 Projects/ISLE/S_PP

# Growth: Logistic

SC-IPM Processes (Predator-Prey)

## Overview

Logistic Growth causes larger populations of prey to grow faster than smaller ones, subject to limitations in available resources.

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)

## Formulas

In this model, the predators (ie, G, grazers) are Zooplankton and the prey (ie, P, producers) are Phytoplankton. This Generic Process uses a single ordinary differential equation (ODE), based on a set of "constant parameters":

• P.conc - Carbon concentration of producers
• growth_rate - growth rate of producers
• saturation - saturation of available resources

As SC-IPM evaluates each candidate model structure against the training data, it estimates appropriate values for all numeric values:

t1 & = & 1 - P.conc * saturation \\{\operatorname{d}\!P.conc\over\operatorname{d}\!t} & = & P.conc * growth\_rate * t1

## Data Structures

:description
"Logistic Growth"

:name      :logistic-growth

:constants {
:growth_rate          [   0,         3      ]
:saturation           [   0,         0.1    ]
}

:entity-roles {
:P { :types [ :producer   ] }
}

:equations {
;; concentration
:P.conc {
:type     :ODE
:infix
"P.conc * growth_rate *
(1 - saturation * P.conc)"
:prefix
"(* P.conc growth_rate
(- 1 (* P.conc saturation)))"
}
}


See Data Structures for details.