 Projects/ISLE/H_AE

# Limit: Logistic

Aquatic Ecosystems, HIPM Processes (Growth)

## Overview

HIPM supports two logistic growth limits (Gompertz and Pearl-Verhulst).

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## Data Structures

```:type    "limited_growth"

:relates {
:E { :generic_entities     [ ee        ] }  ;; Environment

:N { :generic_entities     [ fe no3    ]    ;; Nutrients
:max                  100           }

:P { :generic_entities     [ pe        ] }  ;; Phytoplankton
}
```

### Gompertz

A Gompertz curve or Gompertz function, named after Benjamin Gompertz, is a sigmoid function. It is a type of mathematical model for a time series, where growth is slowest at the start and end of a time period. The right-hand or future value asymptote of the function is approached much more gradually by the curve than the left-hand or lower valued asymptote, in contrast to the simple logistic function in which both asymptotes are approached by the curve symmetrically. It is a special case of the generalised logistic function.

-- Gompertz function (WP)

```:id        "Gompertz_logistic"

:constants { :K              [  1, 10000 ] }  ;; Ross Sea

:algebraic_eqs {
:P.growth_rate   "P.max_growth * (log(K) - log(P.conc))"
; referenced in Rosenzweig 1971
}
```

### Verhulst-Pearl

A typical application of the logistic equation is a common model of population growth, originally due to Pierre-Francois 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. Verhulst derived his logistic equation to describe the self-limiting growth of a biological population. The equation is also sometimes called the Verhulst-Pearl equation following its rediscovery in 1920. Alfred J. Lotka derived the equation again in 1925, calling it the law of population growth.

-- Logistic function (WP)

```:id        "Pearl_Verhulst_logistic"

:constants { :K              [ 20, 10000 ] }  ;; Ross Sea

:algebraic_eqs {
:P.growth_rate   "P.max_growth * (1.0 - P.growth_lim/K)"
}
```

See Data Structures for details.

This wiki page is maintained by Rich Morin, an independent consultant specializing in software design, development, and documentation. Please feel free to email comments, inquiries, suggestions, etc!

Topic revision: r11 - 21 Jul 2013, RichMorin
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