Global Behavior of Solutions

Proposition 6.2. The system (2.4) is uniformly persistent if and only if the rest points E and E2 are unstable. 

In the previous section it was shown that the rest points E and E2, when unstable, had no part of their stable manifolds in the positive cone (Lemma 5.1). This shows that (H) of Theorem D.2 holds. If the covering is taken to be the set of rest points, the flow on the boundary is acyclic (in terms of Appendix D), for the rest points in the faces i = 2 = 0 and the V — V2 = 0 attract all points in that face. An application of Theorem D.2 completes the proof take A = IR+ and E = ( , 2, V, V2) e m, 0 for some / and Vj 0 for some J], 

Recall that hyperbolicity is a generic assumption in this chapter and that dissipativeness has been established. This has the following consequence. 

Proposition 6.3. The rest point exists if and only if Ey and E2 are unstable. 

If El and E2 are unstable, then dissipativeness and uniform persistence (previous proposition) yield the existence of an interior rest point for -k(x, t) (Theorem D.3). If E exists then it is unique and has all eigenvalues negative (Lemma 5.2). Suppose that exists and that Ei is asymptotically stable. Then, since / Ei and both are asymptotically stable, Theorem E.l contradicts the uniqueness of E. A similar argument applies if E2 is asymptotically stable. Note that the computations leading up to Lemma 5.1 explicitly determine the signs of the eigenvalues for linearization about Ei and E2.  

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Recent work of Wolkowicz and Lu [WLu] extends the results of [BWol] described here to include, in some cases, the possibility of population-dependent removal rates. However, at the time of this writing it remains an open problem to describe the global behavior of solutions of the equations modeling n competitors in the chemostat, allowing both for species-specific removal rates and for not necessarily monotone functional responses (e.g., assuming only (iii )). 

The chapter proceeds as follows. In the next section the variable-yield model of single-population growth is derived and analyzed. In Section 3, the competition model is formulated and its equilibrium solutions identified. The conservation principle is introduced in Section 4 in order to reduce the dimension of the system of equations by one local stability properties of the equilibrium solutions are also determined. The global behavior of solutions of the reduced system is treated in Section 5, and the global behavior of solutions of the original competitive system is discussed in Section 6. The chapter concludes with a discussion of the main results. 

These equations are familiar to every reactor physicist and therefore we do not dwell on the physical meaning of the functions and constants involved. We will mainly consider the nonlinear equations (1) and not any linearized versions. Also, we wish to stress that the purpose of this article is not to analyze the physical limitations of Eqs. (1) for athorough treatment of this problem we refer the reader to Gyftopoulos (/). Our problem is, instead, briefly stated as follows Given the equations (1) with a suitable hypothesis on the prescribed functions and constants, what can be said about the qualitative behavior of the solutions Under what conditions are the solutions bounded, when do they tend to equilibrium values, do the equations allow of finite escape times , etc. We make no attempt for completeness and naturally this paper to some extent reflects the interests of the author. We do hope, however, that the results presented here might serve as a guide or a summary to reactor physicists about what is known of the global behavior of solutions of Eqs. (1). 

As in the model of Section 2, the problem can be studied on its omega limit set with three rest points Eq,Ei,E2. A local stability analysis and, for some special cases, the asymptotic behavior of solutions were given in [E]. However, the populations cannot invade each other simultaneously El and E2 cannot be simultaneously unstable), so the persistence theory does not hold [E]. Moreover, for Michaelis-Menten dynamics, when one of the boundary rest points is locally stable and the other unstable, the locally stable one is globally stable [HWE]. In particular, the oscillation observed in the case of system (3.2) does not occur with (3.4). Indeed, the delayed system seems to behave much like the simple chemostat. 

The quantities 2 2 and V2 are the same as z and v, respectively, that were defined in Chapter 2. It is important to note that Z2 is proportional to while z is independent of L. It can be shown that Zj defined similarly for a j-body (J = 4,5,...) cluster is inversely proportional to the — j — 3)/2-fli power of L. Hence, zj for j > 4 asymptotically vanishes as L 00. However, this fact does not mean that only the binaiy and ternary cluster interactions may be taken into account in formulating global behavior of long chains in dilute solution. Inclusion of the higher-order cluster interactions in the analysis offers an interesting problem to theorists. 

The ideally flexible continuous chain generated from the spring-bead chain retains no microscopic feature of actual polymer molecules and hence it is the most abstract of polymer models. The wormlike chain, though a continuous chain, is more realistic since through the parameter q it allows for the stiffness possessed by actual molecules. Up to this point we have seen a number of examples which substantiate the usefulness of these chain models for the quantitative description of global behavior of polymers in dilute solution. But this never means that no other chain model need be considered. 

From a global assessment of these results, it seems inescapable to conclude that mean-field behavior does not remain valid asymptotically close to the critical point. Rather, ionic systems seem to show Ising-to-mean-field crossover. Such a crossover has been a recurring result observed near liquid-liquid consolute points in Coulombic electrolyte solutions, in ternary aqueous electrolyte solutions containing an organic cosolvent, and in binary aqueous solutions of NaCl near the liquid-vapor critical line. 

Another possible electrocatalytic process is that related to a surface-bound molecule which can give rise to a two-electron reaction. In these conditions, the coupling of the catalytic reaction in the presence of an adequate species in solution can lead to different mechanistic schemes from which the elucidation of the global reaction path is not immediate. This situation matches the behavior of a great number of inorganic catalysts (such polyoxometallates or ion complexes) [86, 98] and biological molecules (enzymes, proteins, oligonucleotides, etc.) [79, 80], for which there is a lack of theoretical basis which enables a clear classification of the different possibilities that can be encountered. 

A second, even more speculative point is that the mathematical framework of nonlinear dynamics may provide a basis to begin to bridge the gap between local microstructural features of a fluid flow or transport system and its overall meso- or macroscale behavior. On the one hand, a major failure of researchers and educators alike has been the inability to translate increasingly sophisticated fundamental studies to the larger-scale transport systems of traditional interest to chemical engineers. On the other hand, a basic result from theoretical studies of nonlinear dynamical systems is that there is often an intimate relationship between local solution structure and global behavior. Unfortunately, I am presently unable to improve upon the necessarily vague notion of a connection between these two apparently disparate statements. 

On a global scale, the linear viscoelastic behavior of the polymer chains in the nanocomposites, as detected by conventional rheometry, is dramatically altered when the chains are tethered to the surface of the silicate or are in close proximity to the silicate layers as in intercalated nanocomposites. Some of these systems show close analogies to other intrinsically anisotropic materials such as block copolymers and smectic liquid crystalline polymers and provide model systems to understand the dynamics of polymer brushes. Finally, the polymer melt-brushes exhibit intriguing non-linear viscoelastic behavior, which shows strainhardening with a characteric critical strain amplitude that is only a function of the interlayer distance. These results provide complementary information to that obtained for solution brushes using the SFA, and are attributed to chain stretching associated with the space-filling requirements of a melt brush. 

Unfortunately, there is a disparity between this theoretical convergence result and the practical behavior of the method in general. Thus, modifications of the classic Newton iteration are essential for guaranteeing global convergence, with quadratic convergence rate near the solution. 

Unfortunately, the presence of the term in Eq. (5) makes it much more difficult to extract the equilibrium S J) behavior than it was to find (p T) for an Ising magnet. However, the general approach is the same— fmd the S value at which dQdS= 0 and G is a global minimum. The temperature dependence of 5arising from Eq. (5) for a nematic liquid crystal turns out to be quite complicated since there is no analytic solution to the quartic equation arising from 5 QdS = 0. However, the behavior of 5( 7) for 7 < 7j can still be very well approximated by a power-law expression ... 

The potentiality of hierarchical stratification of complex reactive systems, according to the characteristic times of the involved processes, makes it difficult to use direcdy thermodynamic tools as well as to apply the con cept of stability to very compHcated (in particular, biological) systems. The statistical approach to describe the behavior of a system that contains a large number of particles takes into account the instabihty of mechanical trajectories of individual particles. Indeed, any infinitesimally small distur bances in the particles motion can make it impossible to determine from the starting conditions the trajectory of even one particle s motion. As a result, a global instabihty of mechanical states of individual particles is observed, the system becomes statistical as a whole, and the trajectories of individual particles are no longer predictable. At the same time, the states that correspond to stable solutions of any dynamic (kinetic) problem can only be observed in real systems. In terms of a statistical approach, the dynamic solution of a particular initial state of an ensemble of particles is a fluctuation, while the evolution of instabihty upon destruction of this solution is a relaxation of this fluctuation. 

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