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A logistic function or logistic curve is a common S-shaped curve ( sigmoid curve) with the equation. where. , the value of the function's midpoint; , the supremum of the values of the function; , the logistic growth rate or steepness of the curve. [1] Standard logistic function where. For values of in the domain of real numbers from to , the S ...
Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time ...
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 ...
Like exponential growth and logistic growth, hyperbolic growth is highly nonlinear, but differs in important respects. These functions can be confused, as exponential growth, hyperbolic growth, and the first half of logistic growth are convex functions; however their asymptotic behavior (behavior as input gets large) differs dramatically:
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. The function is sometimes named Richards's curve after F. J. Richards, who proposed the general form for the family of models in 1959.
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 growth curve is used to model mean length from age in animals. [1] The function is commonly applied in ecology to model fish ...
Exponential vs. logistic growth. 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.
In probability theory and statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution in shape but has heavier tails (higher kurtosis ).