We continue to assume IRnis ordered by a cone K having nonempty interior; all notions involving order refer to that defined by K. For this section, the terms “competitive” and
“cooperative” are tacitly understood to mean “K-competitive” and “K-cooperative”, and monotonicity refers to the ordering defined by K.
We first apply results from section 2 to obtain a generic stable convergence theorem for cooperative vector fields.
Let Φ denote the local flow generated by a vector field f on D ⊂ IRn. We assume D is p-convex throughout this section without further mention. When Φt(x) is defined for all (t, x) ∈ [0, ∞) × D, as when all positive semiorbits have compact closure in D, the corresponding positive local semiflow Φ+ is a semiflow. To Φ we associate C, S and E, denoting respectively the sets of convergent, stable and equilibrium points for Φ+. Theorem 3.14 Let f be a cooperative vector field on an open set D ⊂ IRn, generating a local flow Φ such that:
(a) every positive semiorbit of Φ has compact closure in D (b) Condition (ST*) above is satisfied, and D = AC ∪ BC.
Then Φ has the following properties:
(i) C ∩ S contains a dense open subset of D, consisting of points whose trajectories converge to equilibria.
(ii) If E is compact there is a stable equilibrium, and an asymptotically stable equilib-rium when E is finite.
Proof: Assumption (ST*) makes Φ strongly monotone. The hypothesis of Theorem 2.26, with X = D, is fulfilled: D is normally ordered and D = BC ∪ AC. Therefore Theorem 2.26 implies the conclusion.
Theorem 3.14, like Theorem 3.13, holds for some more general domains D, including relatively open subsets of V where V denotes a closed halfspace, a closed order interval, or the cone K.
One of the main results of this subsection is that n-dimensional competitive and cooperative systems behave like general systems of one less dimension. Theorems 3.21 and 3.22 illustrate this principle for n = 2 in a very strong form. In higher dimensions the principle holds for compact limit sets. The key tool in proving this is the following result due to Hirsch [66]:
Theorem 3.15 set of a competitive or cooperative system cannot contain two points related by .
Proof: By time reversal, if necessary, we assume the system is cooperative, hence the local flow is monotone. Now apply Proposition 1.10.
A periodic orbit of a competitive or cooperative system is a limit set and consequently it cannot contain two points related by . The following sharper result will be useful later:
Proposition 3.16 nontrivial periodic orbit of a competitive or cooperative system can-not contain two points related by <.
Proof: By time-reversal we assume the system is cooperative, and in this case the conclusion follows from Proposition 1.10.
Let Φ, Ψ be flows in respective spaces A, B. We say Φ and Ψ are topologically equivalent if there is a homeomorphism Q : A → B that is a conjugacy between them, i.e., Q ◦ Φt = Ψt ◦ Q for all t ∈ IR. The relationship of topological equivalence is an equivalence relation on the class of flow; it formalizes the notion of “having the same qualitative dynamics”.
A system of differential equations y0 = F (y), defined on IRk, is called Lipschitz if F is Lipschitz. That is, there exists K > 0 such that |F (y1) − F (y2)| ≤ K|y1 − y2| for all y1, y2 ∈ IRk. With these definitions, we can state a result of Hirsch [66] that follows directly from Theorem 3.15.
Theorem 3.17 The flow on a compact limit set of a competitive or cooperative system in IRn is topologically equivalent to a flow on a compact invariant set of a Lipschitz system of differential equations in IRn−1.
Proof: Let L be the limit set, v 0 be a unit vector and let Hv be the hyperplane orthogonal to v, i.e, Hv := {x : hx, vi = 0}. The orthogonal projection Q onto Hvis given by Qx = x − hx, viv. By Theorem 3.15, Q is one-to-one on L (this could fail only if L contains two points that are related by ). Therefore, QL, the restriction of Q to L, is a Lipschitz homeomorphism of L onto a compact subset of Hv. We argue by contradiction to establish the existence of m > 0 such that |QLx1 − QLx2| ≥ m|x1 − x2| whenever x1 6= x2 are points of L. If this were false, then there exists sequences xn, yn ∈ L, xn6= yn such that
|Q(xn) − Q(yn)|
|xn− yn| = |(xn− yn) − vhv, xn− yni|
|xn− yn| → 0
as n → ∞. Equivalently, |wn− vhv, wni| → 0 as n → ∞ where wn = xn− yn/|xn− yn|.
We can assume that wn→ w as n → ∞ where |w| = 1. Then, w = vhv, wi and therefore, hv, wi2 = 1 so w = ±v. But then xn− yn/|xn− yn| → ±v as n → ∞ and this implies that xn yn or yn xn for all large n, contradicting Theorem3.15. Therefore, Q−1L is Lipschitz on Q(L). Since L is a limit set, it is an invariant set for (3.4). It follows
that the dynamical system restricted to L can be modeled on a dynamical system in Hv. In fact, if y ∈ Q(L) then y = QL(x) for a unique x ∈ L and Ψt(y) ≡ QL(Φt(x)) is a dynamical system on Q(L) generated by the vector field
F (y) = QL(f (Q−1L (y)))
on Q(L). According to McShane [134], a Lipschitz vector field on an arbitrary subset of Hv can be extended to a Lipschitz vector field on all of Hv, preserving the Lipschitz constant. It follows that F can be extended to all of Hv as a Lipschitz vector field. It is easy to see that Q(L) is an invariant set for the latter vector field. We have established the topological equivalence of the flow Φ on L with the flow Ψ on Q(L). Q(L) is a compact invariant set for the (n − 1)-dimensional dynamical system on Hv generated by the extended vector field.
A consequence of Theorem 3.17 is that the flow on a compact limit set, L, of a com-petitive or cooperative system shares common dynamical properties with the flow of a system of differential equations in one less dimension, restricted to the compact, con-nected invariant set Q(L). Notice, however, that L may be the limit set of a trajectory not in L, and therefore Q(L) need not be a limit set.
On the other hand, the flow Ψ in a compact limit sets enjoys the topological prop-erty of chain recurrence, due to Conley [30, 31], which will be important in the next subsection. The definition is as follows. Let A be a compact invariant set for the flow Φ. Given two points z and y in A and positive numbers and t, an (, t)-chain from z to y in A is an ordered set
{z = x1, x2, . . . , xm+1 = y; t1, t2, . . . , tm} of points xi ∈ A and times ti ≥ t such that
|Φti(xi) − xi+1| < , i = 1, 2, . . . , m. (3.5) A is chain recurrent for Φ if for every z ∈ A and for every > 0 and t > 0, there is an (, t)-chain from z to z in A.
Conley proved that when A is compact and connected, a flow Φ in A is chain recurrent if and only if there are no attractors. This useful condition can be stated as follows: For every proper nonempty compact set S ⊂ A and all t > 0, there exists s > t such that Φs(S) 6⊂ Int S.
Compactness of A implies that chain recurrence of the flow in A is independent of the metric, and thus holds for any topologically equivalent flow.
It is intuitively clear that, as Conley proved, flows in compact alpha and omega limit sets are chain recurrent. Indeed, orbit segments of arbitrarily long lengths through point x repeatedly pass near any point of ω(x) ∪ α(x). Of course these segments do not necessarily belong to ω(x); but by taking suitable limits of points in these segments, one can find enough (, t)-chains in ω(x) and α(x) to prove the flows in these sets chain recurrent. For a rigorous proof, see Smith [193].