Omega limit set

If the omega limit set is particularly simple - a rest point or a periodic orbit - this gives information about the asymptotic behavior of the trajectory. An invariant set which is the omega limit set of a neighborhood of itself is called a (local) attractor. If (3.1) is two-dimensional then the following theorem is very useful, since it severely restricts the structure of possible attractors. 

If an omega limit set contains an asymptotically stable rest point P, then that point is the entire omega limit set. If all of the eigenvalues of the variational matrix have positive real part then the rest point is said to be a repeller such a rest point cannot be in the omega limit set of any trajectory other than itself. If k eigenvalues have positive real part and n-k eigenvalues have negative real part then there exist two sets M P), called the stable manifold and defined by... 

Theorem (Butler-McGehee). Suppose that P is a hyperbolic rest point of (3.1) which is in w(x), the omega limit set of but is not the en-... 

Since all trajectories of the original system are asymptotic to their omega limit set, in analyzing this equation it is sufficient to determine the asymptotic behavior of (2.3). From a more intuitive viewpoint this is merely starting on the manifold 5-f-x= 1, to which all solutions must tend the mathematical support for this is rigorously established later (see the proof of Theorem 5.1 or Appendix F). Define, for aw > 1,... 

Both eigenvalues are positive and the origin is a repeller. In particular, the origin is not in the omega limit set of any trajectory (other than itself). At (1 - Ai, 0), the variational matrix is of the form... 

Since 1 is a local attractor, to prove the theorem it remains only to show that it is a global attractor. This is taken care of by the Poincare-Bendixson theorem. As noted previously, stability conditions preclude a trajectory with positive initial conditions from having 0 or 2 in its omega limit set. The system is dissipative and the omega limit set is not empty. Thus, by the Poincare-Bendixson theorem, the omega limit set of any such trajectory must be an interior periodic orbit or a rest point. However, if there were a periodic orbit then it would have to have a rest point in its interior, and there are no such rest points. Hence every orbit with positive initial conditions must tend to j. (Actually, two-dimensional competitive systems cannot have periodic orbits.) Figure 5.1 shows the X1-X2 plane. 

For the reader unfamiliar with Liapunov theory, the intuition goes like this V is some sort of measure of height. The magnitude of dV/dt = VV-/< 0 is a measure of how fast solutions run downhill. The downhill slide stops at E. The theory of dynamical systems says that the omega limit set is invariant and therefore contained in M. 

Following the same approach as in Chapter 1, we seek to limit the size of the system by restricting ourselves to a set containing the omega limit set (see Appendix F). Let E = 1 — S—x——z. Then system (2.2) may be written as... 

Clearly, lim, E(t) = 0 and hence the omega limit set of any trajectory lies in the set E = 0. (Alternatively, one could appeal to the theory discussed in Appendix F.) Trajectories in the omega limit set are solutions of the following system ... 

As noted in [LH], the ability of Xi to consume the inhibitor (8 > 0) is of crucial importance. Lenski and Hattingh refer to this ability of Xi to detoxify the environment and note that without it (i.e., with 6 = 0 in (2.2)) p t) tends to unity as t tends to infinity. Therefore, the limiting system obtained by dropping the p equation and replacing p by 1 describes the dynamics of (2.2) on the omega limit set. This limiting system is just the equations for competition in the chemostat without an inhibitor and where is replaced by w,/(l). Competitive exclusion must then result. 

A major difference between competitive and cooperative systems is that cycles may occur as attractors in competitive systems. However, three-dimensional systems behave like two-dimensional general autonomous equations in that the possible omega limit sets are similarly restricted. Two important results are given next. These allow the Poincare-Bendix-son conclusions to be used in determining asymptotic behavior of three-dimensional competitive systems in the same manner used previously for two-dimensional autonomous systems. The following theorem of Hirsch is our Theorem C.7 (see Appendix C, where it is stated for cooperative systems). 

Theorem 6.3 [Hi4]. Let L be a compact omega limit set of a competitive system in If L contains no equilibria, then L is a closed orbit. 

By Proposition 4.1, if exists then E2 is unstable and [, if it exists, is also unstable. We begin by showing that if exists, the omega limit set of every solution for which a ,(0) > 0 (/ = 1,2) remains a positive distance away from the boundary of K+ in the terminology of Appendix D, the system (3.2) is persistent. 

The omega limit set of any trajectory lies interior to the positive cone. 

Proof. Note that M (Eq), the stable manifold of Eq, is either the p axis if El exists or the x -p plane if Ei does not exist. The manifold M E2) is the X2 p plane less the p axis if E exists, M (Ei) is the Xi p plane less the p axis. Since (Xi(0), X2(0), p(0)) does not belong to any of these stable manifolds, its omega limit set (denoted by w) cannot be any of the three rest points. Moreover, w cannot contain any of these rest points by the Butler-McGehee theorem (see Chapter 1). (By arguments that we have used several times before, if w did then it would have to contain Eq or an unbounded orbit.) If w contains a point of the boundary of then, by the invariance of w, it must contain one of the rest points Eq,Ei,E2 or an unbounded trajectory. Since none of these alternatives are possible, CO must lie in the interior of the positive cone. This completes the proof. 

Proof. In view of Theorem 7.1, the omega limit set of any trajectory cannot be on the boundary Xj = 0 or X2 = 0. Away from the boundary, the system is irreducible. Since there are no limit cycles, all trajectories... 

The difference between this and the previous conservation results is that it holds in each vessel rather than for the entire biomass. As a consequence of this lemma, the omega limit set of any trajectory is nonempty, compact, connected, and contained in T. Since every trajectory is asymptotic to its omega limit set, it is important to analyze the system on this set. (See Appendix F for a rigorous justification.) Trajectories in the omega limit set satisfy... 

The conservation principle established previously applies here, so one may deal (on the omega limit set) with the system obtained by setting Si(t) = j-Ui(t) and 82(0 = j-U2(t) ... 

Case (b) occurs if and only if one of the inequalities in (5.2a) or (5.2b) is reversed and the other set of inequalities holds. Lemma 6.1 again yields that El belongs to the omega limit set of any trajectory with positive initial conditions. 

Lemma 2.2 says that on the omega limit set, solutions of the system (1.6) satisfy... 

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. 

Clearly lim, E(/) = 0 and trajectories on the omega limit set satisfy E = 0. If all of the rest points of the limiting system are hyperbolic (which will be implied by the conditions stated) and if there are no periodic orbits (which needs to be proved), then the results of Appendix F apply. The limiting system is... 

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. 

According to Theorem C.6, the limit set can be deformed to a compact invariant set A, without rest points, of a planar vector field. By the Poin-care-Bendixson theorem, A must contain at least one periodic orbit and possibly entire orbits which have as their alpha and omega limits sets distinct periodic orbits belonging to A. Using the fact that A is chain-recurrent, Hirsch [Hil] shows that these latter orbits cannot exist. Since A is connected it must consist entirely of periodic orbits that is, it must be an annulus foliated by closed orbits. Monotonicity is used to show... 

Theorem C.7 bears a strong resemblance to the Poincare-Bendixson theorem stated in Chapter 1. It will be used in Chapter 4 for the case where (C.l) is a competitive system, that is, for a system (C.l) where -/ is cooperative. Note that the omega (alpha) limit set of a competitive system is the alpha (omega) limit set of the time-reversed cooperative system, so Theorems C.5, C.6, and C.7 apply to competitive systems. Unlike cooperative systems, competitive systems can have attracting periodic orbits. For more on the Poincare-Bendixson theory of competitive and cooperative systems in see [S3], [SWl], and [ZS]. 

We review the basic definitions and set up the semidynamical system appropriate for systems of the form (D.l). Let A" be a locally compact metric space with metric d, and let be a closed subset of X with boundary dE and interior E. The boundary, dE, corresponds to extinction in the ecological problems. Let tt be a semidynamical system defined on E which leaves dE invariant. (A set B in A" is said to be invariant if n-(B, t) = B.) Dynamical systems and semidynamical systems were discussed in Chapter 1. The principal difficulty for our purposes is that for semidynamical systems, the backward orbit through a point need not exist and, if it does exist, it need not be unique. Hence, in general, the alpha limit set needs to be defined with care (see [H3]) and, for a point x, it may not exist. Those familiar with delay differential equations are aware of the problem. Fortunately, for points in an omega limit set (in general, for a compact invariant set), a backward orbit always exists. The definition of the alpha limit set for a specified backward orbit needs no modification. We will use the notation a.y(x) to denote the alpha limit set for a given orbit 7 through the point x. 

---