# 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:

• Primary producers, usually plants, which are preyed upon, but do not prey. The evolution of their population X follows the equation: Evolution of primary producers

Where :

• pref(P,S) is a function that gives the appetence of species P for species S,

• Eats(P,S) is a relation which contains all pairs (P,S) such that species P eats species S,

• S.X is the population of species S,

• S.X0 is the initial population of species S,

• P.M is the coefficient of spontaneous reproduction of the primary producer, which regulates the speed with which its population increases in the absence of consumers.

• P.N1 is a coefficient which indicate the influence on the population of species S of encounters with other species

• Superpredators, which prey but are not preyed upon. The evolution of their population X follows the equation: Evolution of Superpredators

Where :

• M is the coefficient of inverse resistance to hunger of the superpredator, that regulates the speed with which its population declines in the absence of food.

• P.N2 is a coefficient which indicate the influence on the population of species S of encounters with other species .

• Intermediate consumers, which prey and are preyed upon. The evolution of their population X follows the equation: Evolution of intermediate consumers

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

 Gravitation Ecology Electronics PDEs Other pages

Last modified 21/12/99 by Juan de Lara ( Juan.Lara@ii.uam.es, http://www.ii.uam.es/~jlara) need help for using this courses?.

First model - Second model - The SODA code