sigmoid+curve

=Sigmoid curve=

toc Read here about the sigmoid curve, a quantitative pattern that is found in many phenomena and is a critical concept to comprehending scale.

=Overview=

The Sigmoid curve is a generalized curve that depicts a progression from small beginnings that accelerate to approach a climax over time.

Parts of the sigmoid curve resemble a progress curve, see progress curve.

=Logistic Function=

Often, sigmoid function refers to the special case of the logistic function which is defined by the formula Curve P(t)=1 over 1+e^-t





Sigmoid Curve, public domain.

Makela explains it this way:

During a natural process a j -species populates along a sigmoid course (solid line). Initially, when consuming nascent resources of free energy Aj (dotted line), the specific species Nj will populate its surroundings nearly exponentially. Soon, when the source of free energy begins to narrow, the growth will turn to follow a power law (central section). Eventually when remains of the free energy are being extracted, the growth declines exponentially. The overall course follows closely a logistic curve. The rate of change dNj /dt of the cumulative curve is approximately log-normal (dashed line).

From Makela 2010: "Natural patterns of energy dispersal" by Teemu Mäkelä and Arto Annila in Physics of Life Reviews 7 (2010) 477–498.

=Hockey Stick= The portion of a sigmoid, power, or growth curve that shows a steep rise in the multiplied numbers is commonly called a “hockey stick”, popularized by Al Gore in an Inconvenient Truth. The hockey stick is a function of exponential growth.

=r/K selection theory=

From Wikipedia:

In ecology, r/K selection theory relates to the selection of combinations of traits in an organism that trade off between quantity or quality of offspring. The focus upon either increased quantity of offspring at the expense of individual parental investment, or reduced quantity of offspring with a corresponding increased parental investment, is varied to promote success in particular environments. The theory was popular in the 1970's and 1980's when it was used as a heuristic device, but lost importance in the early 1990's as it was criticized by several empirical studies. The r/K selection paradigm has been replaced by a "life-history" paradigm. However, this continues to incorporate many of the themes important to the r/K paradigm. The terminology of r/K-selection was coined by the ecologists Robert MacArthur and E. O. Wilson based on their work on island biogeography. In r/K selection theory, selective pressures are hypothesised to drive evolution in one of two generalized directions: r- or K-selection. These terms, r and K, are drawn from standard ecological algebra, as illustrated in the highly simplified Verhulst equation of population dynamics:



where r is the maximum growth rate of the population (N), and K is the carrying capacity of its local environmental setting. As the name implies, r-selected species are those that place an emphasis on a high growth rate, and typically exploit less-crowded ecological niches and produce many offspring, each of which has a relatively low probability of surviving to adulthood (i.e. high r, low K). By contrast, K-selected species display traits associated with living at densities close to carrying capacity, and typically are strong competitors in such crowded niches that invest more heavily in fewer offspring, each of which has a relatively high probability of surviving to adulthood (i.e. low r, high K). In the scientific literature, r-selected species are occasionally referred to as "opportunistic", while K-selected species are described as "equilibrium".