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Food Webs

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Food Webs Report

Week 5

MTH / 221

University of Phoenix

Food Webs
It may be difficult to know all the factors which determine an ecological niche, and some factors may be relatively unimportant. Hence it is useful to start with the concept of competition and try to find the minimum number of dimensions necessary for a phase space in which competition can be represented by niche overlap. One approach to this question is to consider the notion of the food web of an ecological community.

Definition 1
A food web of an ecological community is a directed graph with a vertex for each species in the community and a directed edge from the vertex representing species A to the vertex representing species B if and only if A preys on B

We can define competition using the food web. Two species compete if and only if they have a common prey. Thus, a Hawk and Wolf compete (since Rabbit is a common prey); Rabbit and Grasshopper compete, while Deer and Toad do not compete. We use this competition relation to define a graph called the competition graph.

Definition 2
The competition graph of a food web is a simple graph with a vertex for each species. Two vertices are joined by an edge if and only if the species they represent have a common prey.

Definition 3
A graph is an intersection graph for a family of sets if each vertex is assigned a set in such a way that two vertices are joined by an edge if and only if the corresponding sets have non-empty intersection.

Definition 4
A graph is called an interval graph if it is the intersection graph for a family of closed intervals. Our goal is the representation of competition graphs of families of sets in Euclidean n-space. Clearly the simplest case would be that of competition graphs that are interval graphs. This would mean that only one ecological factor is necessary to describe niche overlap.

Definition 5
A graph H is a generated sub graph of a graph G if the vertices of H are a subset of the vertices of G and vertices in H are adjacent in H if and only if they are adjacent in G.
Definition 6
The complement of a graph G is the graph G where the vertices of G are the vertices of G, and two vertices in G are adjacent if and only if they are not adjacent in G.

Definition 7
An orientation of a graph G is an assignment of a direction to each edge in G (which makes G into a directed graph). An orientation is transitive if whenever (u, v) and (v, w) are directed edges, then (u, w) is a directed edge.

Theorem 1
A graph G is an interval graph if and only if it satisfies the following two conditions: The four-cycle C is not a generated sub graph of G, The complement of G is transitively orientable. Our goal in our study of ecological competition is the representation of niches in Euclidean space and competition by niche overlap. It seems desirable in an ideal representation that the factors determining the dimension of the ecological phase space would be independent and the niches would be represented as “boxes”, or Cartesian products of intervals. This leads us to the next part of this discussion, namely, when can we represent a graph as the intersection graph of a family of boxes in n-space.

Conclusion
This is one of the hardest classes I have had to take thus far in University, and I just know that I am going to fail. I do try my hardest, but with kids, a full time job with no set schedule, property to manage and maintain, I am usually up until 2 or 3 am trying to work on school projects. Thank you for your excellent feedback in the discussions, sorry I was not a bit more active. Have a great summer!

Resources
Niche Overlap - dimacs.rutgers.edu
Food Web Graph - www.edusolution.com.
Discrete Mathematics and its Applications, Kenneth H. Rosen, Copyright McGraw-Hill (2012)

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