Monotone dynamical system

H.L. Smith. Monotone dynamical systems. An introduction to the theory of competitive and cooperative systems. AMS Mathematical Surveys and Monographs, 41 31-53, 1995. 

The study of mathematical models of competition has led to the discovery of some very beautiful mathematics. This mathematics, often referred to as monotone dynamical systems theory, was largely developed by M. W. Hirsch [Hil Hi3], although others have made substantial contributions as well. In this section we describe a result that was first obtained in a now classical paper of DeMottoni and Schiaffino [DS] for the special case of periodic Lotka-Volterra systems. Later, it was recognized by Hale and Somolinos [HaS] and Smith [S4 S5] that the arguments in [DS] hold for general competitive and cooperative planar periodic systems. The result says that every bounded solution of such a system converges to a periodic solution that has the same period as the differential equation. 

Three types of new directions are discussed. In two of these, ordinary differential equations are not an adequate model to describe the phenomenon of interest functional differential equations and partial differential equations provide the appropriate setting. In the remaining case ordinary differential equations are appropriate but the modeling is not complete. Improving the model would result in a larger system for which the techniques of monotone dynamical systems are inappropriate. The problems will be described and results indicated, but no proofs are given. In all cases, much more work needs to be done before the problem is appropriately modeled and analyzed. 

The theorem shows clearly that plasmid loss is detrimental (or fatal) to the production of the chemostat. To compensate for this possibility, in commercial production a plasmid that codes for resistance to an antibiotic is added to the DNA that codes for the item to be produced. Thus, if the plasmid is lost then the wild type is susceptible to (inhibited by) the antibiotic. The antibiotic is introduced into the feed bottle along with the nutrient. The dynamics produced by adding an inhibitor to the chemostat was modeled in Chapter 4. A new direction for research on chemostat models would be to include the inhibitor, as in Chapter 4, and the plasmid model of this section (or one of the more general models) into the same model. This is a mathematically more difficult problem to analyze, since the reduced system will not be planar. Moreover, because the methods of monotone dynamical systems do not apply, other techniques would need to be found in order to obtain global results. The model also assumes extremely simple behavior for the plasmid more could be included in a model. 

A more interesting question from the ecological point of view is that of how many different populations can coexist in an -vessel gradostat. [JST] shows that this number cannot exceed n. Some numerical simulations and conjectures appear in [BWu CB], but very little is known about this question. New techniques must be developed to handle this problem, since the resulting equations do not generate a monotone dynamical system when the number of competitors exceeds two (see [JST]). 

Similarly, our analysis of the variable-yield model in Chapter 8 is limited to two competing populations because we rely on the techniques of monotone dynamical systems theory. One would expect the main result of Chapter 8 to remain valid regardless of the number of competitors, just as it did for the simpler constant-yield model treated in Chapters 1 and 2. Perhaps the LaSalle corollary of Chapter 2 can be used to carry out such an extension, using arguments similar to those used in [AM] (described in Chapter 2). As noted in [NG], a structured model in which... 

C.l) has the form (B.9) where F and G are independent oft and (B.12) holds in D, then w is a monotone dynamical system with respect to If in addition, the Jacobian matrix of f is irreducible at every point ofD, then w is strongly monotone with respect to... 

Theorem C.2. Let 7 (a ) be an orbit of the monotone dynamical system (C.l) which has compact closure in D. Then either of the following conditions is sufficient for u(x) to be a rest point ... 

Theorem C.3. A monotone dynamical system cannot have a nontrivial attracting periodic orbit. 

Monotonicity of a dynamical system places restrictions on the basin of attraction of a rest point. Suppose that Xq is a rest point of the monotone dynamical system generated by (C.l). Let B denote the basin of attraction of Xq ... 

Theorem C.4. The stable manifold of an unstable, hyperbolic rest point of a monotone dynamical system cannot contain two points that are related by the strict inequality stable manifold cannot contain two distinct points that are related by stable manifold is unordered. 

A similar assertion to that of Theorem C.4 holds for a compact limit set of a monotone dynamical system. 

Theorem C.6. A compact limit set of a monotone dynamical system in 0 can be deformed by a Lipschitz homeomorphism (with a Lipschitz inverse) to a compact invariant set of a Lipschitz system in IR"" in such a way that trajectories are mapped to trajectories and such that the parameterization of solutions is respected. 

By reversing time - that is, by replacing the vector field / constructed in the previous paragraph by -/ - we see that a monotone dynamical system can have essentially arbitrarily complex, (n —l)-dimensional dynamics. Of course, upon reversing time the invariant set S now becomes a repelling set and the equilibrium points 0 and 00 become the attractors. 

Theorem C.8. Let ir bea strongly monotone dynamical system such that E has no points of accumulation in D. Suppose that 7 (x) has compact closure in D for every xeD. Then the set of points x for which w(x, t) does not converge to an equilibrium has Lebesgue measure zero. 

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