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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.
The logistic function can be calculated efficiently by utilizing type III Unums. [8] An hierarchy of sigmoid growth models with increasing complexity (number of parameters) was built [9] with the primary goal to re-analyze kinetic data, the so called N-t curves, from heterogeneous nucleation experiments [10], in electrochemistry.
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.
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]
F(X) is the instantaneous proliferation rate of the cellular population, whose decreasing nature is due to the competition for the nutrients due to the increase of the cellular population, similarly to the logistic growth rate. However, there is a fundamental difference: in the logistic case the proliferation rate for small cellular population ...
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.
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: =
The Hubbert curve [2] is the first derivative of a logistic function, which has been used for modeling the depletion of crude oil in particular, the depletion of finite mineral resources in general [3] and also population growth patterns. [4] Example of a Hubbert Linearization on the US Lower-48 crude oil production.