3-Geometry-Plane-Curve-Kinds-Logistic

logistic curve

Curves {logistic curve} can model growth with growth factors and limiting resources. Growth can depend on growth factors, which can have weights. Limiting resources {bottleneck, growth} can slow growth.

Amount or percentage P at time t is P(t) = A * (1 + m * e^(-t/T)) / (1 + n * e^(-t/T)), where A = constant growth-factor weight, m = original growth-factor amount, n = original limiting-resource amount, and T = time period or inverse growth rate.

Growth is percentage of factor to resource. Denominator and numerator decrease at same rate.

beginning

Original amount depends on relative m and n amounts and on weight A: P(0) = A * (1 + m) / (1 + n). If n is much greater than m and A, denominator is large, and P is zero.

process

At first, growth is exponential with time. Then growth passes through time when growth has constant rate and is linear. Then growth slows exponentially to zero. Amount is then maximum or 100 percent at maturity.

shape

Logistic curve has sigmoid shape.

comparison

Logistic function inverts natural logit function.

logit curve

For numbers between 0 and 1, functions {logit curve} can be logit(p) = log(p / (1 - p)) = log(p) - log(1 - p). Logarithmic base must be greater than 1, for example, 2 or e. p / (1 - p) is the odds, so logit is logarithm of odds. Logistic function inverts natural logit function. Logit functions can be linear {logit model}: logit(p) = m*x + b. Logit models {logistic regression} can be for linear regression.

probit curve

Inverse cumulative distribution functions, or normal-distribution quantile functions, depend on error-function inverses {probit curve}. It changes probability into function over real numbers: probit(p) = 2^0.5 * erf^-(2*p - 1). Probit functions can be linear over a large real-number middle range {probit model}.

sigmoid curve

Logistic curve can look like S {sigmoid curve}| {standard logistic function}. Sigmoid curve starts at minimum or maximum, always increases or decreases, and ends at maximum or minimum. It has one inflection point, near which it grows linearly. It changes exponentially at beginning and end. Amount or percentage P over time t is P(t) = A * (1 + m * e^(-t/T)) / (1 + n * e^(-t/T)). If A = 1, m = 0, n = 1, and T = 1, P(t) = 1 / (1 + e^-t). dP/dt = P * (1 - P), with P(0) = 1/2 and dP(0) / dt = 1/4.

Arctangent, hyperbolic tangent, and error function make sigmoid curves.

double sigmoid curve

Curves {double sigmoid curve} can look like double S. Double sigmoid curves start at minimum or maximum, always increase or decrease, and end at maximum or minimum. They have two inflection points, near which it grows linearly. Growth rate is zero in middle. Double sigmoid curves change exponentially at beginning and end and near middle. Amount or percentage N over variable x is N(x) = (x - d) * (1 - exp(-((x - d) / s)^2)), where d is average, and s is standard deviation.

Verhulst curve

Logistic curves {Verhulst curve} can have growth rate directly related to current percentage or total amount and directly related to current resource amount. dN / dt = r * N * (1 - N/K), where N is total population, r is growth rate, and K is maximum population possible {carrying capacity, logistic}. Amount or percentage N over time t is N(t) = (K * N(0) * e^(r*t)) / (K + N(0) * (e^(r*t) - 1)), which comes from N(t) = A * (1 + m * e^(-t/T)) / (1 + n * e^(-t/T)), where A ~ 1, m ~ N(0), n ~ N(0) / K, and T = -1/r.

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Date Modified: 2022.0225