Essay About Use Of Discrete Mathematics And Directed Graph

Food Webs
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Food WebsAs Related to Discrete MathematicsDiscrete mathematics can be used to calculate all kinds of relationships between but not limited to both people and animals. It can even be used to calculate and configure accurate descriptions of ecosystems and food chains.  This process is called a food web. A food web is a directed graph modeling the predator-prey relationship in an ecological community. This process is achieved by figuring out the minimum number of parameters needed to describe ecological competition. Graphs can also be used to correctly define intersections of families of sets.  In this particular case study Robert A. McGugan wrote about how mathematical techniques and advanced graphing implementation can be used to adapt and relate ecosystems. This theory basically discusses relationships of elements involving two or more sets and any subsets of these sets.  The graphs that are used track food webs and help establish a requiem of vertices, edges, and links that an information system closely adheres to in modern computing.  This study thoroughly describes the relationship between predator and prey in a particular ecological environment. This relationship can be very competitive but through the use of discrete mathematics the case study calculates the balance in this relationship through the use of assumptions and some mathematical formulas. Each animal and plant occupy’s its own niches that is defined by the availability of resources. These resources are given factors such as temperature, moisture, degree of acidity, amounts of nutrients, and other parameters which regulate resource availability. These niches are beneficial to some species but yet detrimental to others. The graphs used describe the competition within the food web show how each species feed and prey on each other. The graphs show how more than any single species may prey or be prey to more than any single species and still coexist.  It provides an example that restricts the factors down to three parameters such as temperature, nutrients, and pH. It then uses these 3 factors to develop a 3 dimensional Euclidean space which can be used to show how species compete and coexist in their own ecological niche (McGuigan, n.d). One of the purposes in this study was to show how ecological completion is the representation of niches in Euclidean space and competition by niche overlap. These overlaps would be represented as “boxes”, or Cartesian products of intervals. This effect is called Boxicity. Boxiicity is graphed using two different theorems. In the first theorem (Theorem 2) they show that boxicity is well-defined, although there is no efficient algorithm known for determining the boxicity of a general graph. In theorem 3 the study goes on to show how Cohen has studied more than 30 single-habitat food webs published in the ecological literature and has found that the competition graphs of all of them are interval graphs. This shows that in all cases one dimension suffices to represent competition by niche overlap. The theorem then goes on to explain how in some single-habitat communities a single dimension for the space can be identified. This could be explain through some obvious linear factor like temperature, body length or depth in water but it may be that more than one single dimension will work (McGuigan, n.d.).