Difference between revisions of "Carrying Capacity and Logistic Growth Rate"

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* [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(OpenStax)/08%3A_Introduction_to_Differential_Equations/8.4%3A_The_Logistic_Equation The Logistic Equation], Mathematics LibreTexts
 
* [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(OpenStax)/08%3A_Introduction_to_Differential_Equations/8.4%3A_The_Logistic_Equation The Logistic Equation], Mathematics LibreTexts
 
* [https://courses.lumenlearning.com/boundless-biology/chapter/environmental-limits-to-population-growth/ Environmental Limits to Population Growth], Lumen Learning
 
* [https://courses.lumenlearning.com/boundless-biology/chapter/environmental-limits-to-population-growth/ Environmental Limits to Population Growth], Lumen Learning
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== Licensing ==
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Content obtained and/or adapted from:
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* [https://en.wikipedia.org/wiki/Logistic_function Logistic function, Wikipedia] under a CC BY-SA license

Latest revision as of 15:10, 24 October 2021

Standard logistic sigmoid function where

A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation

where

, the value of the sigmoid's midpoint;
, the curve's maximum value;
, the logistic growth rate or steepness of the curve.

For values of in the domain of real numbers from to , the S-curve shown on the right is obtained, with the graph of approaching as approaches and approaching zero as approaches .

Modeling population growth

Pierre-François Verhulst (1804–1849)

A typical application of the logistic equation is a common model of population growth, originally due to Pierre-François Verhulst in 1838, where the rate of reproduction is proportional to both the existing population and the amount of available resources, all else being equal. The Verhulst equation was published after Verhulst had read Thomas Malthus' An Essay on the Principle of Population, which describes the Malthusian growth model of simple (unconstrained) exponential growth. Verhulst derived his logistic equation to describe the self-limiting growth of a biological population. The equation was rediscovered in 1911 by A. G. McKendrick for the growth of bacteria in broth and experimentally tested using a technique for nonlinear parameter estimation. The equation is also sometimes called the Verhulst-Pearl equation following its rediscovery in 1920 by Raymond Pearl (1879–1940) and Lowell Reed (1888–1966) of the Johns Hopkins University. Another scientist, Alfred J. Lotka derived the equation again in 1925, calling it the law of population growth.

Letting represent population size ( is often used in ecology instead) and represent time, this model is formalized by the differential equation:

where the constant defines the growth rate and is the carrying capacity.

In the equation, the early, unimpeded growth rate is modeled by the first term . The value of the rate represents the proportional increase of the population in one unit of time. Later, as the population grows, the modulus of the second term (which multiplied out is ) becomes almost as large as the first, as some members of the population interfere with each other by competing for some critical resource, such as food or living space. This antagonistic effect is called the bottleneck, and is modeled by the value of the parameter . The competition diminishes the combined growth rate, until the value of ceases to grow (this is called maturity of the population). The solution to the equation (with being the initial population) is

where

Which is to say that is the limiting value of : the highest value that the population can reach given infinite time (or come close to reaching in finite time). It is important to stress that the carrying capacity is asymptotically reached independently of the initial value , and also in the case that .

In ecology, species are sometimes referred to as -strategist or -strategist depending upon the selective processes that have shaped their life history strategies. Choosing the variable dimensions so that measures the population in units of carrying capacity, and measures time in units of , gives the dimensionless differential equation

Time-varying carrying capacity

Since the environmental conditions influence the carrying capacity, as a consequence it can be time-varying, with , leading to the following mathematical model:

A particularly important case is that of carrying capacity that varies periodically with period :

It can be shown that in such a case, independently from the initial value , will tend to a unique periodic solution , whose period is .

A typical value of is one year: In such case may reflect periodical variations of weather conditions.

Another interesting generalization is to consider that the carrying capacity is a function of the population at an earlier time, capturing a delay in the way population modifies its environment. This leads to a logistic delay equation, which has a very rich behavior, with bistability in some parameter range, as well as a monotonic decay to zero, smooth exponential growth, punctuated unlimited growth (i.e., multiple S-shapes), punctuated growth or alternation to a stationary level, oscillatory approach to a stationary level, sustainable oscillations, finite-time singularities as well as finite-time death.

Resources

Licensing

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