The Set of Rest Points

As with the chemostat, the basic approach is to locate the rest points, analyze their local stability, and determine the global properties of the dynamical system. The first lemma gives some estimates of quantities that will be important in the analysis. It turns out to be easier to state the results in terms of rest points of the system (2.2) rather than those of the system (2.4). These results will be interpreted as needed for the system (2.4). 

Before proving this technical lemma, we make some observations. One might think that coexistence in the gradostat is possible by having one competitor win in one vessel and the other in the second vessel. Statement (a) shows this is not possible (at least, not as a steady state). The last statements in (c) and (d) provide a crucial key to actual computation of the coordinates of the rest points. The other inequalities provide estimates that will be useful in stability considerations. 

Proof The proofs of these assertions are all relatively straightforward. The results will be established for m the others follow similarly. The equations for the rest point are just the equations with the left-hand side of (2.2) set equal to zero. The equations for m yield the first statement in (a) directly. The second equation in (2.2) at equilibrium becomes 

The term cannot be zero. Since / is monotone increasing, (b) follows. To prove (c), note that 

The inequalities then follow from (b). This completes the proof of the lemma.  

---

We turn now to the question of competition. In the previous chapter, we established a classification of the dynamic behavior based on the set of rest points. Unfortunately, our computations - which established the stability of any interior rest point and thereby led to the conclusion that such an equilibrium is unique in the case of two vessels and Michaelis-Menten uptake functions - are extremely difficult for n vessels and general uptake functions [HSo], so the results in this case are not as simple as in that chapter. In the present context we attempt to classify the dynamics in terms of both the set 0 of rest points and the sign of the stability modulus of certain key matrices. The theory of monotone dynamics is then used to resolve global questions. The principal result is Theorem 4.4. There are three obvious candidates for equilibria ... 

Now we state without proof some results that require tt to be strongly monotone. They are all due to Hirsch. Let E be the set of rest points of w E=[x f(x)=0. ... 

In the mathematical theory of dynamical systems, the more general concept of (w-limit sets has been developed. These limit sets help in gaining an overall understanding of how a dynamical system behaves, particularly in the long term, by providing an asymptotic description of the dynamics. An (w-limit set combines a set of rest points with a set of specific phase trajectories, particularly trajectories around the rest points. In the simplest case, the (W-limit set (w(k, Co) consists of just a rest point. 

A detailed study of the flavonoid chemistry of the island endemics, the closely related G. tinctoria, and live additional species from the mainland provided additional evidence pointing toward G. tinctoria as the ancestral species (Pacheco et al., 1993). The flavonoid profiles of all species consisted of flavonol glycosides as major components with an unidentified flavone glycoside and several unidentified phenolic compounds (presumably not flavonoids). The pattern of distribution of the flavonol glycosides and unidentified flavones within the set of nine species proved to be extremely informative. (The phenols were ubiquitous and are not considered further.) Kaempferol glycosides were seen in neither the island species nor G. tinctoria, but were present, in several combinations, in the rest of the mainland taxa. The isorhamnetin glycosides showed the reverse pattern, with one exception the island endemics and G. tinctoria exhibited these compounds, whereas four of the other mainland species did not. The sole exception is G. boliviari, which exhibited one of the isorhamnetin derivatives. 

Fig. 1. Examples of w-limit sets, (a) Rest point (b) limit cycle (c) Lorenz attractor [projection on the (Cj, c3) plane, a = 10, r = 30, b = 8/3].

One of the most commonly used measures of central tendency is the mean, more correctly (but rarely) called the arithmetic mean, a term that unambiguously distinguishes it from the geometric mean. While very informative in some circumstances, the geometric mean is less commonly used, and, in the absence of the prefix arithmetic or geometric, the default interpretation of the term mean is the arithmetic mean. This is the convention followed in the rest of this book. The mean of a set of data points is therefore defined as their sum divided by the total number of data points. 

The equilibrium state of a system can be represented by a point in a (c + 2)-dimensional diagram. Phase diagrams are state diagrams for open systems, in which a particular state of equilibrium is specihed by the values of c + 2 independent state variables (the rest ate hxed dependent variables), with additional information on the phases present under various conditions. Although one typically sees phase diagrams with potentials (T) P, /a) chosen as the state variables, there are actually many possible ways to select the set of independent variables. 

For a fixed set of parameters, let 2 denote the rest point set of the system (2.4) in r. There are four possible types of rest points, which we denote as follows ... 

Enough information has now been collected to classify the behavior of all solutions of the gradostat equations. This classification will be given as a function of the set Q of rest points. The existence and stability of the rest points has already been established. Indeed, the coordinates of the rest... 

By a nontrivial periodic orbit we mean a periodic orbit that is not a rest point. Such an orbit is attracting if the omega limit set of each point of some neighborhood of the periodic orbit is the periodic orbit. 

The theorem states that almost all trajectories converge to one of the asymptotically stable rest points (x, 0). In most of our applications, r = 1 there is exactly one asymptotically stable rest point. In this case, an observer would conclude that all trajectories converge to (x, 0), since the probability of an initial condition being in the exceptional set is zero. 

The Sections on applications in this review are preceded by a brief explanation of ESR spectroscopy, the purpose of which is to help the newcomer to this technique to understand the content of the rest of this article. It will not enable him to read ESR work critically and still less to perform ESR experiments himself for these purposes, reference should be made to one of the several textbooks which now exist on ESR spectroscopy, or better still, to obtain the collaboration of an ESR spectroscopist. It should be pointed out that the ESR spectrum obtained is dependent on the state of the sample and on the settings of the various controls of the spectrometer, and that neglect of these factors in the past has led to some ESR work, particularly in the medical field, being of less value than it might have been. 

A/j for the components (C//4, H O, H2, CO, CO2) at the collocation points (B.41-B.45) are found by a subroutine called FLUX. The subroutine FLUX evaluates ch4 Xcm the collocation points by solving the set of 2N linear algebraic equations (B.41-B.42) — excluding the centre of the pellet where the fluxes are known — by Gauss elimination with partial pivoting using the subroutine called GAUSL (Villadsen and Michelsen, 1978). The rest of the fluxes of the components are found from the stoichiometric equations (5.215). The roots (Uj) of the Jacobi polynomial (w) and the discretization... 

The initial conditions are satisfied by setting the value of every point in the concentration grid to 1 before the simulation begins. We must now consider how the rest of these conditions are represented in discrete form when they are treated implicitly (boundary perpendicular to the implicit direction) and explicitly (boundary parallel to the implicit direction). 

Examples of ta-limit sets (A) rest point, (B) limit cycle, (C) Lorenz attractor (projection on the x,y) plane a = 0, p = 30, f = 8/3). Reprinted from Yablonskii, G.S., Bykov, V.i, Gorban, A.N., Elokhin, V.I., 1991. Kinetic models of catalytic reactions. In Compton, R.G. (Ed.), Comprehensive Chemical Kinetics, vol. 32. Elsevier, Amsterdam, Copyright (1991), with permission from Elsevier. 

The state of the ideal rubber can be specified by the locations of all the junction points, ij, and by fce end-to-end vectors for all tire chains connecting the junction points,. The first postulate of the statistical theory of rubber elasticity is that, in the rest state with no external constraints, the distribution fimction for the set of chain end-to-end vectors is a Gaussian distribution witii a mean-squared end-to-end distance that is proportional to the molecular weight of the chains between jimcnons ... 

---