6.5 QUALITATIVE STABILITY The Routh-Hurwitz criteria outlined in Section 6.4 are an exact but cumbersome method for determining stability of a large system. For communities of five or more species, the technique proves so computationally involved that it is of diminishing practical value. Shortcuts, when available, can be quite useful. In this section we explore a shortcut method for investigating large systems that needs little if any computation. Because this method is not universally applicable, its importance is viewed as secondary. Furthermore, to understand exactly why the method works requires knowledge of matrices beyond elementary linear algebra. Nevertheless, what makes the technique of qualitative stability appealing is that it is easy to explain, easy to test, and thus a refreshing change from intensive computations. The technique of qualitative stability analysis applies ideally to large complicated systems in which there is no quantitative information about the interrelationship of species or subsystems. Motivation for this method actually came from economics. A paper by the economists Quirk and Ruppert (1965) was followed later by further work and application to ecology by May (1973), Levins (1974) , and Jeffries (1974). In a complex community composed of many species , numerous interactions take place. The magnitudes of the mutual effects of species on each other are seldom accurately known , but one can establish with greater certainty whether predation , competition , or other influences are present. This means that technically the functions appearing in equations that describe the system [such as equation (11)] are not known. What is known instead is the pattern of signs of partial derivatives of these functions [contained , for example , in the Jacobian of equation (16)]. We encountered a similar problem in the context of a plant-herbivore system (Chapter 3 ) and of a glucose-insulin model (Chapter 4). Here the problem consists of larger systems in a continuous setting , and even the magnitudes of partial derivatives may not be known. There are two equivalent ways of representing qualitative information. A more obvious one is to assign the symbols + , 0 , and – to the (i,j)th entry of a matrix if the species j has respectively a positive influence , no influence , or a negative influence on species i. An alternate , visual representation captures the same ideas in a directed graph (also called digraph) in which nodes represent species and arrows between them represent the mutual interactions , as shown in Figures 6.10 and 6.11 The question is then whether it can be concluded. From this graph or sign pattern only , that the system is stable. If so , the system is called qualitatively stable Figure 6.10 Signed directed graphs (bigraphs) can be used to represent species interactions in a complex ecosystem. The graphs shown here are equivalent to the matrix representation of sign patterns given in the text (a) example 2, (b) example 3, and (c) example 4. Figure 6.11 Properties of signed directed graphs can be used to deduce whether the system qualitatively stable (stable regardless of the magnitudes of mutual effects) The Jeffries color test and the Quirk-Ruppert conditions are applied to these graphs to conclude that (a), which corresponds to example 2, and (c) are stable communities, whereas (b), which corresponds to example 4, is not. Systems that are qualitatively stable are also stable in the ordinary sense. (The converse is not true.) Systems that are not qualitatively stable can still be stable under certain conditions (for example, if the magnitudes of interactions are appropriately balanced.) Following Quirk and Ruppert (1965), May (1973) outlined five conditions for Example 2 Here we study the sign pattern of the community described in equations (21a,b,c) of Section 6.4 From Jacobian (24) of the system we obtain the qualitative matrix Q = สัญญาณ J = This means that close to equilibrium, the community can also be represented by the graph in Figure 6.10 Reading entries in Q from left to right, top to bottom: Species 1 gets positive feedback from species 2 and 3. Species 2 gets negative feedback from species 1 Species 3 gets negative feedback from species 1 and from itself. Example 3 (Levins, 1977) In a closed community, three predators or parasitoides, labeled P1, P2, and p3, attack three different stages in the life cycle of a host, H1, H2, and H3. The presence of hosts is a positive influence for their predators but predators have a negative influence on their prey. Figure 6.10(b) and the following matrix summarize the interactions: Note that H1, H2, and H3 each exert negative feedback on themselves. Example 4 (Jeffries, 1974) In a five-species ecosystem, species 2 preys on species 1, species 3 on species 2, and so on in a food chain up to species 5. Species 3 is also self-regulating. A qualitative matrix for this community is See Figure 6.10(c) Qualitative stability. Suppose aij is the ij th element of the matrix of signs Q. Then it is necessary for all of the following conditions to hold: • aij < 0 for all i. • Aij < 0 for at least one i. • Aijaji< 0 for all i = j. • Aijaik...aqrari=0 for any sequences of three or more distinct indices i, j, k, ... ,q, r. • Det Q = 0. These con ditions can be interpreted in the following way: • No species exerts positive feedback on itself. • At least one species is self- regulating. • The members of any given pair of interacting species must have opposite effects on each other. • There are no closed chains of interactions among three or more species. • There is no species that is unaffected by interactions with itself or with other species. For mathematical proof of these five necessary conditions, consult Quirk and Ruppert (1965). May (1973) and Pielou (1969) comment on the biological significance, particularly of conditions 3 and 4. The conditions can be tested by looking at graphs representing the communities. One must check that these graphs have all the following properties: 1. No + loops on any single species (that is, no positive feedback). 2. At least one - loop on some species in the graph. 3. No pair of like arrows connecting a pair of species. 4. No cycles connecting three or more species. 5. No node devoid of input arrows. These five condition are eQuivalent to the orginal algebraic statement Example 5 For example 2 to 4 we check off the conditions given earlier: Condition Number Example 2 Example 3 Example 4 It was shown Jeffries (1974) that these five conditions alone cannot distinguish between neutral stability (as in the Lotka-Volterra cycles) and asymptotic stability wherein the steady state is a stable node or spiral. (In other words, the conditions are necessary but not sufficient to guarantee that the species will coexist in a constant steady state). In example 4 Jeffries notes that pure imaginary eigenvalues can occur , so that even though the five conditions are met, the system will oscillate. To weed out such marginal cases, Jeffries devised an auxiliary set of conditions, which he called the color test, that replaces conditions 2. Before describing the color test, it is necessary to define the following: A predation link is a pair of species connected by one + line and one - line. A predation community is a subgraph consisting of all interconnected predation links. If one define a species not connected to any other by a predation link as a trivial predation community, then is a possible to decompose any graph tito a set of distinct predation communities. The systems shown in Figure 6. 10 have predation communities as follows: (a) {2, 1, 3}; (a) {H1, P1}, {H2, P2}, {H3, P3}; and (c) {1, 2, 3, 4, 5}. InFigure 6.11(c) there are three predation communities: {1, 2, 3}, {4, 5}, and{6, 7, 8}. The following color scheme constitutes the test to be made. A predation community is said to fail the color test if it is not possible to color each node in the subgraph black or white in such a way that 1. Each self-regulating node is black 2. These is at least one white point. 3. Each white point is connected by a predation link to at least one other white point. 4.Each black point connected by a predation link to one white not is also connected by a predation like to one other white nod. Jeffries (1974) proved that for asymptotic stability, a community must satisfy the original Quirk-Ruppert condition 1, 3, 4, and 5, and in addition must have only predation communities that fail the color test. Example 5 (continued) Example 2 and 4 satisfy the original conditions. In Figure 6. 11 the color test is applied to their communities. We see that example 2 consists of a single predation community that fails part 4 of the color test. Example 4 satisfies the test. A final example shown in Figure 6.11(c) has three predation communities, and each one fails the test. We conclude that Figure 6.11(a) and (c) represent systems consist of species that coexist at a stable fixed steady state without sustained oscillations. A proof and discussion of the revised conditions is to be found in Jeffries (1974). For other applications and properties of graphs, you are encouraged to peruse Roberts (1976).
Posted on: Tue, 30 Sep 2014 09:31:04 +0000
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