Definitions

# Ecosystem model

Ecosystem models, or ecological models, are mathematical representations of ecosystems. Typically they simplify complex foodwebs down to their major components or trophic levels, and quantify these as either numbers of organisms, biomass or the inventory/concentration of some pertinent chemical element (for instance, carbon or a nutrient species such as nitrogen or phosphorus).

## Overview

### Complexity

Ecosystem models are a development of theoretical ecology that aim to characterise the major dynamics of ecosystems, both to synthesise the understanding of such systems and to allow predictions of their behaviour (in general terms, or in response to particular changes).

Because of the complexity of ecosystems (in terms of numbers of species/ecological interactions), ecosystem models typically simplify the systems they are studying to a limited number of pragmatic components. These may be particular species of interest, or may be broad functional types such as autotrophs, heterotrophs or saprotrophs. In biogeochemistry, ecosystem models usually include representations of non-living "resources" such as nutrients, which are consumed by (and may be depleted by) living components of the model.

This simplification is driven by a number of factors:

• Ignorance: while understood in broad outline, the details of a particular foodweb may not be known; this applies both to identifying relevant species, and to the functional responses linking them (which are often extremely difficult to quantify)
• Computation: practical constraints on simulating large numbers of ecological elements; this is particularly true when ecosystem models are embedded within other models (such as physical models of terrain or ocean bodies, or idealised models such as cellular automata or coupled map lattices)
• Understanding: depending upon the nature of the study, complexity can confound the analysis of an ecosystem model; the more interacting components a model has, the less straightforward it is to extract and separate causes and consequences; this is compounded when uncertainty about components obscures the accuracy of a simulation

### Structure

The process of simplification described above typically reduces an ecosystem to a small number of state variables. Depending upon the system under study, these may represent ecological components in terms of numbers of discrete individuals or quantify the component more continuously as a measure of the total biomass of all organisms of that type, often using a common model currency (e.g. mass of carbon per unit area/volume).

The components are then linked together by mathematical functions that describe the nature of the relationships between them. For instance, in models which include predator-prey relationships, the two components are usually linked by some function that relates total prey captured to the populations of both predators and prey. Deriving these relationships is often extremely difficult given habitat heterogeneity, the details of component behavioral ecology (including issues such as perception, foraging behaviour), and the difficulties involved in unobtrusively studying these relationships under field conditions.

Typically relationships are derived statistically or heuristically. For example, some standard functional forms describing these relationships are linear, quadratic, hyperbolic or sigmoid functions. The latter two are known in ecology as type II and type III responses, named by C. S. Holling in early, groundbreaking work on predation in mammals. Both describe relationships in which a linkage between components saturates at some maximum rate (e.g. above a certain concentration of prey organisms, predators cannot catch any more per unit time). Some ecological interactions are derived explicitly from the biochemical processes that underlie them; for instance, nutrient processing by an organism may saturate because of either a limited number of binding sites on the organism's exterior surface or the rate of diffusion of nutrient across the boundary layer surrounding the organism (see also Michaelis-Menten kinetics).

After establishing the components to be modelled and the relationships between them, another important factor in ecosystem model structure is the representation of space used. Historically, models have often ignored the confounding issue of space, utilising zero-dimensional approaches, such as ordinary differential equations. With increases in computational power, models which incorporate space are increasingly used (e.g. partial differential equations, cellular automata). This inclusion of space permits dynamics not present in non-spatial frameworks, and illuminates processes that lead to pattern formation in ecological systems.

## Examples

One of the earliest, and most well-known, ecological models is the predator-prey model of Alfred J. Lotka (1925) and Vito Volterra (1926). This model takes the form of a pair of ordinary differential equations, one representing a prey species, the other its predator.

$frac\left\{dX\right\}\left\{dt\right\} = alpha . X - beta . X . Y$
$frac\left\{dY\right\}\left\{dt\right\} = gamma . beta . X . Y - delta . Y$

where,

 $X$ is the number/concentration of the prey species; $Y$ is the number/concentration of the predator species; $alpha$ is the prey species' growth rate; $beta$ is the predation rate of $Y$ upon $X$; $gamma$ is the assimilation efficiency of $Y$; $delta$ is the mortality rate of the predator species

Volterra originally devised the model to explain fluctuations in fish and shark populations observed in the Adriatic Sea after the First World War (when fishing was curtailed). However, the equations have subsequently been applied more generally. Although simple, they illustrate some of the salient features of ecological models: modelled biological populations experience growth, interact with other populations (as either predators, prey or competitors) and suffer mortality.