A Unified Approach to Persistence

Josef Hofbauer Institut für Mathematik, Universität Wien A-1090 Wien, Strudlhofgasse 4, Austria

In this paper^{1} we present a simplified approach to persistence. We show in §1 that essentially all known persistence theorems, including those of Butler, Freedman and Waltman [3,4], Gard and Hallam [9], Fonda [6], Hofbauer [11], Hutson [15] are all rather immediate consequences of a classical result of Ura and Kimura [20] (stated as Theorem 1 below) on the flow near a compact invariant set, see chapter VI of Bhatia and Szegő [1]. In §2 we apply this to give a new and direct proof of the geometric permanence condition for Lotka-Volterra equations, and in §3 we illustrate the whole approach with a concrete two-predator two-prey system.

1. Repellers in dynamical systems Let f^{t} be a flow on a compact metric space X and M be a closed invariant

subset of X.

Theorem 1. If M is isolated then one of the following three alternatives hold.

(a) M is an attractor.

(b) M is a repeller.

(c) M is a ‘saddle’: there exist x, y ∉ M such that ω(x) ⊂ M and α(y) ⊂ M.

Remark. ‘Attractor’ stands here for the ‘stable attractor’ of [1, ch.V], or as used by Conley [5]. In case (b), when M is a repeller, there is a dual attractor which attracts all orbits in X \ M (see [5, ch.II.5]).

Actually, we need only the following special case.

Theorem 2. M is a repeller if and only if

(1) M is isolated, and

(2) W^{s}(M) ⊂ M, so that no orbit from X \ M converges to M.

Remark. For completeness we sketch the (simple) proof. Choose a compact isolating neighborhood U of M. ω(x) of any x ∉ M must meet X \ U if neither (1) nor (2) is violated. Then X \ U is a weakly attracting region for X \ M. Its forward invariant closure γ^{+}(\overline{X \ U}) is then still compact and contains an attractor for X \ M whose dual repeller is M. [Consult the ‘weak attractor theorem’ of [1, ch. V, 1.25] or, in a more modern and concise language, Conley [5, II.5.1.D] (with reversed time) or Hutson’s [15] Lemma 2.1 for details]. The general result, Theorem 1, follows immediately from Theorem 2 and its time reversal.

1This paper was presented at a conference on Selected Topics in Biomathematics at IIASA, Laxenburg in December 1987. Unfortunately, in the publication in Acta Applicandae Mathematicae 14 (1989), 11–22 the introductory paragraph was thrown out by the editors.

Corollary 1. Let P : X → IR be a continuous function on X satisfying the following conditions.

(a) P (x) = 0 for x ∈ M and P (x) > 0 for x ∈ X \ M.

(b) For every x in a neighbourhood U of M there exists a time t > 0 such that P (xt) > P (x). Then M is a repeller.

Proof. Suppose, for some x ∈ U \ M, γ^{+}(x) ⊂ U. Then there is a y ∈ \overline{γ^{+}(x)} such that P (y) ≥ P (z) for all z ∈ \overline{γ^{+}(x)} and in particular P (y) ≥ P (yt) for t ≥ 0. This contradicts (b). Hence U is an isolating neighbourhood of M and for no x ∈ U \ M, ω(x) ⊂ U. Hence, by Theorem 2, M is a repeller. □

As shown by Fonda [6] this condition readily implies those given earlier by Gard and Hallam [9] in terms of ‘persistence functions’ and Hofbauer [11] and Hutson [15] in terms of ‘average Ljapunov functions’ which are more accessible to concrete applications, since they involve only conditions for x ∈ M:

Corollary 2. If P is differentiable along orbits then condition (b) can be replaced by (b’) There is a continuous function ψ : X → IR, such that Ṗ (x) ≥ P (x)ψ(x)

for all x ∈ X, and for each x ∈ M there is a time T > 0 such that

∫_{0}^{T}ψ(x(t)) dt > 0.  (1)

It is also sufficient that (1) holds for all x ∈ ω(M).

Up to now no information on the flow on M was assumed or required. However, sometimes it is useful to exploit the structure of the flow on M. The main idea is, if an orbit x ∈ X \ M converges to M then its ω−limit cannot be any closed invariant subset of M, but has the crucial property of being chain transitive (see [2]). To explain this we need some notation.

An (ε, T)−pseudoorbit or (ε, T)−chain for f^{t} is the union ⋃_{i=1}^{n} x_{i}[0, τ_{i}] of pieces of orbits of length τ_{i} ≥ T and the jumps obeying d(x_{i}τ_{i}, x_{i+1}) < ε. If the flow f^{t} is given by a C^{1} vector field ẋ = g(x) then one could use ε−approximate solutions {y(t) : 0 ≤ t ≤ T′} satisfying |ẏ(t) − g(y(t))| < ε instead. We say x is chained to y if for all (small) ε > 0 and all (large) T > 0 there is an (ε, T)-chain with x_{0} = x and x_{n+1} = y. If x is chained to x, then it is said to be a chain recurrent point, and we write x ∈ R. Now x ∉ R iff there exists an open set U such that f^{t}Ū ⊂ U ∀t > 0 and x ∈ U \ f^{t}U for some t > 0, or equivalently, iff there exists an attractor A, such that x ∈ W^{s}(A) \ A. Thus R = ⋂ A ∪ A^{∗}, the intersection over all attractor-repeller pairs (A, A^{∗}). The connected components Λ_{i} of R, sometimes called the basic sets of the flow, are the maximal chain transitive subsets of X: for all x, y ∈ Λ_{i}, x is chained to y, with chains lying completely in Λ_{i}.

This is the essence of the structure theorem of Conley [5] for general dynamical systems. In the following theorem we assume that the structure of the flow f^{t} on M is given.

Theorem 3. Let Λ_{i} be the basic sets of M. Then M is a repeller iff each basic set Λ_{i} of M satisfies the following two conditions.

(1) Λ_{i} is M-isolated in X, in the sense that some neighbourhood U of Λ_{i} in X contains a full orbit γ(x) only if x ∈ M.

(2) W^{s}(Λ_{i}) ⊂ M.

In some sense this result is best possible since a priori each basic set in M is a candidate for an ω-limit set of an orbit x ∈ X \ M. Proof. That (1) and (2) are necessary conditions, is clear. For the converse we have to show that (1) and (2) of Theorem 2 are fulfilled. As already pointed out, (2) is a consequence of the chain transitivity of ω-limit sets. It remains to show that M is isolated. Suppose not. Then there exists a sequence of compact invariant sets Γ_{k} ⊄ M which can be chosen to be chain transitive such that Γ_{k} ⊂ \{x : d(x, M) ≤ \frac{1}{k} \}. A subsequence of (Γ_{k}) will converge in the Hausdorff metric to a compact set Γ ⊂ M which is again invariant and chain transitive. (See the appendix in [18] for details.) Thus Γ must be contained in a single basic set Λ_{i}. But then any neighborhood of Λ_{i} contains some Γ_{k} for large k. Hence Λ_{i} is not M-isolated. □

Basic sets need not be isolated in M (not even generically) and there may even be uncountably many of them. Hence it is often difficult to determine the complete decomposition into basic sets. Usually it is more convenient to find a finite covering {M_{1}, M_{2}, ..., M_{n}} such that each basic set lies in one M_{i}. Such a Morse decomposition into closed invariant sets {M_{1}, …, M_{n}} is characterized by the property that for each x ∈ M, either x ∈ M_{i} for some i, or ω(x) ⊂ M_{i} and α(x) ⊂ M_{j} for some i < j.

Corollary 3. Let {M_{1}, …, M_{n}} be a Morse decomposition for M such that each Morse set M_{i} is also isolated in X. Then M is a repeller if and only if no M_{i} attracts orbits from X \ M, i.e. for each M_{i} , W^{s}(M_{i}) ⊂ M.

Remark. This is essentially the Butler-Freedman-Waltman persistence theorem (see Theorem 3.1 of [4]). Their acyclic covering of the ω-limit sets of M is indeed a Morse decomposition, as shown e.g. in [7]. The proof of this uses the so–called Butler–McGehee lemma, which by itself is a simple consequence of the Ura–Kimura theorem. We conclude with a simple combination of Corollary 1 and Theorem 3.

Corollary 4. Suppose that for each basic set Λ (or each Morse set M_{i} of a Morse decomposition of M) there is a function P which is defined in a neighbourhood U of Λ (resp. M_{i}) and satisfies the conditions of Corollary 1 or 2 in this neighbourhood U. Then M is a repeller.

In applications to ecological problems one usually encounters the nonnegative orthant IR_{+}^{n} as state space, and a dissipative flow thereon. Considering the one-point compactification X = IR_{+}^{n} ∪ {∞}, the system is permament or uniformly persistent iff the closed invariant subset M = bdIR_{+}^{n} ∪ {∞} is a repeller. Another way would be to take X_{1} as the set of all bounded orbits in IR_{+}^{n}, that is the dual attractor of the repeller {∞}, and M_{1} the dual attractor to {∞} in bdIR_{+}^{n}.

Corollary 4 above provides a natural combination of the two dual approaches to persistence problems, opened by Freedman and Waltman [8] and Schuster and Sigmund [19], respectively. It seems the most effective for concrete applications: Determine first the basic sets of M, or find at least a suitably fine Morse decomposition. Then use Ljapunov type arguments to show that no orbit x ∈ X \ M converges to or even stays near one of these basic sets. For a typical application see section 3.

Theoretically, the problem of persistence is thus reduced to the study of the local behaviour around basic sets. In ecological applications such basic sets - additionally to the ‘classical’ ones (fixed points, periodic orbits, suspensions of subshifts, etc.) - frequently occur as ‘heteroclinic cycles’ (a preliminary definition might be C^{1}-robust chain transitive set which is not contained in the interior of a single face of bdIR_{+}^{n}). In simple cases the stability and bifurcation of such heteroclinic cycles in arbitrary dimensions has been discussed in [12], but further analysis is necessary.

2. Time averages and permanence of Lotka-Volterra equations In this section we consider the familiar Lotka-Volterra equations

ẋ_{i} = x_{i}(r_{i} + ∑_{j=1}^{n} a_{ij}x_{j})  i = 1, …, n  (2)

and assume throughout that they are dissipative, i.e. their orbits are uniformly bounded for t → +∞: there is a constant K such that

lim sup_{t→+∞} x_{i}(t) ≤ K for all x ∈ IR_{+}^{n}.

Explicit criteria for this are given in [14, ch. 21.2].

There are two sufficient conditions for permanence for such Lotka-Volterra systems, see Jansen [16] and Hofbauer and Sigmund [13,14].

Theorem 4. If the system of linear inequalities

∑_{i=1}^{n} p_{i}(r_{i} + (Ax)_{i}) > 0,  (3)

where x runs through all boundary fixed points of (2), admits a positive solution p_{i} > 0 then (2) is permanent.

Theorem 5. If the convex hull C of all boundary fixed points is disjoint from the set D = {x ∈ IR_{+}^{n} : r + Ax ≤ 0} then (2) is permanent.

The proof of Theorem 4 consists in checking that P (x) = ∏ x_{i}^{p_{i}} is an average Ljapunov function and satisfies (b’) of Corollary 2. Theorem 5 can be reduced to Theorem 4. It is the aim of this section to give a new, independent proof of this ‘geometric condition’ for permanence, without recourse to average Ljapunov functions.

The key is to study the asymptotic behaviour of time averages for Lotka-Volterra equations for which we introduce the following notation:

m_{T} (x) = \frac{1}{T} ∫_{0}^{T}x(t) dt

μ(x) = {y : y = lim_{T_{n}→+∞} m_{T_{n}}(x)}  (4)

μ(A) = ⋃_{x∈A} μ(x)

A classical result (going back to Volterra) states that for a ‘persistent’ orbit x(t), μ(x) consists only of interior fixed points (in general a unique one). This follows from the identity

\frac{log x_{i}(T) − log x_{i}(0)}{T} = r_{i} + (Am_{T} (x))_{i}.  (5)

If lim inf_{t→+∞} x_{i}(T) > 0, the left hand side goes to 0. Hence any z ∈ μ(x) is a fixed point of (2).

For arbitrary x this argument yields only

μ(x) ⊂ D.  (6)

If one ignores the trivial case ω(x) = {0}, then 0 ∉ ω(x) and hence μ(x) ⊂ bdD.

To get further information on μ(x) we first need a general fact on the limit sets μ(x) of time averages, valid for dynamical systems on a convex subset of a normed space.

Lemma.

μ(x) ⊂ conv μ(ω(x))

Proof. We show that for any given ε > 0, μ(x) is contained in the ε-neighbour-hood of conv μ(ω(x)). Obviously, for each z there is a time T = T (ε, z) such that d (m_{T} (z), μ(z)) < \frac{ε}{2}. Although T is in general not a bounded function of z (e.g. near an unstable fixed point), we still can conclude the existence of a constant τ = τ(ε, K) such that

∀z ∈ K ∃ T ∈ [1, τ(ε, K)] : d(m_{T} (z), μ(K)) < \frac{ε}{2}

for any compact set K which in our case will be ω(x) or \overline{γ^{+}(x)}. Now choose δ > 0 such that

|y − z| < δ ⟹ |y(t) − z(t)| < \frac{ε}{2} for 0 ≤ t ≤ τ(ε) and y, z ∈ \overline{γ^{+}(x)}

and hence

|m_{T} (y) − m_{T} (x)| < \frac{ε}{2} for 0 ≤ T ≤ τ(ε).

Now consider the orbit x(t), and find a t_{0} > 0 such that d(x(t), ω(x)) < δ for all t ≥ t_{0}. Then there exists a z_{0} ∈ ω(x) such that |x(t_{0}) − z_{0}| < δ and hence a time T_{1} ∈ [1, τ(ε)] with

|m_{T_{1}}(x(t_{0})) − m_{T_{1}}(z_{0})| < \frac{ε}{2} and |m_{T_{1}}(z_{0}) − μ(ω(x))| < \frac{ε}{2}.

Thus we have a T_{1} ∈ [1, τ(ε)] such that

d(m_{T_{1}}(x(t_{0})), μ(ω(x))) < ε.

Repeating this argument, we find a point z_{1} ∈ ω(x) near x(t_{0} + T_{1}), etc. Hence m_{T} (x) can be represented as a convex linear combination of vectors m_{t_{0}}(x), m_{T_{1}}(x(t_{0})), m_{T_{2}}(x(t_{0} + T_{1})), … each of which (up to the first and the last one, whose weight goes to 0 as T → ∞) is ε-close to μ(ω(x)). So m_{T} (x) itself is ε-close to conv μ(ω(x)) for large T. □

Now let F_{x} denote the set of all open faces of IR_{+}^{n} met by ω(x):

F_{x} = {I ⊂ {1, …, n} : ∃z ∈ ω(x), supp z = I},

and let G_{x} = {F_{I} : I ∈ F_{x}} consist of all rest points z = F_{I} with r_{i} + (Az)_{i} = 0 and z_{i} ≥ 0 for i ∈ I and z_{i} = 0 for i ∉ I in those faces, and let

C_{x} = conv G_{x}

be the convex hull of those fixed points. Then we have Theorem 6. For Lotka-Volterra equations: μ(x) ⊂ C_{x}.

Proof. By induction on the size of the support of x. Suppose w.l.o.g. that the statement is true for all y ∈ bdIR_{+}^{n}, and let x ∈ intIR_{+}^{n}. If ω(x) ⊂ bdIR_{+}^{n} then by induction hypothesis μ(y) ⊂ C_{y} ⊂ C_{x} for all y ∈ ω(x), since ω(y) ⊂ ω(x). Hence conv μ(ω(x)) ⊂ C_{x}, and together with the lemma, μ(x) ⊂ C_{x}. If ω(x) ⊄ bdIR_{+}^{n} then we have to recall (5): if log x_{i}(T)/T > −δ or x_{i}(T) > e^{−δT} for all i then

|r + Am_{T} (x)| < δ  for large T.  (7)

For any given ε there is then a δ such that (7) implies that m_{T} (x) is ε-close to an interior fixed point of (2). For a large time T consider now the last instance T′ ≤ T when x_{i}(T′) ≥ e^{−δT′} holds for all i. For the remaining time T′ < t ≤ T at least one x_{i}(t) < e^{−δt}, so x(t) is very close to bdIR_{+}^{n}. For this time interval we repeat the construction from the proof of the Lemma to find m_{T −T′}(x(T′)) ε-close to conv μ(ω(x) ∩ bdIR_{+}^{n}). Joining the two pieces puts m_{T} (x) ε-close to C_{x}. □

As a consequence, μ(x) ⊂ C_{x} ∩ D. Now if ω(x) ⊂ bdIR_{+}^{n} then C_{x} ⊂ C, the convex hull of all boundary fixed points, and hence μ(x) ⊂ C ∩ D. This shows that no interior orbit can converge to the boundary if C ∩ D = ∅, i.e. the second part (2) of Theorem 2. The isolatedness might be shown by a similar argument. We omit this here since we hope to replace it one day by a simple robustness argument.

Hence altogether the proof of Theorem 5 presented here won’t be shorter than the original one. But the result on time averages might be of interest per se. A good example to illustrate it is the May-Leonard case where μ(x) = C_{x} ∩bdD equals the three sides of a triangle in intIR_{+}^{3}, see [14, ch. 9.5].

It has been shown in [14, ch. 22] that the conditions of Theorems 4 and 5 characterize robust permanence for n = 3. As shown in the next section, this is no longer true for n ≥ 4. Like in section 1, the condition can be weakened, however. For a basic set Λ of bdIR_{+}^{n}, let C_{Λ} = ⋃_{x∈Λ} C_{x}. Then the above proof yields

Theorem 7. Suppose for each basic set Λ of the boundary bdIR_{+}^{n}, C_{Λ} ∩D = ∅, or equivalently that (3) holds for all fixed points x ∈ C_{Λ}. Then (2) is permanent.

There is some evidence that Theorem 7 might even be a characterization of robust permanence for Lotka-Volterra equations.

3. An example Let us consider now the following two-prey two-predator system

\begin{aligned}ẋ_{1} & = x_{1} (1 − \frac{2}{3}x_{1} − \frac{1}{3}x_{2} − by_{1} − y_{2}) \\ ẋ_{2} & = x_{2} (1 − \frac{1}{3}x_{1} − \frac{2}{3}x_{2} − y_{1} − by_{2}) \\ ẏ_{1} & = y_{1}(−1 + 2x_{1} + x_{2}) \\ ẏ_{2} & = y_{2}(−1 + x_{1} + 2x_{2}) \end{aligned}  (8)

This example nicely illustrates the method for proving permanence described in section 1. Furthermore it shows that the conditions given in Theorems 4 and 5 do not characterize permanence of Lotka-Volterra equations if n ≥ 4. I realized this example while reading between the lines of Kirlinger’s thesis [17].

(8) has the following fixed points:

F_{1} = (\frac{3}{2}, 0, 0, 0), F_{12} = (1, 1, 0, 0), F_{1}^{1} = (\frac{1}{2}, 0, \frac{2}{3b}, 0), F_{1}^{2} = (1, 0, 0, \frac{1}{3}),

F_{12}^{12} = (\frac{1}{3}, \frac{1}{3}, \frac{2}{3(b + 1)}, \frac{2}{3(b + 1)})

and their symmetric images F_{2}, F_{2}^{1}, F_{2}^{2}. From the external eigenvalues

\frac{ẋ_{2}}{x_{2}} (F_{1}^{1}) = \frac{5b − 4}{6b}, \frac{ẋ_{1}}{x_{1}} (F_{2}^{1}) = \frac{2 − b}{3}, \frac{ẏ_{1}}{y_{1}} (F_{12}) = 2,

we see that, for b > 2, F_{2}^{1} is the only saturated fixed point in the subsystem {y_{2} = 0} and hence globally stable there (see Fig 1a). Since

\frac{ẏ_{1}}{y_{1}} (F_{2}^{2}) = −\frac{1}{2} < 0, \frac{ẏ_{2}}{y_{2}} (F_{2}^{1}) = 1 > 0,

F_{2}^{2} attracts all orbits in {x_{1} = 0} (Fig 1b).

In particular this shows the existence of a heteroclinic cycle γ: F_{1}^{1} → F_{2}^{1} → F_{2}^{2} → F_{1}^{2} → F_{1}^{1}. Since all orbits on the boundary of IR_{+}^{4} converge to fixed points we obtain the following graphic description of the boundary flow (Fig 2), or after identifying the irreducible component corresponding to γ (Fig 3). It shows the topological structure of the boundary flow, i.e. the basic sets O, F_{1}, F_{2}, F_{12}, and γ, and the connections between them.

Since γ is a “planar” heteroclinic cycle (at each fixed point there is just one positive and one negative external eigenvalue), its stability is easily determined by the product of the ratios of the outgoing versus the incoming eigenvalues (see e.g. [11] or [14, ch. 22]). In view of the symmetry we only need to look at two fixed points, say F_{1}^{1} and F_{2}^{1}):

ρ = \frac{5b − 4}{6b} · 2 · \frac{3}{b − 2} = \frac{5b − 4}{b(b − 2)}.

γ is attracting if ρ < 1, i.e. if b > b_{1} = \frac{7+\sqrt{33}}{2} ≈ 6.4, and repelling for b < b_{1}.

Hence the system (8) is permanent for 2 < b < b_{1}, as a consequence of Theorem 3 since neither of the basic sets attracts interior orbits.

An alternative way to prove permanence of (8) would be to construct an average Ljapunov function P for the whole system and apply Corollary 2 (this way we need not know the complete structure of the basic sets, but use only the fact that all boundary orbits converge to fixed points):

P = (x_{1} + x_{2})^{p_{0}}x_{1}^{p_{1}}x_{2}^{p_{2}}y_{1}^{q_{1}}y_{2}^{q_{2}}

(this choice is motivated by Hutson [15]). Then

\frac{Ṗ}{P} = p_{0} \frac{ẋ_{1} + ẋ_{2}}{x_{1} + x_{2}} + p_{1} \frac{ẋ_{1}}{x_{1}} + p_{2} \frac{ẋ_{2}}{x_{2}} + q_{1} \frac{ẏ_{1}}{y_{1}} + q_{2} \frac{ẏ_{2}}{y_{2}}

Using the estimate

\frac{ẋ_{1} + ẋ_{2}}{x_{1} + x_{2}} ≥ min (\frac{ẋ_{1}}{x_{1}} , \frac{ẋ_{2}}{x_{2}} )

near the origin (the only point where Ṗ /P is not continuous), we need a solution of the following set of inequalities. (Those for F_{1}, F_{2}, and F_{12} are trivially satisfied.)

\begin{matrix}(O)  p_{0} + p_{1} + p_{2} − q_{1} − q_{2} > & 0 \\ (F_{1}^{1})  \frac{5b − 4}{6b}p_{2} − \frac{1}{2}q_{2} > & 0 \\ (F_{2}^{1})  − \frac{b − 2}{3}p_{1} + q_{2} > & 0 \\ (F_{2}^{2})  \frac{5b − 4}{6b}p_{1} − \frac{1}{2}q_{1} > & 0 \\ (F_{1}^{2})  − \frac{b − 2}{3}p_{2} + q_{1} > & 0 \end{matrix}  (9)

Because of the symmetry, we can set w.l.o.g. p_{1} = p_{2} = p and q_{1} = q_{2} = q. Then (9) leads to

\frac{5b − 4}{6b} > \frac{q}{p} > \frac{b − 2}{3}  and  p_{0} + 2p − 2q > 0.

The choice p_{0} = 0, which corresponds to Theorem 4, is good only for b < 5. For 5 ≤ b < b_{1} we need p_{0} ≫ 0 to satisfy the inequality for the origin.

Geometrically (see Jansen [16] or the proof of Theorem 19.6.2 in [14]) this means that for 5 ≤ b < b_{1}, the interior fixed point F_{12}^{12} is contained in the convex hull of the five fixed points O, F_{1}^{1}, F_{1}^{2}, F_{2}^{1}, F_{2}^{2} (but the cone D does not yet meet C_{γ}, the hyperplane through the four points F_{i}^{j}, and hence Theorem 7 applies). For b = b_{1}, F_{12}^{12} lies exactly on this hyperplane. For larger values of b, F_{12}^{12} lies in the hull of the F_{i}^{j} and F_{12}. So only for b < 5, F_{12}^{12} lies outside the convex hull of all boundary fixed points and Theorem 5 yields permanence.

Acknowledgements. I am indebted to J. Reineck for discussions on Conley’s theory. This work was partially supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, Project P 5994. Part of it was done during a stay at IIASA, Laxenburg, Austria. After presenting this lecture again at SISSA in Trieste in February 1988, F. Zanolin showed me a paper by B.M. Garay [10], who had the very same idea of applying the Ura-Kimura theorem and Conley’s structure theorem to the question of persistence. I recommend his paper for more details.

References

1 N. P. Bhatia and G. P. Szegő: Stability Theory of Dynamical Systems. Grundlehren math. Wissensch. 161. Berlin - Heidelberg - New York: Springer. 1970.

2 R. Bowen: ω-limit sets for Axiom A diffeomorphisms. J. Diff. Equ. 18, 333-339 (1975).

3 G. Butler, H. I. Freedman and P. Waltman: Uniformly persistent systems. Proc. Amer. Math. Soc. 96, 425-430 (1986).

4 G. Butler, P. Waltman: Persistence in dynamical systems. J. Diff. Equ. 63, 255-263 (1986).

5 C. Conley: Isolated invariant sets and the Morse index. CBMS 38. Providence, R.I.: Amer. Math. Soc. 1978.

6 A. Fonda: Uniformly persistent semi-dynamical systems. Proc. Amer. Math. Soc. To appear.

7 H. I. Freedman and J. W.-H. So: Persistence in discrete semi-dynamical systems. Preprint (1987).

8 H. I. Freedman and P. Waltman: Mathematical analysis of some threespecies food-chain models. Math. Biosci. 33, 257-276 (1977).

9 T. C. Gard and T. G. Hallam: Persistence of food webs: I. Lotka-Volterra food chains. Bull. Math. Biol. 41, 877-891 (1979).

10 B. M. Garay: Uniform persistence and chain recurrence. J. Math. Anal. Appl. To appear.

11 J. Hofbauer: A general cooperation theorem for hypercycles. Monatsh.

Math. 91: 233-240 (1981). 12 J. Hofbauer: Heteroclinic cycles on the simplex. Proc. Int. Conf. Nonlinear

Oscillations. Budapest 1987. 13 J. Hofbauer and K. Sigmund: Permanence for replicator equations. In:

Dynamical Systems. Ed. A. B. Kurzhansky and K. Sigmund. Springer Lect. Notes Econ. Math. Systems 287. 1987.

14 J. Hofbauer and K. Sigmund: Dynamical Systems and the Theory of Evolu-

tion. Cambridge Univ. Press 1988. 15 V. Hutson: A theorem on average Ljapunov functions. Monatsh. Math. 98,

267-275 (1984). 16 W. Jansen: A permanence theorem for replicator and Lotka-Volterra sys-

tems. J. Math. Biol. 25, 411-422 (1987). 17 G. Kirlinger: Permanence of some four-species Lotka-Volterra systems. Dis-

sertation. Universit”at Wien. 1987. 18 C. Robinson: Stability theorems and hyperbolicity in dynamical systems.

Rocky Mountain J. Math. 7, 425-434 (1977). 19 P. Schuster, K. Sigmund and R. Wolff: Dynamical systems under constant

organization. III. Cooperative and competitive behaviour of hypercycles. J. Diff. Equ. 32, 357-368 (1979).

20 T. Ura and I. Kimura: Sur le courant exterieur a une region invariante. Theoreme de Bendixson. Comm. Math. Univ. Sanctii Pauli 8, 23-39 (1960).