modelling intraspecific competition despite its simplicity, and has been used to model many real biological systems. At low population densities, N (t) is much smaller than K and so the main determinant for population growth is just the per capita growth rate.
With logistic growth, this point, called the maximum sustainable yield, is where the population size is half the carrying capacity (or ). The maximum sustainable yield is the largest yield that can be taken from a population at equilibrium.
Malthusian growth model. 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 ...
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 ...