Omega limit point

However, if (5.2a) is satisfied then iim, co a ,(0 = 0 i — 1,2), so the omega limit point p must be of the form (0,0, , ) that is, the u population becomes extinct. Similarly, if (5.2b) holds then p is of the form (, , 0,0) and the v population is eliminated. Note that this is independent of initial conditions and hence is a global result. The component marked will be zero or positive depending on the other inequality. 

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 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. 

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. 

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... 

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. 

OMEGAF and siGMAF Contain the estimates of omega and sigma, which in this case will be the same as the values specified on the omega and sigma rows since maxeval is set to 0. The limits of the prediction intervals are stored in the variables com (i) and COM (2), which are tabulated. The input data file has to include all time points for which the interval is to be computed (including any doses). 

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