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Alfred J. Lotka | A Wisdom Archive on Alfred J. Lotka | | Alfred J. Lotka A selection of articles related to Alfred J. Lotka | |
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| | | Alfred J. Lotka | | |
| ARTICLES RELATED TO Alfred J. Lotka | |
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| | |  Alfred J. Lotka: Encyclopedia II - Logistic function - The logistic functionThe logistic function is defined by the mathematical formula:
for real parameters a, m, n, and τ. These functions find applications in a range of fields, from biology to economics.
For example, in the development of an embryo, a fertilized ovum splits, and the cell count grows: 1, 2, 4, 8, 16, 32, 64, etc. This is exponential growth. But the fetus can grow only as large as the uterus can hold; thus other factors start slowing down the increase in the ...
See also:Logistic function, Logistic function - The logistic function, Logistic function - The Verhulst equation, Logistic function - Sigmoid function, Logistic function - Properties of the sigmoid function, Logistic function - History, Logistic function - Critics Read more here: » Logistic function: Encyclopedia II - Logistic function - The logistic function |
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| | | | Alfred J. Lotka: Encyclopedia II - Lotka-Volterra equation - Dynamics of the system In the model system, the predators thrive when there are plentiful prey but, ultimately, outstrip their food supply and decline. As the predator population is low the prey population will increase again. These dynamics continue in a cycle of growth and decline.
Lotka-Volterra equation - Population equilibrium.
Population equilibrium occurs in the model when neither of the population levels are changing, i.e. when both of the differential equations are equal to 0.
x(α − βy) = 0
See also:Lotka-Volterra equation, Lotka-Volterra equation - The equations, Lotka-Volterra equation - Physical meanings of the equations, Lotka-Volterra equation - Prey, Lotka-Volterra equation - Predators, Lotka-Volterra equation - Solutions to the equations, Lotka-Volterra equation - Dynamics of the system, Lotka-Volterra equation - Population equilibrium, Lotka-Volterra equation - Stability of the fixed points, Lotka-Volterra equation - Bibliography Read more here: » Lotka-Volterra equation: Encyclopedia II - Lotka-Volterra equation - Dynamics of the system |
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| | | | Alfred J. Lotka: Encyclopedia II - Logistic function - Sigmoid function The special case of the logistic function with a = 1,m = 0,n = 1,τ = 1, namely
is called sigmoid function or sigmoid curve. The name is due to the sigmoid shape of its graph. This function is also called the standard logistic function and is often encountered in many technical domains, especially in artificial neural networks as a transfer function, probability, statistics, biomathematics, and economics.
Logisti ...
See also:Logistic function, Logistic function - The logistic function, Logistic function - The Verhulst equation, Logistic function - Sigmoid function, Logistic function - Properties of the sigmoid function, Logistic function - History, Logistic function - Critics Read more here: » Logistic function: Encyclopedia II - Logistic function - Sigmoid function |
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| | | | Alfred J. Lotka: Encyclopedia II - Lotka-Volterra equation - Physical meanings of the equations When multiplied out, the equations take a form useful for physical interpretation.
Lotka-Volterra equation - Prey.
The prey equation becomes:
The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growth is represented in equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by β ...
See also:Lotka-Volterra equation, Lotka-Volterra equation - The equations, Lotka-Volterra equation - Physical meanings of the equations, Lotka-Volterra equation - Prey, Lotka-Volterra equation - Predators, Lotka-Volterra equation - Solutions to the equations, Lotka-Volterra equation - Dynamics of the system, Lotka-Volterra equation - Population equilibrium, Lotka-Volterra equation - Stability of the fixed points, Lotka-Volterra equation - Bibliography Read more here: » Lotka-Volterra equation: Encyclopedia II - Lotka-Volterra equation - Physical meanings of the equations |
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| | | | Alfred J. Lotka: Encyclopedia II - Logistic function - Critics Despite its persistent popularity, the logistic function has been heavily criticised in the field of population dynamics. One such critic is demographer, and Professor of Population, Joel E. Cohen (How Many People Can The Earth Support, 1995). Cohen explains that Verhulst attempted to fit a logistic curve based on the logistic function to 3 separate censuses of the population of the United States of America in order to predict fut ...
See also:Logistic function, Logistic function - The logistic function, Logistic function - The Verhulst equation, Logistic function - Sigmoid function, Logistic function - Properties of the sigmoid function, Logistic function - History, Logistic function - Critics Read more here: » Logistic function: Encyclopedia II - Logistic function - Critics |
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| | | | Alfred J. Lotka: Encyclopedia II - Lotka-Volterra equation - The equations The usual form of the equations is:
where
y is the number of some predator (for example, dingoes);
x is the number of its prey (for example, wallabies);
t represents the growth of the two populations against time; and
α, β, γ and δ are parameters representing the interaction of the two species.
...
See also:Lotka-Volterra equation, Lotka-Volterra equation - The equations, Lotka-Volterra equation - Physical meanings of the equations, Lotka-Volterra equation - Prey, Lotka-Volterra equation - Predators, Lotka-Volterra equation - Solutions to the equations, Lotka-Volterra equation - Dynamics of the system, Lotka-Volterra equation - Population equilibrium, Lotka-Volterra equation - Stability of the fixed points, Lotka-Volterra equation - Bibliography Read more here: » Lotka-Volterra equation: Encyclopedia II - Lotka-Volterra equation - The equations |
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| | | | Alfred J. Lotka: Encyclopedia II - Logistic function - The Verhulst equation A typical application of the logistic equation is a common model of population growth, which states that:
the rate of reproduction is proportional to the existing population, all else being equal
the rate of reproduction is proportional to the amount of available resources, all else being equal. Thus the second term models the competition for available resources, which tends to limit the population growth.
Letting P represent population size (N is often used in ecology instead) and t represent time, this model is fo ...
See also:Logistic function, Logistic function - The logistic function, Logistic function - The Verhulst equation, Logistic function - Sigmoid function, Logistic function - Properties of the sigmoid function, Logistic function - History, Logistic function - Critics Read more here: » Logistic function: Encyclopedia II - Logistic function - The Verhulst equation |
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| | | | Alfred J. Lotka: Encyclopedia II - Logistic function - Sigmoid function The special case of the logistic function with a = 1,m = 0,n = 1,τ = 1, namely
is called sigmoid function or sigmoid curve. The name is due to the sigmoid shape of its graph. This function is also called the standard logistic function and is often encountered in many technical domains, especially in artificial neural networks as a transfer function, probability, statistics, bio ...
See also:Logistic function, Logistic function - The logistic function, Logistic function - The Verhulst equation, Logistic function - Sigmoid function, Logistic function - Properties of the sigmoid function, Logistic function - History, Logistic function - Critics Read more here: » Logistic function: Encyclopedia II - Logistic function - Sigmoid function |
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