INTRODUCTION A food web is a directed graph modeling the predator-prey relationship in an ecological community. Let the vertices of a directed graph (digraph) be species in an ecosystem. Include an arc from x to y if x preys on y.
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5 COMPETITION GRAPH The competition graph C(D) has vertex set V and an edge between a and b if there is an x with arcs (a,x) ε A and (b,x) ε A. Consider a corresponding undirected graph, where Vertices = the species in the ecosystem Edge between a and b if they have a common prey
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A key idea in the study of competition graphs is the notion of interval graph. A graph is an interval graph if we can find intervals on the line so that two vertices are joined by an edge if and only if their corresponding intervals overlap. INTERVAL GRAPH
8 OWL ANT DEERGRASS This is an interval graph. FOX OWL ANT DEER GRASS FOX
BOXICITY In graph theory, boxicity is a graph invariant, introduced by Fred S. Roberts in The boxicity of a graph is the minimum dimension in which a given graph can be represented as an intersection graph of axis- parallel boxes.
EXAMPLE The figure shows a graph with four vertices, and a representation of this graph as an intersection graph of rectangles (two-dimensional boxes). This graph cannot be represented as an intersection graph of boxes in any lower dimension, so its boxicity is two a b c d b c d a
Relation to other classes A graph has boxicity at most one if and only if it is an interval graph. Every outer planar graph has boxicity at most two Every planar graph has boxicity at most three.
ECOLOGICAL NICHE The range of temperatures, moisture, pH, capturable prey, size, etc., in which a species lives is called "ecological niche.“ Two species compete if and only if their ecological niches overlap.
13 FACTORS AFFECTING BIODIVERSITY Temp t t0t0 t1t1 Moisture m m1m1 m0m0
14 Temp t t0t0 t1t1 Moisture m m1m1 m0m0 p0p0 p1p1 pH p
15 OWL ANT DEERGRASS This is an interval graph. Thus, the boxicity is one. FOX OWL ANT DEER GRASS FOX
OPEN PROBLEMS A community food web includes all predation relations among species. A community food web can have a competition graph that is an interval graph while some source food web contained in it has a competition graph that is not an interval graph. In practice, it’s impossible to gather all the data about food webs.
Some of the difficulties are involved in understanding the structure of competition graphs. It remains a challenge (dating back to 1968) to understand what properties of food webs give rise to competition graphs of boxicity 1, i.e., interval graphs. It is NP-complete to determine the boxicity of a graph