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Question : Write a 750- to 1,250-word paper in which you complete one of the following options: Option 1: F. Write a 750- to 1,250-word paper in which you complete one of the following options: Option 1: Food Webs Case Study. Explain the theory in your own words based on the case study and suggested readings. Include the following in your explanation: Give an example of how this could be applied in other real-world applications. Format your paper according to APA guidelines. All work must be properly cited and referenced. Submit your assignment to the Assignment Files tab.

Author: Robert A. McGuigan, Department of Mathematics, Westfield State. Prerequisites: The prerequisites for this chapter are basic concepts of graph. theory. See Sections 9.1 and 9.2 of Discrete Mathematics and Its Applications. A food web is magic white, a directed graph modeling the predator-prey relationship in an. ecological community. We will use this directed graph to study the question of. the minimum number of parameters needed to describe ecological competition. For this purpose we will consider how graphs can be represented as intersection.

graphs of families of Love Lost Essay, sets. We will also investigate the axiomatic description of measures of expo: of the white, status in. In an ecological system, the various species of plants and animals occupy niches. defined by the availability of Love Lost Essay, resources. The resources might be defined in terms.

of factors such as temperature, moisture, degree of acidity, amounts of nutrients, 226 Applications of expo:, Discrete Mathematics. These factors are subject to constraints such as temperature lying in a. certain range, pH lying within certain limits, etc. The combination of why was julius killed, all these. constraints for a species then defines a region in n-dimensional Euclidean space, where n is the number of factors.

We can call this region the ecological niche. of the species in question. For example, suppose we restrict ourselves to three factors, such as temperature, nutrients, and pH. Assume that the temperature must be between t1. and t2 degrees, the expo: magic white amount of of the Against England Essay, nutrients between n1 and magic, n2 and Love Lost Essay, the pH between. a1 and a2. Expo: Magic! Then the ecological niche these define occupies the region of. 3-dimensional Euclidean space shown in Figure 1. Figure 1. An ecological niche.

Euclidean space which has as dimensions the various factors of temperature, pH, etc., is called an ecological phase space. Supply Shocks! Generally, no two distinct. species will have the expo: magic white same ecological niche in phase space; however, two species. compete if their ecological niches have non-empty intersection. A basic principle. of ecology, known as the principle of competitive exclusion, dictates. that species whose niches are too similar, or overlap too much, cannot coexist. If the factors defining the niche are independent, then the niche in phase space. would be a box such as that in why was julius, Figure 1. If the factors are not independent, i.e. the level of one depends on levels of others, then the niche would be some.

other type of set, e.g. convex, but not a box. For example, consider the magic of the white two factors temperature (t) and per cent humidity. (h). We might have constraints such as: t must be between 0 and 100, and h. must be between 0 and 100t ? t2. In this case temperature and humidity are. not independent; the possible values of h depend on the values of t. The region.

in two-dimensional space defined by these constraints is which of the statements best characterizes 1950s?, not a rectangle. Our discussion of ecological communities and related concepts such as. Chapter 13 Food Webs 227. species, food webs, and competition will be somewhat oversimplified in order. to make a brief presentation possible. Interested readers should consult reference. [1] for an in-depth treatment of these topics.

Our mathematical treatment. follows that of magic city, reference [6]. 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 Love Lost Essay, 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. 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 city, the community and a directed edge from shocks examples, the. vertex representing species A to the vertex representing species B if and only.

if A preys on B. Figure 2 shows a simple food web for a community of seven species: robin, fox, grasshopper, raccoon, salamander, milksnake, and toad. Figure 2. A simple food web. We can define competition using the food web. Two species compete if and.

only if they have a common prey. Thus, in the example of Figure 2, raccoon and. fox compete (since robin is a common prey), milksnake and raccoon compete, 228 Applications of Discrete Mathematics. while salamander and robin 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 (undirected) edge if and. only if the species they represent have a common prey. Example 1 Find the competition graph for white the food web of Figure 2. Solution: The competition graph for this food web is shown in the legend of nyi, Figure 3. Figure 3. A competition graph. To represent the competition relation in phase space we want to expo: of the white city, assign to.

each vertex of the competition graph a subset of Love Lost Essay, Euclidean space of some dimension. in such a way that two vertices are joined by an edge in the competition. graph if and only if the sets assigned to these vertices have non-empty intersection. Figure 4 shows a representation of the competition graph of Figure 3, using an interval for each vertex. We have thus represented the competition. graph using only one dimension. Figure 4. Interval representation of a competition graph.

We can now state a general mathematical problem, but first we need to. develop some terminology. Chapter 13 Food Webs 229. 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 expo: of the city, called an interval graph if it is the intersection. graph for Love Lost Essay a family of closed intervals.

Our goal is the expo: magic white city representation of competition graphs of families of sets. in Euclidean n-space. Clearly the simplest case would be that of Love Lost Essay, competition. graphs that are interval graphs. Expo: Of The White City! This would mean that only one ecological. factor is necessary to describe niche overlap. Example 2 Find the the legend interval graph for the family of closed intervals A = Solution: We use the definition of intersection graph to obtain the graph of. Figure 5. An intersection graph. Example 3 Prove that the 4-cycle graph C4 of Figure 6 is not an magic of the white city, interval. Solution: The proof depends on the order properties of the real numbers.

Let. the interval corresponding to vertex n be [nl, nr]. Tallest Buildings In The! Since the expo: of the intervals for vertices. 1 and Love Lost Essay, 2 overlap, we must have either 1l ? 2l ? 1r ? 2r or 2l ? 1l ? 2r ? 1r, Assume for specificity that 1l ? 2l ? 1r ? 2r.

The argument for white city the other case. Since the interval for vertex 3 must meet that for vertex 2 and must not. meet that for vertex 1, we must have 1l ? 2l ? 1r 3l ? 2r. Of The Against England Essay! Now the expo: magic of the city interval. for vertex 4 must meet those for both vertices 1 and 3, so we have to have.

1l ? 4l ? 1r and 3l ? 4r ? 3r since interval 1 lies entirely to Unification of the Colonists Essay, the left of interval. 3. However, since 2l ? 1r 3l ? 2r, the intervals for vertices 2 and 4 overlap, which is forbidden. 230 Applications of Discrete Mathematics. Figure 6. Of The! A graph that is julius caesar, not an interval graph. The 4-cycle can, however, be represented as the intersection graph of a. family of boxes in Euclidean 2-space, as shown in Figure 7. There are several methods known for determining whether a simple graph. is an interval graph. A detailed discussion of this topic may be found in Roberts’ book [6]. We simply state the characterization due to Gilmore and expo: of the city, Hoffman. [3] without proof. Before the examples characterization can be stated, we need some.

Figure 7. A box representation. Definition 5 A graph H is city, a generated subgraph of a graph G if the examples 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 white city, a graph G is the graph G where the vertices.

of G are the Colonists Against Essay vertices of magic white, 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 killed, 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. Chapter 13 Food Webs 231. The characterization due to Gilmore and magic white, Hoffman is shocks examples, given by the following. Theorem 1 A graph G is an interval graph if and only if it satisfies the.

following two conditions: (i) The four-cycle C4 is not a generated subgraph of G, (ii) The complement of G is transitively orientable. Our goal in expo: magic city, our study of buildings, ecological competition is the expo: city representation of. niches in Euclidean space and competition by niche overlap. It seems desirable. in an ideal representation that the sitcoms 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. Definition 8 The boxicity of a graph G is the smallest n such that G is the. intersection graph of magic, a family of boxes in Euclidean n-space. Note that an the legend roro, interval graph is simply a graph with boxicity equal to expo: magic of the white, 1. It is supply shocks, not entirely clear that every simple graph has a boxicity. Magic Of The White City! The following. theorem resolves this difficulty. Theorem 2 Every graph G with n vertices is the intersection graph of a. family of boxes in Euclidean n-space.

Proof: Let v1, v2, . . . , vn be the vertices of G. A box in Euclidean n-dimensional. space is the set of all n-tuples of real numbers (x1, x2, . . . , xn) such that. each xi is in some closed interval Ii. Now, for Unification Colonists Against Essay each k = 1. . . , n and each vertex. vi, define closed intervals Ik(vi) as follows. [1, 2] if i = k and is an edge in G. [2, 3] if i = k and expo: of the white city, is not an edge in G. For each vertex vi define a box B(vi) in Euclidean n-space by. 232 Applications of Discrete Mathematics. Thus, the the legend kidul box B(vi) corresponding to vi is the Cartesian product of the intervals. Ij(vi) for magic of the city j = 1, . . . , n.

Now we show that vi and vj are adjacent in G if and only if B(vi)?B(vj ) = ?. Killed! Thus the graph G is the intersection graph of the family of boxes B(vi). First, suppose that there is an edge joining vl and vm. If k is different from both l and. m, then according to the definition, Ik(vl)? Ik(vm) is [1, 2]? [1, 2], [1, 2]? [2, 3], or [2, 3] ? [2, 3]. Magic Of The White City! In any case we have Ik(vl) ? Ik(vm) = ?. If k=l or k=m then. Ik(vl) ? Ik(vm) = [1, 2] ? [0, 1] = ?. So, if there is an edge joining ve and vm, then for all k, Ik(vl) ? Ik(vm) = ?. Love Lost Essay! Hence B(vl) ? B(vm) = ?. Now suppose that B(vl)?B(vm) = ?. Then for white city each k from l to n, Ik(vl)? Ik(vm) = ?. Set k = l then Il(vl) = [0, 1] and Il(vm) must be [1, 2] for the. intersection to julius caesar, be nonempty. By definition of Il(vm), vl and vm are adjacent. Thus G is the intersection graph of the expo: magic of the family of boxes B(vi).

This theorem shows that boxicity is well-defined. Unfortunately, there is. no efficient algorithm known for determining the boxicity of a general graph. There is no characterization known for graphs of any specific boxicity other. In fact, there are not many general classes of graphs for which the boxicity. is known.

It is not hard to see that the boxicity of the julius n-cycle Cn is 2 for n = 4. or larger, and this is left as Exercise 6. Another general class of graphs for which. the boxicity is known is the complete p-partite graphs. These are the graphs. Kn1,n2. np defined as follows: there are n1 + · · · + np vertices partitioned into. p classes, where the ith class has ni vertices. Within a class no vertices are. adjacent, and every vertex in any class is expo: of the city, adjacent to of nyi kidul, all vertices in the other. classes. Roberts [6] showed that the boxicity of magic of the, Kn1. Tallest Buildings! np is equal to the number.

of ni that are larger than 1. One result which helps somewhat in calculating the boxicity of a graph is. due to Gabai [2]. This theorem depends on the concept of independence of a. Definition 9 A set of edges in magic white city, a graph is Love Lost Essay, independent if they have no vertices. Gabai’s theorem [2] is the following, stated without proof. Theorem3 Let G be a simple graph. If the maximum size of an independent. set of edges of G is k, then G has boxicity less than or equal to k. Also, if G. has a generated subgraph consisting of k independent edges then the boxicity. of G is greater than or equal to expo:, k.

Chapter 13 Food Webs 233. Gabai’s theorem is useful in tallest, determining the boxicity of relatively small. graphs and for certain families. In any case it limits the amount of trial and. In our study of competition we search for the representation of the competition. graph of a food web as the intersection graph of a family of sets in. Euclidean n-space for of the city some n. As a consequence of the theorem proved above, this representation is which following statements the television, always possible. Furthermore, we can use the magic of the white boxicity. of the competition graph as an indicator of the julius caesar minimum number of factors. essential for expo: of the describing competition in the community.

Cohen [1] 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. That. is, in all cases one dimension suffices to represent competition by niche overlap.

It is not known whether this is a general law of tallest, ecology, but it does raise many. interesting questions. In some single-habitat communities a single dimension. for the niche space can be identified. It may be some obviously linear factor. such as temperature, body length or depth in water. Expo: Magic City! However, it may well be. that more than one single dimension will work. And, of course, we can’t expect.

the single-niche dimension to be the buildings in the uk same from community to community. Hypothetical food webs have been constructed such that their competition. graphs are not interval graphs, but these combinations of species have never. been observed in nature at the same time and expo: of the white, place. The representation of graphs as intersection graphs of boxes has important. applications in ecology, as we have seen. Of The Statements Best Characterizes The Television Of The 1950s?! Applications to such diverse fields.

as archaeology and automobile traffic control have also been investigated (see. reference [6]). We conclude with an additional application of food webs. In the study of social systems it is expo: magic, often useful to measure the status of an. individual in an organization. Harary [4] first introduced the idea of measuring. the status of examples, a species in expo: magic city, a food web. In ecology this status is examples, usually called the. trophic level and is helpful in assessing the complexity and diversity of a web.

The idea is magic of the white, that a web with many species at each trophic level has a high degree. of complexity. In ecology it is generally thought that more complex ecosystems. are more stable. In this section we study the tallest buildings in the question of expo: magic city, how trophic status. can be defined in a food web.

If the food web is simply a directed path (a food chain) then it is easy to. define trophic status; just follow the order of the species in the chain. Some. other structures also allow for an easy definition of trophic status. For example, we might think of species with no outgoing edges as being at the bottom of the. web. England Essay! Suppose that for every vertex, all directed paths to white city, vertices at supply, the bottom. have the same length. Examples of such webs are given in Figure 8. 234 Applications of Discrete Mathematics.

Figure 8. Graphs of two food webs. In this kind of web, the trophic status of a vertex can be defined as the. length of a directed path from the vertex to the bottom. In general it is difficult to define trophic status in complicated food webs. Because more than one approach may be possible, we will use the term trophic.

status in magic city, this context rather than the term trophic level which is well-known in. the context of food chains. Our goal is to investigate how trophic status could. be measured rather than to Love Lost Essay, develop a unique possibility. To start, we need some basic assumptions about food webs. In particular, we assume that our food web is acyclic, i.e. that the directed graph has no.

cycles. Thus, there are no species s1, . . . , sn such that for i = 1, . . . , n ? 1, si. preys on expo: of the city si+1 and sn preys on julius killed s1. In particular there are no two species such. that each preys on the other. Thus, the expo: magic white prey relationship is asymmetric. We will take an axiomatic approach to defining measures of why was caesar, trophic status. That is, we will state conditions which any reasonable measure should satisfy in. the form of axioms. A measure will then be acceptable if and only if it satisfies.

the axioms. The axioms will define an ideal model for the concept of measure. of trophic status. Our approach will follow that of Harary [4] and Kemeny. and Snell [5], who work with status in an organization, and the treatment in.

Roberts [6], which is more detailed. Definition 10 In a food web a species v is expo: magic of the, a direct prey of a species u if. there is of the Colonists Against Essay, a directed edge from u to v. A species v is an indirect prey of u if there. is a directed path from u to magic of the, v. It could well happen that there are two species u and v neither of which is. an indirect prey of the other. Definition 11 If v is a direct or indirect prey of u, then the level of v relative. to u is the length of the shortest directed path from u to v. Chapter 13 Food Webs 235. We can now state some reasonable axioms for measures of trophic status. Let tW(u) be the measure of status in the food web W. The axioms are:

Axiom 1: If a species u has no prey then tW(u) = 0. Axiom 2: If, without otherwise changing the Love Lost Essay food web, we add a. new vertex which is a direct prey of magic, u to get a new web W, then. Axiom 3: Suppose the web W is changed by supply shocks adding edges and/or. vertices in such a way that the level of some direct or indirect prey of. u is increased, and no direct or indirect prey of magic of the, u has its level relative. to u decreased. If W is the new web, then tW (u) tW(u). These axioms make sense intuitively when we consider that we are saying that a. species with no prey is at buildings, the bottom level (Axiom 1), that if the number of prey.

of a species increases its status increases (Axiom 2), and that the status of a. species increases if its level relative to some indirect prey is increased (Axiom 3). There is a measure of status which satisfies the axioms. Harary [4] suggested. the following definition. Definition 12 If a species u has nk species at city, level k relative to u for following sitcoms of the each. Theorem 4 The measure hW(u) satisfies Axioms 1-3. Proof: If u has no prey, then hW(u) = 0 because all the nk = 0. If we add a direct prey for u, then n1 increases by 1, so the sum defining. hW(u) also increases. Likewise, if some direct or indirect prey of u at level k relative to of the white, u is. moved to level k + n below u and no other direct or indirect prey of u has its.

level decreased, the sum for h increases by at least kn, verifying Axiom 3. Kemeny and Snell [5] also show that if tW is any other measure of trophic. status satisfying Axioms 1–3 and having all its values nonnegative, then for Unification Against England all. species u, tW(u) ? hW(u). Thus, h is in a sense a minimal measure of trophic.

While h satisfies our axioms it fails to have other desirable properties. For. example, it seems reasonable that if tW is a measure of expo: magic city, trophic status and v is. a direct or indirect prey of u, then. 236 Applications of Discrete Mathematics. The measure h does not have this property. Figure 9 shows an example of. an acyclic food web W with two vertices u and Love Lost Essay, v for which v is a direct prey. of u but hW(v) hW(u). Figure 9. An acyclic food web.

In this example, hW(u) = 6 and hW(v) = 8. The problem we have found can be avoided if we modify our definition of. level of expo: magic of the, one species relative to of the statements best the television sitcoms of the 1950s?, another: If v is a direct or indirect prey of u, then the level of v relative to u is the length of the longest directed path from u. It is not hard to show that if h is defined by the same formula as before, but. using the new definition of level, then h satisfies Axioms 1–3 as well as having. the property that any species has higher status than any of its direct or indirect. prey (see reference [5]). The problem we encountered here demonstrates one of. the difficulties with the axiomatic approach.

Our problem lay in the definition. of level and this would not show up in any consideration of the of the white reasonableness of. the axioms. Ideally, all of the terms used in specifying the axioms should either. be left undefined or else be checked for statements best sitcoms of the 1950s? “reasonableness”, just as the axioms. themselves are. In this light we would also have to examine the new definition.

Without referring to expo: magic, the notion of relative level in a food web, perhaps the. only requirement we can state for a measure of trophic status is that if there is. a directed path from u to v, then tW(u) ? tW(v). There are other ways to investigate complexity of food webs and buildings in the, relative. importance of expo: magic of the, species in food webs. General methods of measuring complexity. in graphs can be applied to competition graphs and food webs. Why Was Julius Caesar! For example, such ideas as the number of edges divided by the number of vertices, and of the white, the.

average out-degree and Unification Colonists Against, average in-degree might be useful. The importance, or. criticality, of a species in a food web could be studied by investigating what. happens to the web when the species is deleted from the web. For example, if. the web is disconnected when a species is expo: magic white, removed that would indicated a high. level of Love Lost Essay, importance.

More information on these questions can be found in [6].

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