Modelling an ecosystem

Ecosystems can be represented, in their simplest form, by the nonlinear Volterra equations, proposed over half a century ago by Vito Volterra (1860-1940).

Portrait of Volterra (by courtesy of SEDS).

The equations are :

X' = -mX + nXY
Y' = pY - qXY

where X and Y represent the populations of the higher/lower ecological levels (the predator and the prey, in the common terminology).The four constants m, n, p and q, are positive.
The prey is assumed to have an unlimited source of food. Therefore, in the absence of the predator, its number (Y) would be incremented indefinitely at the rate indicated by constant p. In the presence of the predator, its number would be diminished in a proportion to the number of encounters (constant q).
In the absence of the prey, the predator (X) will dwindle at the rate given by constant m. In its presence, the number of predators will be incremented proportionally to the number of encounters (constant n).
The above equations can be generalised to make them applicable to multi-level multi-species ecosystems. In addition, these ecosystems are divided in niches, which means that neighboring species may neither belong to the same trophic chain, nor compete for the same resources.
We consider three different types of species:

The following is a simulation of an ecosystem with three species which is not in equilibrium.

Course pages:
Modelling an ecosystem
An ecosystem with three species in equilibrium
The introduction of a predator breaks the equilibrium
The introduction of a herbivore breaks the equilibrium
An ecosystem with five species in equilibrium
Let's experiment
The African savanna ecosystem
Three ecosystems with migrations
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Last modified 21/12/99 by Juan de Lara (, need help for using this courses?.

First model - Second model - The SODA code