Heteroclinic Cycles in Ecological Differential Equations

Josef Hofbauer

Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A–1090 Vienna, Austria Josef.Hofbauer @ univie.ac.at

Abstract. Differential equations on IR^{n} that leave certain hyperplanes invariant, arise as models in mathematical biology and in systems with symmetry. In such systems heteroclinic cycles occur in a robust way. We survey examples from the literature and propose a classification into ‘planar’, simple, and multiple heteroclinic cycles (or heteroclinic networks). We associate a characteristic matrix to such objects, consisting of certain eigenvalues at the fixed points, and show how to read off stability properties from this matrix. Instead of Poincaré sections we use average Lyapunov functions to obtain stability results.

1. Motivation.

In mathematical biology ODE models often take one of the following forms. Ecological differential equations

ẋ_{i} = x_{i}f_{i}(x_{1}, …, x_{n}), i = 1, …, n,  (1)

are defined on the nonnegative orthant IR_{+}^{n}. Such equations are used to model ecological interactions of species. The x_{i} are interpreted as densities of species. The most prominent and important special case are the Lotka–Volterra equations

ẋ_{i} = x_{i} (r_{i} + (Ax)_{i}) .  (1’)

If the independent variables x_{i} are frequencies or there is a conservation of total mass, (1) is replaced by the replicator equation

ẋ_{i} = x_{i} (f_{i}(x) − f̄(x)) , i = 1, …, n,  f̄(x) = ∑_{i=1}^{n} x_{i}f_{i}(x),  (2)

Its state space is the simplex S_{n} = {x ∈ IR^{n} : x_{i} ≥ 0, ∑ x_{i} = 1}. Again a much studied particular case of (2) is the replicator equation with linear f_{i} which models the evolution of gene frequencies and game dynamics (see [HS]):

ẋ_{i} = x_{i}[(Ax)_{i} − x · Ax].  (2’)

There are analogues of (2) and (2’) where the state space is a product of simplices, e. g. for the dynamics of bimatrix games (see [HS]).

1Presented as invited lecture at the EQUADIFF 8 (Bratislava 1993), published in Tatra Mountains Math. Publ. 4 (1994), 105–116.

1991 Mathematics Subject Classification. 34C37, 92D25.

Mathematically, these classes of ODEs are characterized by the invariance of the boundary faces of the state space.

It should be noted that some of these equations occur also in other settings outside biomathematics. In particular, the Lotka–Volterra equation is quite ubiquitous (see e. g. [PM]). It occurs as truncated normal form of the codimension n bifurcation with n pairs of purely imaginary eigenvalues. Generally, equations on IR^{n} which have a reflectional ℤ_{2}^{n} symmetry must leave the coordinate planes invariant and hence take the form (1).

Therefore, it is desirable to discuss the generic behaviour of such systems. One type of behaviour which is not possible in general dynamical systems is the occurrence of robust (= structurally stable within this class) attracting heteroclinic cycles on the boundary of the state space. The first example was given by May and Leonard [ML] and studied further by [C], [CPC], [S1], [HS]. It consists of three competing species, where 1 beats 2, 2 beats 3, 3 beats 1 (in the simplest case with LV dynamics and with cyclic symmetry). There is a robust heteroclinic cycle connecting the three one-species fixed points F_{1} ⟶ F_{2} ⟶ F_{3} ⟶ F_{1} (see fig. 1a). Whether this cycle is attracting or repelling, depends on the eigenvalues at the fixed points, see section 4. The same example, now in the setting of systems with symmetry, was presented again in [GH]. An equivalent dynamics occurs in the so-called ‘rock-scissors-paper’ game (see [Z, p. 482, 490] or [HS, p. 130]), a replicator equation (2’) on the triangle S_{3} (see fig. 1b). Analogous cycles in higher dimensions were considered in [K2], [K3], [HS2], …

Figure 1: The May–Leonard system. The rock–scissors–paper game. The characteristic matrix.

A biologically plausible example of a heteroclinic cycle was given by Kirlinger [K1]. This is a two–prey two–predator system with Lotka–Volterra dynamics. The two prey species x_{1} and x_{2} live in severe competition, i.e. the dynamics of the x_{1}–x_{2} plane is bistable: There are two stable one-species fixed points F_{1} and F_{2}. At F_{1} predator y_{1}, which is specialized on prey 1, is able to invade, leading to a coexistence equilibrium F_{1}^{1}. At this equilibrium, the number x_{1} of prey 1 is reduced due to predation so much, that it will be outcompeted by prey 2. There is a heteroclinic orbit in the x_{1}–x_{2}–y_{1} subsystem which connects F_{1}^{1} to F_{2}. Assuming some symmetry between 1 and 2, there is a heteroclinic cycle on the boundary of IR_{+}^{4} connecting the four equilibria F_{1} ⟶ F_{1}^{1} ⟶ F_{2} ⟶ F_{2}^{2} ⟶ F_{1} (see fig. 2). Similar cycles were considered in [K3], [HS2], [MR], [SR].

2. The characteristic matrix of a heteroclinic cycle

Suppose the state space of our dynamical system is a polyhedron X in some IR^{N}, defined as the intersection of finitely many halfspaces {x ∈ IR^{N} : x_{j} ≥ 0}, j = 1, … n. Here x_{j} denotes a linear functional on IR^{N} which vanishes on one of the supporting hyperplanes of X. Let ∂X = ⋃_{j}{x_{j} = 0} ∩ X denote the boundary of X, and int X = X \ ∂X the interior of the polyhedron. More generally X could be a manifold with corners.

If ∂X is invariant under the smooth flow then we can write our differential equation in the form

ẋ_{j} = x_{j}f_{j}(x), j = 1, …, n,  (3)

(there will be some relations between the f_{j} if n > N). We can always assume that n ≥ N and that a point x ∈ X is uniquely determined by its ‘coordinates’ x_{j}, j = 1, …, n. If not, we simply add some additional variables to achieve that.

Consider a fixed point x̄ of (3), and rearrange indices such that the zero coordinates come first. Then the Jacobian at x̄ takes the form ( \begin{matrix}E O \\ ∗ ∗ \end{matrix} ) where E is a diagonal matrix whose

entries are the external eigenvalues \frac{ẋ_{j}}{x_{j}} |x̄ = f_{j}(x̄). They describe the motion transverse to the hyperplane j.

Now let Γ ⊂ ∂X be a heteroclinic cycle which consists of m fixed points F_{k} and heteroclinic orbits between them. Then we associate to Γ a rectangular scheme of external eigenvalues: The entry in row k and column j is the external eigenvalue f_{j}(F_{k}). (It is 0 if x_{j} > 0 at F_{k}.) We call this scheme the charactereristic matrix C of Γ. E. g. in Fig. 1, at F_{1}, there are two external eigenvalues, f_{2}(F_{1}) > 0 and f_{3}(F_{1}) < 0. Since f_{1}(F_{1}) = 0, these three numbers give the first row of the characteristic matrix in Figure 1c.

It seems that this matrix C contains the essential information about Γ. In particular the stability properties of Γ can be read off from C in all known cases, as shown in section 4. Apriori C is an m × n matrix (m = the number of fixed points, n = the number of boundary hyperplanes of the polyhedron X), but we can omit columns of zeroes, thus ignoring those hyperplanes which do not touch Γ. Each row of C contains at least one positive entry (under the generic assumption that the fixed points F_{k} are hyperbolic).

Classification of heteroclinic cycles. If each row and each column of C contains only one positive entry then we call Γ a simple heteroclinic cycle. Note that there are heteroclinic cycles where each row contains only one positive entry (one unstable transverse direction at each F_{k}), but m > n (e. g. on the cube (m = 8, n = 6) or on any polyhedron with more corners than faces) so that there are several positive entries in one column. We do not consider such heteroclinic cycles here, since we don’t have general stability results for them.

If furthermore there is only one negative entry in each row and column (so that there are just two external directions at each F_{k}, one stable and one unstable) then we speak of a ‘planar’ heteroclinic cycle. We will see below (Corollary 2) that they behave like heteroclinic cycles in the plane.

If some row of C contains at least two positive entries (so at least one F_{k} has an unstable manifold of dimension ≥ 2) then Γ is a multiple heteroclinic cycle or a heteroclinic network.

Since the numbering of the hyperplanes (or columns) and of the fixed points F_{k} (or rows of C) is arbitrary, the characteristic matrix is determined only up to permutations of rows and columns. Our stability criteria (Thm. 1 and 2) will therefore be independent against such permutations.

Examples. 1) The May–Leonard cycle is ‘planar’. Its characteristic matrix is given in figure 1c.

2) Kirlinger’s cycle is not ‘planar’ but still simple. See figure 2c for its characteristic matrix.

3) The simplest robust heteroclinic cycle is probably the one shown in figure 3. Two coordinate planes in IR^{3} are supposed to be invariant, they contain the two connecting orbits. The two fixed points lie in the invariant line of intersection and are stable inthere. This is again a ‘planar’ heteroclinic cycle. The third column in the characteristic matrix will be omitted before applying the stability criteria of section 4.

4) Heteroclinic cycles in the plane. X is essentially an n−gon in this case. The characteristic matrix can be arranged such that it contains the positive eigenvalues λ_{k} in the main diagonal and the negative eigenvalues −μ_{k} right next to it. Robust examples from biology are the rock–scissors–paper game (figure 1b) and the ‘battle of the sexes’ on the square [0, 1]^{2} (see [HS]).

5) Heteroclinic cycles on the simplex. Let A be an n×n matrix with a_{ii} = 0. If a_{ij} and a_{ji} have different sign for sufficiently many pairs (i, j), then A will be the characteristic matrix of a heteroclinic cycle or network on S_{n}. The simplest dynamics (a ‘normal form’ for the heteroclinic cycle?) is given by the replicator equation (2’). The cycle is simple if a_{i,i+1} > 0 and all other entries are ≤ 0. This case was studied in [AH], [H2], [HS], etc. The classical ‘hypercycle’ equation from prebiotic evolution [S2, H1, HS] is the prototype special or limiting case, for which the method of proof indicated in section 5 was originally develloped. Networks were recently treated in [Br] and [KS].

6) In the same way as the rock–scissors–paper game and the May–Leonard example (figure 1) are essentially equivalent, all the above examples on the simplex S_{n} can be imbedded to heteroclinic cycles or networks in equations (1) or (1’) on IR_{+}^{n}. Thereby the n corners of the simplex are replaced by fixed points on the n coordinate axes. The dynamics on IR_{+}^{n} could be chosen to be competitive in the sense of Hirsch.

7) Heteroclinic cycles on a cube and other polyhedra occur naturally in the dynamics of n person games, whenever there is a cycle of best responses, see [GaH].

8) Another example can be found in [SR], again a two–prey two–predator system as in ex. 2: It is a simple heteroclinic cycle of length three: F_{1} ⟶ F_{1}^{1} ⟶ F_{12} ⟶ F_{1}. The last connection is not in the x_{1}–x_{2} plane (the boundary face spanned by F_{12} and F_{1}), but in the x_{1}–x_{2}–y_{2} subsystem. Therefore y_{2} > 0 along this connection, but not at any of the equilibria, so that y_{2} = 0 on the average along this cycle. Note also that the first column of C is zero (and hence can be ignored), since x_{1} > 0 along the cycle. (See figure 5).

There are also heteroclinic cycles whose minimal center of attraction consists of more complicated invariant sets than fixed points. See [S3] for a first example (of ‘planar’ type), where one of the fixed points is replaced by a periodic orbit. In Kirlinger’s two-prey twopredator systems it is easy to imagine the fixed points F_{i}^{i} being replaced by predator–prey limit cycles (if the restrictive class of Lotka–Volterra dynamics is left). For examples in the context of symmetry see [MCG].

9) A May–Leonard like system in IR_{+}^{4} with three limit cycles in the x_{i}x_{4} planes (i = 1, 2, 3). Such a ‘planar’ heteroclinic cycle would still be robust even if one of the three limit cycles is internally unstable, since there may be transverse intersections of their stable and unstable manifolds.

Our approach applies also to such heteroclinic cycles, as long as the components Λ_{k} of the minimal center of attraction are uniquely ergodic invariant sets. The entries of the characteristic matrix are then given as the external Lyapunov exponents

λ_{i}(Λ_{k}) = lim_{T ⟶∞} \frac{1}{T} ∫_{0}^{T} f_{i}(x(t))dt = ∫_{Λ_{k}} f_{i}(x)dμ_{k}(x)  (4)

with respect to the unique invariant measure μ_{k} on the invariant set Λ_{k}.

3. Permanence and repelling sets.

In all these examples it is of interest to know the stability properties of this heteroclinic cycle. In particular, one would like to know whether Γ does attract interior orbits or repells them all. This is closely related to the question of persistence or permanence of ecological systems ([HS], [HuS]). A system (1), (2) or (3) is called permanent if the boundary of the state space X is a repellor in the sense that there is a δ > 0 such that lim inf_{t⟶∞} x_{i}(t) > δ > 0 holds for each solution x(t) > 0 of (1), (2) or (3). If X is unbounded (e. g. IR_{+}^{n}) we also assume that the (semi)flow is dissipative.

We call a compact invariant subset Λ of ∂X repelling (a better terminology might be: ∂X is repelling near Λ), whenever both of the following two conditions are satisfied:

(i) There is no x ∈ int X with ω(x) ⊂ Λ, and

(ii) There is a neighbourhood U of Λ in X such that for each x ∈ U \ ∂X ∃t ∈ IR: x(t) ∉ U. ( ∂X is isolated near Λ.)

The following characterization of permanence follows from classical stability results in [BS].

A system on X is permanent iff ∂X is repelling in the above sense.

This can be sharpened to (see [G, H3]):

A system on X is permanent iff every chain–recurrent invariant subset Λ ⊂ ∂X is repelling in the above sense.

Heteroclinic cycles are such chain–recurrent subsets and it is therefore of importance to characterize when they are repelling.

4. Stability conditions for heteroclinic cycles.

Let Λ ⊂ ∂X be a compact invariant subset of the boundary and Λ_{k} ⊂ Λ (k = 1, …, m) be such that for each x ∈ Λ there is a k with ω(x) ⊂ Λ_{k}. We assume further that each set Λ_{k} is a compact invariant, uniquely ergodic subset of a single boundary face. Then the external eigenvalues are uniquely defined at Λ_{k} by (4) and can be arranged in the characteristic matrix C of the set Λ. We omit those columns which consist of zero entries only, thus ignoring those hyperplanes which do not touch Λ, as in examples 3 and 8. We need not assume for the following that Λ is a heteroclinic cycle or network in the sense that each Λ_{k} can be reached by a sequence of heteroclinic connections from any other Λ_{l}.

Theorem 1. a) If there is a vector p ∈ IR^{n} such that p > 0 and Cp > 0 then Λ is repelling.

b) If Λ is asymptotically stable within ∂X and there is a p < 0 such that Cp > 0 then Λ is asymptotically stable in X.

c) If Λ is asymptotically stable within ∂X and there is a p ∈ IR^{n} such that p_{i} < 0 for at least one i and Cp > 0 then Λ attracts at least one (actually an open set of) interior orbit(s) from X.

The proof is deferred to section 5. A consequence is

Theorem 2. Let Λ be asymptotically stable within ∂X; m = n′ (= number of nonzero columns) so that the (reduced) characteristic matrix C is a square matrix, and let det C ≠ 0. Then Λ is repelling iff C^{−1} ≥ 0, i. e. the inverse of C has only nonnegative entries.

Proof. If Λ is repelling then case c) of Theorem 1 must not apply and hence for any positive vector q > 0 in IR^{n}, we must have p = C^{−1}q ≥ 0. This implies C^{−1} ≥ 0. The converse follows immediately from case a). □

Matrices C satisfying condition a) of Thm. 1 are called semipositive. The step from Theorem 1 to Theorem 2 is closely related to a result of [FP] on irreducibly semipositive matrices. The case where C is not square (n′ ≠ m) is largely unresolved. Thm. 1 gives only a partial result in this case. Also the stability assumption within the boundary should be relaxed. Sometimes it can be replaced by assuming Λ to have nontrivial index within ∂X, as in [CG].

Note that this property of C is invariant under interchanges of rows or columns of the characteristic matrix C. If one entry of C^{−1} is negative then some interior orbit will converge to (some subset of) Λ. If Λ is a heteroclinic network, a more detailed analysis is necessary to decide which parts of it can attract interior orbits. See [Br] for some interesting results in this direction: He identifies (for a 4 × 4 matrix C) those 3 × 3 minors which determine the stability of the cycles of length 3.

For simple heteroclinic cycles the situation is considerably simpler because case c) can be ignored: We can apply the theory of M-matrices to solve the arising linear inequalities.

Corollary 1. Let Λ be a simple heteroclinic cycle, which is asymptotically stable within ∂X. (This is automatically satisfied if the cycle is robust and all Λ_{k} are fixed points.) Then C is a square matrix (after elimination of superfluous columns) with positive entries occurring only in the main diagonal (after a suitable rearrangement of the rows or columns). Let det C ≠ 0.

If C is an M−matrix (all leading principal minors of C are positive) then Λ is repelling. If C is not an M−matrix (at least one leading principal minor is negative) then Λ is

asymptotically stable.

Corollary 2. Let Λ be a ‘planar’ heteroclinic cycle, which is asymptotically stable within ∂X. Let λ_{k} > 0 and −μ_{k} < 0 be the two external eigenvalues at Λ_{k}. If ∏ λ_{k} > ∏ μ_{k} then Λ is repelling. If ∏ λ_{k} < ∏ μ_{k} then Λ is asymptotically stable.

Corollary 2 was known for heteroclinic cycles in the plane (ex. 4) to Dulac, and rediscovered by [R], [H1], and others. It applies to our examples 1,3 and 9. Corollary 1 covers most of the examples of robust heteroclinic cycles, forced by invariance of hyperplanes, considered in the literature so far. In particular it applies to examples 1–4, 7–9 above. The essential special case of example 5 was treated in [AH, Thm. 5] and [HS, ch. 20.5]. The stability of simple heteroclinic cycles was analyzed again with Poincaré sections in [H2], [HS, ch. 29.3], [FS], [KM, Thm. 7.1]. For the behaviour of time–averages near (planar and simple) attracting heteroclinic cycles see [A], [Gau], [GaH] and [T].

The method of Poincaré sections is more involved (if done rigorously). First it requires the existence of a smooth linearization near the fixed points in order to compute the Poincaré map. This point was ignored in most of the above papers. According to Sternberg, smooth linearizations exist if sufficiently many non-resonance conditions are satisfied. However, the existence of n invariant hyperplanes near a hyperbolic fixed point in IR^{n} already implies the existence of a C^{1}−linearization, even if there are resonances between the n real eigenvalues. This follows from the work of [Be] and [Sa].

Also the conditions obtained by Poincaré maps are far less explicit (in higher dimensions at least, in terms of a spectral radius) and it is not obvious how to derive our explicit characterization on C from them. Moreover Poincaré sections would be more difficult to apply if the Λ_{k} are not fixed points but more complicated invariant sets. In this respect our method seems superior. On the other hand, the Poincaré map contains more information on the dynamics and one can treat also the induced bifurcations.

5. Appendix: Proof of Theorem 1.

Part a) is essentially contained (although not explicitely stated in this form) in [H1], [Hu], [H3]. The basic idea is to use P (x) = ∏ x_{k}^{p_{k}} as an average Ljapunov function: The assumption Cp > 0 means that P increases most of the time along orbits as long as they are close to Λ. Hence they finally have to move away from Λ.

Part b) generalizes Thm. 3 in [AH]. The idea to consider p ∈ IR^{n} with both positive and negative entries as in case c) was first exploited by [J].

For the proof of b) and c) let P (x) = ∏ x_{i}^{−p_{i}}. Let I = {i : p_{i} < 0} and X_{0} = {x ∈ X : x_{i} = 0 for some i ∈ I\} be the part of ∂X where P vanishes. Now \frac{Ṗ}{P} = − ∑_{j} p_{j} \frac{ẋ_{j}}{x_{j}} extends continuously to ∂X and takes the (average) value − ∑_{j} p_{j}λ_{j}(Λ_{k}) = −(Cp)_{k} < 0 at the fixed point (resp. along the uniquely ergodic set) Λ_{k}. Intuitively this means that P decreases exponentially (in the average) near Λ and we expect that orbits close to Λ will converge to X_{0}. More precisely, with Λ_{0} = Λ ∩ X_{0} we have the following

Lemma. Suppose there is a p ∈ IR^{n} such that p_{i} < 0 for at least one i and Cp > 0 and that Λ_{0} is asymptotically stable within X_{0}. Then all orbits starting in a set {x ∈ int X : P (x) < δ} and close enough to Λ_{0} will converge to Λ_{0}.

If all p_{i} < 0 then X_{0} = ∂X and the Lemma implies that Λ is asymptotically stable in X. This shows part b) of Theorem 1. If p has positive and negative components then the Lemma isolates a partially attracting part Λ_{0} of Λ. (It is easy to see that Λ_{0} ≠ ∅.) The situation in the lemma and its proof can be illustrated by the behaviour of the planar system in fig. 6 which has an elliptic sector at 0: X_{0} corresponds to the x–axis, Λ to the y–axis, Λ_{0} to the origin, and the assumptions on C are replaced by the fact that P = y/x decreases in the positive quadrant. It may be interesting to remark that the return map near a heteroclinic network may have such elliptic sectors for an open set in parameter space, see [Br].

Proof. Since Λ_{0} is asymptotically stable within the invariant subset X_{0} one can find a neighbourhood U ⊂ X_{0} of Λ_{0} in X_{0} which is forward invariant and its smooth boundary is transverse to the vector field in X_{0}. (If Λ_{0} = X_{0}, then we set U = X_{0}.) For small enough ε, the cylinder U_{ε} = ⋃_{i∈I}{x ∈ X : 0 ≤ x_{i} < ε and π_{i}(x) ∈ U} (where π_{i} is a suitable projection onto the face {x_{i} = 0}), with base set U and height ε, is then still forward invariant along its side surface (but not necessarily at its top). Let U_{ε}^{δ} = U_{ε} ∩ {x ∈ X : P (x) < δ}, with δ so small that min{x_{i} : i ∈ I} < ε holds for all x with P (x) ≤ δ. As in [H1] or [Hu] we can find constants T̄ > 1, k < 1 and K > 0 such that for all x ∈ X close to Λ there is a time T with 1 < T < T̄ such that P (x(T)) < kP (x) and P (x(t)) < KP (x) for all 0 < t < T̄. So for x ∈ U_{ε}^{δ/K} and 0 < t < T̄, x(t) ∈ U_{ε}^{δ} while x(T) ∈ U_{ε}^{kδ/K}. Iterating this argument we see that the forward orbit of x cannot leave U_{ε}^{δ} and P (x(t)) will converge exponentially to 0, as t ⟶ ∞. Hence ω(x) ⊂ Λ_{0} (the maximal compact invariant subset of U_{ε}^{δ}) and Λ_{0} is stable for the semiflow restricted to U_{ε}^{δ/K}. □

An alternative way to prove the Lemma is to apply the approach of [G,H3] for backward time and show that there is no negative orbit in the set U_{ε}^{δ}.

References

[A] E. Akin: The General Topology of Dynamical Systems. Amer. Math. Soc. 1993.

[AH] E. Amann, J. Hofbauer: Permanence in Lotka–Volterra and replicator equations. In: Lotka-Volterra Approach to Cooperation and Competition (Proc. Wartburg 1984). Ed. W. Ebeling and M. Peschel. pp. 23-34. Akademie-Verlag Berlin 1985.

[Be] G. R. Belitskiĭ: Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class. Funct. Anal. Appl. 7 (1973) 268–277.

[BS] N. P. Bhatia, G. P. Szegő: Stability Theory of Dynamical Systems. Springer (1970).

[Br] W. Brannath: Heteroclinic networks on the tetrahedron. Nonlinearity 7 (5), 1367-1384 (1994).

[C] A. Chenciner: Comportement asymptotique de systemes differentiels du type ‘competition d’especes’. Comptes Rendus Ac. Sc. Paris 284 313–315 (1977).

[CG] A. Capietto, B. M. Garay: Saturated invariant sets and the boundary behaviour of differential systems. J. Math. Anal. Appl. 176, 166–181 (1993).

[CPC] J. Coste, J. Peyraud, P. Coullet: Asymptotic behaviour in the dynamics of competing species. SIAM J. Appl. Math. 36 (1979) 516–542.

[FP] M. Fiedler, V. Ptak: Some generalizations of positive definiteness and monotonicity. Numer. Math. 9 (1966) 163–172.

[FS] M. Field, J. W. Swift: Stationary bifurcation to limit cycles and heteroclinic cycles. Nonlinearity 4 (1991) 1001–1043.

[G] B. M. Garay: Uniform persistence and chain recurrence. J. Math. Anal. Appl. 139 (1989) 372–381.

[Gau] A. Gaunersdorfer: Time averages for heteroclinic attractors. SIAM J. Appl. Math. 52 (1992) 1476–1489.

[GaH] A. Gaunersdorfer, J. Hofbauer: Fictitious play, Shapley polygons, and the replicator equation. Games Economic Behaviour 11 (1995) 279–303.

[GH] J. Guckenheimer, P. Holmes: Structurally stable heteroclinic cycles. Math. Proc. Cambridge Philos. Soc. 103 (1988) 189–192.

[H1] J. Hofbauer: A general cooperation theorem for hypercycles. Monatsh. Math. 91 (1981) 233–240.

[H2] J. Hofbauer: Heteroclinic cycles on the simplex. In: Proc. 11th International Conference on Nonlinear Oscillations (ICNO). Eds. M. Farkas, V. Kertész, G. Stépán. pp. 828–831. Budapest: Janos Bolyai Math. Society. 1987.

[H3] J. Hofbauer: A unified approach to persistence. Acta Applicandae Math. 14 (1989) 11–22.

[HS] J. Hofbauer, K. Sigmund: The Theory of Evolution and Dynamical Systems. Cambridge University Press (1988).

[HS2] J. Hofbauer, K. Sigmund: On the stabilizing effect of predators and competitors on ecological communities. J. Math. Biol. 27 (1989) 537–548.

[Hu] V. Hutson: A theorem on average Liapunov functions. Monatsh. Math. 98, 267–275 (1984).

[HuS] V. Hutson, K. Schmitt: Permanence and the dynamics of biological systems. Math. Biosciences 111, 1–71 (1992).

[J] W. Jansen: Über Permanenzbedingungen von Populationssystemen. Preprint. Osnabrück. 1991.

[K1] G. Kirlinger: Permanence in Lotka–Volterra equations: Linked predator–prey systems. Math. Biosciences 82, 165–191 (1986).

[K2] G. Kirlinger: Permanence of some ecological systems with several predator and one prey species. J. Math. Biol. 26, 217–232 (1988).

[K3] G. Kirlinger: Two predators feeding on two prey species: A result on permanence. Math. Biosciences 96, 1–32 (1989).

[KM] M. Krupa, I. Melbourne: Asymptotic stability of heteroclinic cycles in systems with symmetry. Ergodic Theory Dyn. Syst. 15 (1), 121–147 (1995).

[KS] V. Kirk, M. Silber: A competition between heteroclinic cycles. Nonlinearity 7 (6), 1605–1621 (1994).

[LR] R. Lauterbach, M. Roberts: Heteroclinic cycles in dynamical systems with broken spherical symmetry. J. Diff. Equs. 100, 2–48 (1992).

[MCG] I. Melbourne, P. Chossat, M. Golubitsky: Heteroclinic cycles involving periodic solutions in mode interaction with O(2) symmetry. Proc. Roy. Soc. Edinburgh 113A (1989) 315–345.

[ML] R. May, W. J. Leonard: Nonlinear aspects of competition between three species. SIAM J. Appl. Math. 29, 243–253 (1975).

[MR] D. Mukherje, A. B. Roy: Uniform persistence and global stability of two prey two predator pairs linked by competition. Math. Biosciences 99, 31–45 (1990).

[PM] M. Peschel, W. Mende: The Predator–Prey Model. Do We Live in a Volterra World? Berlin: Akademie– Verlag. 1986.

[R] J. W. Reyn: A stability criterion for separatrix polygons in the phase plane. Nieuw Archiv Wisk. 27 (1979) 238–254.

[Sa] V. S. Samovol: Linearization of a system of differential equations in the neighbourhood of a singular point. Dokl. Akad. Nauk SSSR 206 (1972) 545–548.

[S1] P. Schuster, K. Sigmund, R. Wolff: On ω−limits for competition between three species. SIAM J. Appl. Math. 37, 49–57 (1979).

[S2] P. Schuster, K. Sigmund, R. Wolff: Dynamical systems under constant organization. Part 3: Cooperative and competitive behaviour of hypercycles. J. Diff. Equations 32, 357–368 (1979).

[S3] K. Sigmund: Permanence and heteroclinic cycles for ecological equations. In: see [H2], pp. 210–216.

[SR] A. Sikder, A. B. Roy: Persistence of a generalized Gause–type predator prey pair linked by competition. Math. Biosciences 122, 1–23 (1994).

[T] F. Takens: Heteroclinic attractors: Time averages and moduli of topological conjugacy. Bol. Soc. Bras. Mat. 25, 107–120 (1994).

[Z] E. C. Zeeman: Population dynamics from game theory. In: Global Theory of Dynamical Systems. Proc. Conf. Northwestern Univ. 1979. Springer Lecture Notes in Mathematics 819, 472–497 (1980).