## Definition of Cycle

A very common way of describing ecosystems is by means of graphs. Graphs are constituted by nodes (representing species or functional groups of species) connected by arrows (or edges, arcs, links, representing relationships between species).

The simplest way of sketching ecosystems using graphs is the food web representation. In this way of drawing species relations, edges connect prey to their predators (see Figure 1).

This food web representation can be associated with a matrix that expresses the relationships between species. This is the so-called 'adjacency matrix', A. If the row species is a food source of the column species then the corresponding coefficient will be 1. More generally, this relation is a consumer-resource relation, as nodes can represent nutrient pools, etc. Elsewhere, the coefficients will be 0. The food web in Figure 1 can therefore be represented by this adjacency matrix:

(0 0 1 1 0 1 0 0 10 0 A = 0 0 0 1 1 0 10 0 1 \0 0 0 0 0/

The adjacency matrix represents direct interactions between species. These direct interactions, however, yield chains of indirect interactions. These will be sequences of nodes and edges that are called 'paths'. We can discriminate between different kinds of paths:

1. Open paths connect two different nodes. They can be subdivided into 'simple paths', containing no repeated nodes (e.g., A! B! C, Figure 2a) and 'compound

Figure 1 Example of food web containing five species and seven feeding relations (arrows, edges).

Figure 2 Classification of pathways in (a) simple paths (open pathways start and end at different nodes); (b) compound paths (open pathways start and end at different nodes, contain repeated nodes); (c) simple cycles (closed pathways start and end at the same node); and (d) compound cycles (same cycle traversed more than once).

paths', which contain repeated nodes (e.g., A ! B ! C ! B ! D, Figure 2b). 2. Closed paths start and end at the same node. Also closed paths can be divided into 'simple cycles', containing no repeated nodes except the initial one (e.g., A ! B ! C ! A, Figure 2 c) and 'compound cycles', representing repeated cycles (e.g., A ! B ! A ! B ! A, Figure 2d, where double arrows mean that the cycle is traversed twice).

All kinds of paths, other than simple paths, contain at least one cycle. For example, the graph in Figure 1 contains just the simple cycle 2 ! 3 ! 4 ! 2. A graph containing no cycles is said to be acyclic. Every pathway can be classified according to its length that is given by the number of nodes involved.

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