# Studies on food webs using matrices associated with graphs.

Introduction

Ecological systems are extremely complex networks, consisting of many biological species that interact in many different ways, such as mutualism, competition, parasitism and predator-prey relationships. 'Predators' mean organisms which acquire their nutritional energy by eating other organisms; while a 'Prey' is any creature that is eaten by the predators for consuming food. There may be hierarchies of predators; for example, though small birds prey on insects, they may in turn be prey for rats, which may in turn be prey for snakes and so on.

A community food webs(or simply food webs or food networks) describes the feeding habits of a set of organisms ,i.e., which kind of organisms in a community eat which other kinds, if any. So food webs describe the feeding relationships between species (prey- predator relations) in a biotic community. In other words, they show the transfer of material and energy from one species to another within an ecosystem. In this paper by "food web" we shall mean a network (a directed graph) in which species are represented by nodes, and information like 'species u is eaten by species v' is indicated by drawing a directed edge (or an arc ) from the species 'u' to species 'v', i.e., from prey to predator [Cohen et al].

We know diagrammatic representation of a graph is really convenient for visual study and understanding the problem. But when the problem becomes large, it is really difficult to have non-trivial information by studying the underlying properties of the graph. It is seen that matrix representation of a graph is very convenient for computer processing. In many applications of graph theory matrices are natural way of expressing the problem [Deo]. To represent the relation between predators and preys in matrix form, we have taken preys along the rows and predators along the columns of the matrix. The (i,j)th element of the matrix is taken 1 or 0 according as species i is preyed by species j (i.e., prey-predator link). The matrix based on the prey-predator link, a binary relation, is, in fact, the adjacency matrix (defined below) of the digraph associated with a food web.

Some Basic concepts

In the following paragraphs we explain some terminologies of food webs and graph theory. A specific geographic region characterized by the local landforms, the various plants and animals living there, and the prevailing climatic conditions such as temperature and rainfall etc. is called an ecosystem. Animals, and in rare cases plants, that eat other animals and/or plants to fulfill their own nutritional energy requirements are called consumers. The path which the organisms follow to obtain, use, and transfer energy is called food chain. For example, plants are eaten by a grazing animal; and grazer; grass. rabbit etc., are eaten different types of predators. A food web is a collection of multiple interwoven food chains.

An animal that eats plants and animals is called omnivore. Organisms that obtain their nutritional energy by killing and eating other organisms are called predators. Any creature that is hunted and caught to be eaten for food is called a prey.

Considering the species of a food web as vertices of an undirected graph the competition graph in a food web is defined by joining the vertices corresponding to the species (predators) having at least one common prey. The competition graph is also known as consumer graph or "trophic niche overlap graph" (or niche graph or overlap graph). Similarly, the resource graph in a food web is defined by joining the vertices corresponding to the species (preys) having at least one common predator.

Let G = (V,E) be a simple connected graph without self-loops and parallel edges where V={[v.sub.1],[v.sub.2], ... ,[v.sub.n]} is the set of vertices and E = {[e.sub.1], [e.sub.2], ... ,[e.sub.m]} the set of edges. When each edge is given a direction, G is called a directed graph or digraph; and the edges of G are called directed edges or arcs. A subgraph H of G is called complete if every pair of distinct vertices in H is adjacent. A complete subgraph H of G is called a clique or maximal complete subgraph of G if no other complete subgraph of G contains H. If [absolute value of H] = h, then H is called a clique of order h or a h-clique.

If A = ([a.sub.ij]) be the adjacency matrix of order n x n corresponding to a graph G, then [a.sub.ij] = [a.sub.ji] = 1 or 0 according as there is an edge between [v.sub.i] and [v.sub.j] or not. In case of digraph [a.sub.ij] = 1 if there is an arc (directed edge) from [v.sub.i] to [v.sub.j] , otherwise [a.sub.ij] = 0. In this case the out degree of [v.sub.i] is 1 and the in degree of [v.sub.j] is 1. Let [d.sub.i.sup.+] and [d.sub.i.sup.- ] denote respectively the out degree and in degree of the vertex [v.sub.i] . Let the out- neighbourhood of a vertex v be denoted by [N.sup.+] (v), then [N.sup.+] (v) = {u: (v,u) [epsilon] E}. Clearly, [absolute value of ([N.sup.+] (v))] is the sum of the number of 1's in the v th row of the adjacency matrix A. We call the sum C-sum. So [absolute value of [N.sup.+] (v)] = C-sum. Similarly, the in- neighbourhood of the vertex v can be defined by [N.sup.-] (v) = {u: (u,v) [epsilon] E}. We write R-sum for the corresponding sum and get [absolute value of ([N.sup.-] (v))] = R-sum. We next consider an example to explain the results.

Example

Let us consider the following food web consisting of species 1,2,...,12,13 . The food web is taken from MacDonald with a minor change. The prey-predator relationships are depicted by drawing arcs from preys to predators. This means energy flows from the species j to the species i.

[ILLUSTRATION OMITTED]

The adjacency matrix A of the above digraph i.e., the predation matrix of the food web is

where C-sum = [summation] [d.sub.i.sup.+] and R-sum = [summation] [d.sub.i.sup.- ] for i = 1, 2,..., 13.

From the first column of A , it can be said that species '1' preys upon the species '6' and 8'. So the in neighbourhood set [N.sup.-] ([v.sub.1]) = {6,8} and the corresponding R-sum is 2. Again from the sixth row of A, it can be said that species '6' is preyed by the species (predators) 1,4,5. So the out neighbourhood set [N.sup.+] ([v.sub.6]) ={1,4,5} and the corresponding C-sum is 3. [N.sup.+] ([v.sub.2]) = 0 means species '2' has no predator and [N.sup.-] ([v.sub.12]) = 0 means species '12' has no prey. The members of an out-neighbourhood set [N.sup.+] (v) fight among themselves for the common prey v. Thus it is a competitive situation and by taking 'competition for consuming the common species by the predators' as the adjacency relation we can draw the competition graph or common enemy graph or consumer graph (cohen) for the predators. Ecologists call this niche overlap graph.

We can construct the adjacency matrix B of the competition graph as follows.

The th element of the competition graph is taken '1' if the i th and the j th predators compete for a common prey, i.e., there exists at least one species k such that the arcs (k,i) and (k,j) are present in the web. For example, from [N.sup.+] ([v.sub.6]) = { 1,4,5} we see that (6,1),(6,4) and (6,5) arcs are present in the food web. Moreover, we can say that the species 1,4,5 compete for the prey '6'. Thus considering the species as vertices we can construct a complete subgraph formed by these set of vertices {1,4,5}. So '1' is taken in the (1,4),(1,5) and (4,5) positions of the matrix B given below.

Similarly, each of the set ofvertices {3,4,5}, {1,3,4,5,6},{2,3,6,7}, {2,3,4}, {2,3,5,6,7}, {2,6,7,8,9,10,11} and {7,8,9,10,11} forms a complete subgraph and '1' is taken in the corresponding positions of the matrix B. Among these {1,3,4,5,6}, {2,3,4}, {2,3,5,6,7}, {2,6,7,8,9,10,11} are the cliques of the consumer graph.

The members of an in-neighbourhood set [N.sup.-] (k) are preyed by the predator k i.e., there exists at least two species i and j which are preyed by k. So at least two arcs (i,k) and (j,k) must be present in the food web. In this case we join the species i and j by an edge, and the graph thus formed is called the Resource graph or Common Enemy graph of the food web.

The (i,j) th element of the Resource grraph is taken if the i th and the j th species have a common predator. For example, from [N.sup.-] (v1) = {6,8} we see that (6,1) and (8,1) arcs are present in the food web i.e., the species 6,8 have a common predator 1. Moreover, we can say that the species '6' and '8' are the resources for the predator 1. Thus considering the species as vertices we can construct a complete subgraph formed by these set of vertices {6,8}. So is taken in the (6,8) positions of the matrix B.

Similarly, each of the set of vertices {9,10,11,12}, {7,8,9,10,11}, { 6,7,8,10}, {6,7,8,11},{8,9,11,12},{9,11,12,13} and{12,13} forms a complete subgraph and is taken in the corresponding positions of the matrix B. Among these, except {12,13}, there are five 4-cliques and one 5-clique of the resource graph.

If a cell of B contains both '1' and just to avoid cluttering of symbols, we have written '2'in that cell. Thus the presence of a '2' in the (i,j) position indicates that both the species i and j have at least one common prey as well as at least one common predator. For example, (6,7), (6,8) and (7,8) cells contain '2', hence S={6,7,8}. It is seen that any two species of S(say S1) have a common prey and any two species of S (say [S.sub.2]) have a common predator, where [S.sub.1] [intersection] [S.sub.2] [not equal to] [phi]. Now we observe that

(6,7) have common preys '9', '11' and a common predator '4',

(6,8) have a common prey '12' and a common predator '1', and

(7,8) have common preys '12', '13' and common predators '3', '4' and '5'.

Similarly, the species contained in the set { 7,8,9,10,11} have these properties. Since the set of species determined by S have common preys and common predators, they should be equally affected by some kind of ecological changes in the food web. Again, the extinction of a small number of species contained in S should not have a remarkable effect on the food web. For example, the extinction of the species '6' does not have a remarkable effect on the food web.

Conclusion

In this paper we have used matrices associated with graphs to study some properties of trophic level 4 omnivorous food web. We have identified a set of species from a new matrix and observed some interesting properties. We have also tried to interpret the result from ecological point of view of a food web.

References

[1] Cohen.JE, Briand.F and Newman.CM- Community Food Webs( Biomathematics , vol. 20)- Springer-Verlag(1990).

[2] Deo.N- Graph Theory with Applications to Engineering and Computer Science- PHI- 1974.

[3] Drossel.B and McKane.AJ - Modelling Food Webs (A review paper from website).

[4] Harary.F - Graph theory .Addison -Wesely publishing co,1972.

[5] Mackenzie. A, Ball.AS & Virdee SR- Instant Notes in Ecology-Viva Books Private Limited-New Delhi(1999).

[6] MacDonald.N- Trees and Networks in Biological Models, John Wiley & Sons(1983).

[7] Roberts.FS-Graph Theory and its Applications to Problems of Society- SIAM(1978).

[8] Svirezhev. YM and Logofet. DO-Stability of Biological Communities-Mir Publishers(1983).

Phalguni Mukherjee

Department of Mathematics & Post Graduate Department of Environmental Science,

Asutosh College, kolkata, India

Email: phalguni.mukherjee@gmail.com
Matrix A

1 2 3 4 5 6 7 8 9 10 11 12 13 C-sum

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 1 0 0 1 1 0 0 0 0 0 0 0 0 3
7 0 0 1 1 1 0 0 0 0 0 0 0 0 3
8 1 0 1 1 1 1 0 0 0 0 0 0 0 5
9 0 1 1 0 0 1 1 0 0 0 0 0 0 4
10 0 1 1 1 0 0 0 0 0 0 0 0 0 3
11 0 1 1 0 1 1 1 0 0 0 0 0 0 5
12 0 1 0 0 0 1 1 1 1 1 1 0 0 7
13 0 0 0 0 0 0 1 1 1 1 1 0 0 5
R-sum 2 4 5 4 4 4 4 2 2 2 2 0 0 35

Matrix B

C 1 2 3 4 5 6 7 8 9 10 11 12 13
1 1 1 1 1
2 1 1 1 1 1 1 1 1 1
3 1 1 1 1
4 1 1
5 1 1
6 2 2 1 2 2
7 2 2 2 2
8 2 2 2 -1
9 2 2 -1 -1
10 2 -1
11 -1 -1
12 -1
13