Global Stability of Competition in a Spatially Heterogeneous Environment^{†}

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

Joseph W.-H. So Department of Mathematics University of Alberta Edmonton, Alberta, Canada T6G 2G1

Yasuhiro Takeuchi Department of Applied Mathematics Shizuoka University Hamamatsu, Japan

Abstract. It is shown that under certain assumptions on the dispersal rates, competition of two species which disperse between several patches, leads to a globally stable equilibrium, even in a heterogeneous environment.

† Research partially supported by the Natural Sciences and Engineering Research Council of Canada, grant number NSERC OGP36475 and by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, project P8043.

Introduction. One promising application of mathematical population biology is to verify the role of the heterogeneity of the environment on the species coexistence.

In the present paper, we will consider the problem whether a system composed of multiple competitive patches, which are connected by species dispersion, is globally stable or not, if each isolated patch is globally stable. It is known under some biologically reasonable assumptions that global stability continues to hold if all the patch dynamics are identical and the dispersion is species-independent. It turns out that this need not be the case when the environment is heterogeneous, that is, when the dynamics are patch dependent. When the number of the patches is two or three, it will be shown that global stability still holds under some particular types of the species-dispersion.

1. Competition of Two Species in an n–Patch Environment

We consider competition of two species dispersing between n patches governed by the equations:

\{ \begin{matrix}u̇_{i} & = ∑_{j=1}^{n} d_{ij}u_{j} + u_{i}f_{i}(u_{i}, v_{i}) \\ v̇_{i} & = ∑_{j=1}^{n} d′_{ij}v_{j} + v_{i}g_{i}(u_{i}, v_{i}) \end{matrix}  (i = 1, …, n) .  (1.1)

Here u_{i} and v_{i} are the densities of the two competing species in patch i. The constants d_{ij}, d′_{ij} ≥ 0 (i ≠ j) denote the dispersal rate of species u, v from patch j to patch i. The constants d_{ii}, d′_{ii} < 0 are the rates at which the two species in the i–th patch move out of the patch. Therefore

d_{ii} + ∑_{j≠i} d_{ji} ≤ 0  and  d′_{ii} + ∑_{j≠i} d′_{ji} ≤ 0  (1.2)

with equality holding if all individuals from patch i arrive safely at some other patch and with strict inequality holding if there is loss along the way.

More specifically, the d_{ij}’s and d′_{ij}’s are assumed to satisfy

(H1) d_{ij}, d′_{ij} ≥ 0 for 1 ≤ i ≠ j ≤ n and the matrices D = [d_{ij}] and D′ = [d′_{ij}] are irreducible.

The functions f_{i}(u_{i}, v_{i}) and g_{i}(u_{i}, v_{i}) describe the per capita growth due to the competition of species u and v in the i–th patch in the absence of dispersion. As such, they are assumed to satisfy

(H2) For all i = 1, …, n, \frac{∂f_{i}}{∂v_{i}} < 0 and \frac{∂g_{i}}{∂u_{i}} < 0.

The vast majority of research has been focused on the case of equal patches (f_{i} and g_{i} are independent of i) and species-independent dispersal rates d_{ij} = d′_{ij}. It has essentially been shown, that if the single patch dynamics exhibits (a certain strong form of) global stability, then also the n–patch model is globally stable, see [Ha], [T]. In general, local stability of an equilibrium can be destroyed by diffusion, and stable spatially non-constant equilibria will arise. For Lotka–Volterra interactions, at least three species are needed for this Turing effect to arise, see [K]. On the other hand, if the single patch dynamics is bistable, then the system with dispersion will have lots of stable equilibria (2^{n}) and even more unstable ones, at least for small d_{ij}’s, see [L].

In the more general case when the dynamics is patch dependent, but each patch separately is still globally stable, one cannot expect global stability for the full system, as we will show in examples below. This is in sharp contrast to the one species case ([FT], [Ho]). However, we will show in §2 and §3 that for certain Lotka–Volterra type interactions where the basic growth rates may depend on the patch but the interaction matrix is patch independent, that is,

\begin{aligned}f_{i}(u_{i}, v_{i}) & = r_{i} − αu_{i} − βv_{i} \\ g_{i}(u_{i}, v_{i}) & = s_{i} − γu_{i} − δv_{i}, \end{aligned}  (1.3)

global stability results are available. These results are related to resp. confirm conjectures based on numerical simulations suggested by Goh [G, sect. 5.10, in particular Example 5.10.5].

For the rest of the section, we will review some general properties of system (1.1) which follows from the theory of monotone dynamical systems.

If one of the species (say v) is absent in all the patches, i.e. v_{i} = 0 for all i, then (1.1) reduces to a standard model of a single species dispersing among n patches:

u̇_{i} = ∑_{j=1}^{n} d_{ij}u_{j} + u_{i}f_{i}(u_{i}, 0)  (i = 1, …, n).  (1.4)

Equation (1.4) is a cooperative system and it defines a strongly monotone semiflow on IR_{+}^{n} (c.f. [Hi]).^{1} The only equilibrium on ∂IR_{+}^{n} is the origin E_{0} = (0, 0) and its stability can be read from the Jacobian matrix, J_{u}(E_{0}) = [d_{ij} + δ_{ij}f_{i}(0, 0)], of (1.4) evaluated at E_{0}.

Under the usual assumptions, namely

(H3) For all i = 1, …, n, \frac{∂f_{i}}{∂u_{i}} (u_{i}, 0) < 0, \frac{∂g_{i}}{∂v_{i}} (0, v_{i}) < 0, and there exists K > 0 such that f_{i}(K, 0), g_{i}(0, K) < 0,

it can be shown (c.f. [FT], [Ho]) that if the spectral modulus, s(J_{u}(0)), of the quasi– positive matrix J_{u}(0) is positive, there is a unique positive equilibrium û of (1.4) which is globally stable. This is certainly the case if equality holds in (1.2) and f_{i}(0, 0) > 0 for all i. On the other hand, if s(J_{u}(0)) ≤ 0 then the origin E_{0} is globally stable. Of course, this extinction is possible only if either migration is very dangerous (strict inequality in (1.2)) or some of the basic growth rates f_{i}(0, 0) are negative.

There are analogous statements for the case when u is absent and we will denote by ṽ the corresponding positive equilibrium when s(J_{v}(0)) > 0, where J_{v}(0) = [d′_{ij} +δ_{ij}g_{i}(0, 0)].

The full system (1.1) defines a semiflow on IR_{+}^{2n} and E_{0} = (0, 0), Ê = (û, 0) and Ẽ = (0, ṽ) are the only equilibria on ∂(IR_{+}^{2n}). The Jacobian matrix of (1.1) is given by the following 2n × 2n matrix (where i, j = 1, …, n in each of the four blocks)

1An ODE u̇_{i} = F_{i}(u) is called cooperative, if \frac{∂F_{i}}{∂u_{j}} ≥ 0 for i ≠ j. Monotonicity means that for two solutions u, ũ of this ODE, u_{i}(0) ≤ ũ_{i}(0) for all i implies that u_{i}(t) ≤ ũ_{i}(t) for all i and t > 0, and strong monotonicity, that whenever for one i the inequality is strict at time 0, it will be strict for all i for all positive times t.

J = ( \begin{matrix}d_{ij} + δ_{ij}(f_{i} + u_{i} \frac{∂f_{i}}{∂u_{i}}) & δ_{ij}u_{i} \frac{∂f_{i}}{∂v_{i}} \\ δ_{ij}v_{i} \frac{∂g_{i}}{∂u_{i}} & d′_{ij} + δ_{ij}(g_{i} + v_{i} \frac{∂g_{i}}{∂v_{i}}) \end{matrix} ) .  (1.5)

Clearly, J(E_{0}) = diag{J_{u}(0), J_{v}(0)}. At Ê, the Jacobian matrix J takes the block triangular form

J(Ê) = ( \begin{matrix}d_{ij} + δ_{ij}(f_{i}(û_{i}, 0) + û_{i} \frac{∂f_{i}}{∂u_{i}} (û_{i}, 0)) & δ_{ij}û_{i} \frac{∂f_{i}}{∂v_{i}} (û_{i}, 0) \\ 0 & d′_{ij} + δ_{ij}g_{i}(û_{i}, 0) \end{matrix} )  (1.6)

with the lower left block being zero. The upper left block, J_{u}(û), corresponds to the Jacobian of (1.4) at û and is a stable matrix. Hence the stability of Ê is determined by the lower right block C = [d′_{ij} + δ_{ij}g_{i}(û_{i}, 0)]. Since C is a quasi–positive matrix, Ê is stable if s(C) < 0 and is unstable (in the v direction) if s(C) > 0. Analogous statements hold for Ẽ.

The global analysis of (1.1) on IR_{+}^{2n} is made possible by the observation that it defines a monotone semiflow with respect to the following cone in IR_{+}^{2n},

P = {(u, v) ∈ IR^{2n} : u_{i} ≥ 0 and v_{i} ≤ 0 for all i = 1, …, n}  (1.7)

(c.f. [S2]), and induced ordering (u, v) ≼ (ū, v̄) if (ū, v̄) − (u, v) ∈ P, (u, v) ≺ (ū, v̄) if (ū, v̄) − (u, v) ∈ P \ {(0, 0)}, and (u, v) ≺≺ (ū, v̄) if (ū, v̄) − (u, v) ∈ int P.

The flow of (1.1) is actually strongly monotone in int IR_{+}^{2n}, but not on IR_{+}^{2n}, since Ẽ ≺ E_{0} ≺ Ê, but not Ẽ ≺≺ E_{0} ≺≺ Ê.

Still, this implies the following: If there is no interior equilibrium in IR_{+}^{2n}, then one of the boundary equilibria must be a global attractor.

More precisely,

(i) If s(J_{u}(0)) ≤ 0 and s(J_{v}(0)) ≤ 0 (i.e. neither Ê nor Ẽ exists), then the origin E_{0} is globally stable. (ii) If s(J_{u}(0)) > 0 and s(J_{v}(0)) ≤ 0 (i.e. Ê exists but not Ẽ), then Ê is globally stable. Similarly if s(J_{v}(0)) > 0 and s(J_{u}(0)) ≤ 0, then Ẽ is globally stable. (iii) If both Ê and Ẽ exist and there is no interior equilibrium, then one of them is globally stable.

The last assertion follows from a small modification of [DH, Prop. 1 and Remark 1.2]. They show that, if the interval [E_{1}, E_{2}] spanned by two equilibria E_{1}, E_{2} contains no third equilibrium, then there is a monotone connecting orbit from one to the other, say from E_{1} to E_{2}. This implies that all orbits starting in [E_{1}, E_{2}] converge to E_{2}. Note that [DH] do not require strong monotonicity, as e.g. [Hi]. In our situation this result does not apply directly, since the origin E_{0} lies in the interval [Ẽ, Ê]. However, E_{0} is a repellor in case (iii). More precisely, there exist numbers c_{i}, c′_{i} > 0 (e.g., the components of the left Perron– eigenvectors of J_{u}(0) and J_{v}(0)) such that L(u, v) = ∑_{i}(c_{i}u_{i} + c′_{i}v_{i}) is a monotonically increasing Ljapunov function near E_{0}. Hence we can excise a neighbourhood U of E_{0}, such that the set [Ẽ, Ê] \ U is positively invariant and convex. The global stability of either Ẽ or Ê follows now by repeating the argument by [DH, Prop. 1] for this set.^{2}

2Another proof of (iii) was recently given by [HSW].

If both Ê and Ẽ are unstable, then (1.1) is uniformly persistent and there exist interior equilibria E_{∗} and E^{∗} of (1.1) with Ẽ ≺ E_{∗} ≼ E^{∗} ≺ Ê and all interior orbits of (1.1) converge to the ordered interval [E_{∗}, E^{∗}]. In particular, if there is only one interior equilibrium (call it E^{∗} say), then E^{∗} is globally stable.

On the other hand, it is also possible that both Ê and Ẽ are stable. In that case, there will be (at least one) an unstable interior equilibrium (c.f [DH]) and possibly complicated dynamics on a hypersurface (c.f. [Hi]).

Remark. If (1.1) has an (exponentially) unstable interior fixed point, then it must have at least two stable equilibria. This follows from the fact that any strongly monotone system defined on an order interval has at least one stable (but not necessarily asymptotically stable) equilibrium, see [Hi, Thm. 10.2] and [DH, Thm. 3].

2. Global Stability for Two Patches

When n = 2, system (1.1) becomes

\{ \begin{matrix}u̇_{1} & = d_{11}u_{1} + d_{12}u_{2} + u_{1}f_{1}(u_{1}, v_{1}) \\ u̇_{2} & = d_{21}u_{1} + d_{22}u_{2} + u_{2}f_{2}(u_{2}, v_{2}) \\ v̇_{1} & = d′_{11}v_{1} + d′_{12}v_{2} + v_{1}g_{1}(u_{1}, v_{1}) \\ v̇_{2} & = d′_{21}v_{1} + d′_{22}v_{2} + v_{2}g_{2}(u_{2}, v_{2}) . \end{matrix}  (2.1)

To simplify matters, we employ the notations:

α_{i} = −\frac{∂f_{i}}{∂u_{i}} , β_{i} = −\frac{∂f_{i}}{∂v_{i}} , γ_{i} = −\frac{∂g_{i}}{∂u_{i}} , δ_{i} = −\frac{∂g_{i}}{∂v_{i}} .

Moreover, it will be assumed that

\{ \begin{matrix}α_{i}, β_{i}, γ_{i}, δ_{i} > 0 \\ α_{i}δ_{i} ≥ β_{i}γ_{i} \end{matrix}  (i = 1, 2) .  (2.2)

The second condition was called tolerant competition in [G]. It implies the stability of an interior equilibrium in patch i in case of no dispersal.

Let J denote the Jacobian matrix of (2.1) evaluated at a positive equilibrium E^{∗} of (2.1). By using the equilibrium equations of (2.1), more specifically,

\{ \begin{matrix}f_{1} = −d_{11} − d_{12} \frac{u_{2}}{u_{1}} , f_{2} = −d_{22} − d_{21} \frac{u_{1}}{u_{2}} \\ g_{1} = −d′_{11} − d′_{12} \frac{v_{2}}{v_{1}} , g_{2} = −d′_{22} − d′_{21} \frac{v_{1}}{v_{2}} , \end{matrix}

J can be simplified to

J = ( \begin{matrix}−d_{12} \frac{u_{2}}{u_{1}} − u_{1}α_{1} & d_{12} & −u_{1}β_{1} & 0 \\ d_{21} & −d_{21} \frac{u_{1}}{u_{2}} − u_{2}α_{2} & 0 & −u_{2}β_{2} \\ −v_{1}γ_{1} & 0 & −d′_{12} \frac{v_{2}}{v_{1}} − v_{1}δ_{1} & d′_{12} \\ 0 & −v_{2}γ_{2} & d′_{21} & −d′_{21} \frac{v_{1}}{v_{2}} − v_{2}δ_{2} \end{matrix} ) .

We would like to find out under what conditions will E^{∗} be asympotically stable. In order to show that J is a stable matrix, it suffices to demonstrate that the leading principal minors of J alternate in sign (starting from negative) (c.f. [S2]). It is easily seen that the one by one leading principal minor of J is negative and the two by two principal minor of J is positive. Using Maple, one can show that the three by three leading principle minor of J is given by:

\begin{aligned}D_{3} & = −\frac{1}{u_{1}u_{2}v_{1}} (d_{12}d′_{12}u_{2}^{3}v_{2}α_{2} + d_{12}u_{2}^{3}v_{1}^{2}α_{2}δ_{1} + d_{21}d′_{12}u_{1}^{3}v_{2}α_{1} \\ & + d′_{12}u_{1}^{2}u_{2}^{2}v_{2}α_{1}α_{2} + d_{21}u_{1}^{3}v_{1}^{2}ε_{1} + u_{1}^{2}u_{2}^{2}v_{1}^{2}α_{2}ε_{1}) , \end{aligned}

where ε_{i} = α_{i}δ_{i} − β_{i}γ_{i} ≥ 0 (i = 1, 2), which is negative. Finally, the determinant of J is given by:

D_{4} = \frac{1}{u_{1}u_{2}v_{1}v_{2}} ((u_{1}v_{2}γ_{1}/δ_{1} − u_{2}v_{1}γ_{2}/δ_{2})(d_{21}d′_{12}u_{1}^{2}v_{2}^{2}β_{1}δ_{2} − d_{12}d′_{21}u_{2}^{2}v_{1}^{2}β_{2}δ_{1})
+ ε_{1}d_{21}d′_{12}u_{1}^{3}v_{2}^{3}δ_{2}/δ_{1} + ε_{2}d_{12}d′_{21}u_{2}^{3}v_{1}^{3}δ_{1}/δ_{2}
+ ε_{1} (d_{21}d′_{21}u_{1}^{3}v_{1}^{3} + d′_{21}u_{1}^{2}u_{2}^{2}v_{1}^{3}α_{2} + d_{21}u_{1}^{3}v_{1}^{2}v_{2}^{2}δ_{2})
+ ε_{2} (d_{12}d′_{12}u_{2}^{3}v_{2}^{3} + d′_{12}u_{1}^{2}u_{2}^{2}v_{2}^{3}α_{1} + d_{12}u_{2}^{3}v_{1}^{2}v_{2}^{2}δ_{1})
+ ε_{1}ε_{2}u_{1}^{2}u_{2}^{2}v_{1}^{2}v_{2}^{2}).

To proceed further we will now assume that (1.3) holds. Then

α_{1} = α_{2} = α, β_{1} = β_{2} = β, γ_{1} = γ_{2} = γ, δ_{1} = δ_{2} = δ .  (2.3)

Under this assumption, the determinant of J can be simplified to

D_{4} = \frac{βγ}{u_{1}u_{2}v_{1}v_{2}} ((u_{1}v_{2} − u_{2}v_{1})(d_{21}d′_{12}u_{1}^{2}v_{2}^{2} − d_{12}d′_{21}u_{2}^{2}v_{1}^{2}) + non–negative terms) ,

where the ‘non-negative’ terms contain ε = αδ − βγ ≥ 0.

We will use the term coordinated dispersal if

d_{12}/d_{21} = d′_{12}/d′_{21}.  (2.4)

This covers the case of equal dispersal rates (i. e. d_{12} = d′_{12} and d_{21} = d′_{21} ) as well as the case of symmetric dispersal rates (i. e. d_{21} = d_{12} and d′_{21} = d′_{12}). Under the assumption of coordinated dispersal, we finally obtain

D_{4} = \frac{d_{21}d′_{12}βγ}{u_{1}u_{2}v_{1}v_{2}} ((u_{1}v_{2} − u_{2}v_{1})^{2}(u_{1}v_{2} + u_{2}v_{1}) + non–negative terms)

so that D_{4} is positive as long as ε_{1} = ε_{2} = ε > 0.

Theorem 2.1. Under the assumptions: (1.3), tolerant competition ε = αδ − βγ > 0 and coordinated dispersal (2.4), the system (2.1) has a unique, globally stable equilibrium.

Proof. Following the discussion of Section 1, it suffices to consider the case when there is a positive equilibrium E^{∗}. Since Ẽ ≺ E^{∗} ≺ Ê, the conclusion follows from the trichotomy of [DH, Prop. 2.1] and the fact, that any interior equilibrium E^{∗} of (2.1) is asymptotically stable. □

Remark. If (2.3) or (2.4) is not satisfied, then the leading term in the expansion of D_{4} will be indefinite. Hence it is possible to construct numerical examples (with small ε) for which D_{4} < 0. In that case E^{∗} will be unstable and (2.1) will have at least two stable equilibria.

3. Three Patches

We now extend the global stability results of Section 3 to symmetric dispersal rates as well as linear migration pattern in three patches. For cyclic migration patterns we will construct a numerical example with multiple stable equilibria.

We consider competition of two species dispersing between 3 patches under certain Lotka–Volterra interactions. System (1.1) in this case becomes:

\{ \begin{matrix}u̇_{i} & = ∑_{j=1}^{3} d_{ij}u_{j} + u_{i}f_{i}(u_{i}, v_{i}) \\ v̇_{i} & = ∑_{j=1}^{3} d′_{ij}v_{j} + v_{i}g_{i}(u_{i}, v_{i}) \end{matrix}  (i = 1, 2, 3) .  (3.1)

where f_{i} and g_{i} are given by (1.3).

We will mainly consider the following two cases:

(i) linearly linked: d_{13} = d_{31} = d′_{13} = d′_{31} = 0, and (ii) cyclically linked: d_{12} = d_{23} = d_{31} = 0 = d′_{12} = d′_{23} = d′_{31} = 0.

Theorem 3.1. Let n = 3, (1.3) hold and ε = αδ − βγ > 0. Assume that d_{ij}, d′_{ij} > 0 for i ≠ j except for d_{13} = d_{31} = d′_{13} = d′_{31} = 0 and that we have coordinated dispersion in the sense of (2.4) between patches 1 and 2 and also patches 2 and 3. Then system (3.1) has at most one interior equilibrium E^{∗}. If E^{∗} exists, it is globally stable. If E^{∗} does not exist, then one of the three boundary equilibria is globally stable.

Proof. We first show that E^{∗}, if exists, is locally stable. For this we compute the Jacobian matrix J of (3.1) at E^{∗}. Using (1.5), J is the 6 by 6 matrix

J = ( \begin{matrix}d_{ij} + δ_{ij}(f_{i} − αu_{i}) & −δ_{ij}βu_{i} \\ −δ_{ij}γv_{i} & d′_{ij} + δ_{ij}(g_{i} − δv_{i}) \end{matrix} )  (3.2)

At E^{∗}, (3.1) implies

\begin{matrix}f_{i} & = −\frac{∑_{j=1}^{3} d_{ij}u_{j}}{u_{i}} \\ g_{i} & = −\frac{∑_{j=1}^{3} d′_{ij}v_{j}}{v_{i}} \end{matrix}  (i = 1, 2, 3)  (3.3)

which we use to simplify the diagonal terms of J.

According to [S2], it suffices to show that the leading principal minors of J alternate in sign. We use Maple to prove this. For instance, the last leading principal minor D_{6} of J is the determinant of J and can be expressed as:

(δ^{2}u_{1}u_{2}u_{3}v_{1}v_{2}v_{3})^{−1}(D_{60} + D_{61}ε + D_{62}ε^{2} + D_{63}ε^{3})

where ε = αδ − βγ ≥ 0 is the determinant of the interaction matrix. It turns out that

D_{60} =βγδd_{12}d_{32}
(xd_{21}(u_{1}v_{2} + u_{2}v_{1})(u_{1}v_{2} − u_{2}v_{1})^{2}(yd_{32}δu_{2}v_{2} + yβγu_{3}^{2}v_{2} + δ^{2}u_{2}v_{3}^{2})
+ yd_{23}(u_{2}v_{3} + u_{3}v_{2})(u_{2}v_{3} − u_{3}v_{2})^{2}(xd_{12}δu_{2}v_{2} + xβγu_{1}^{2}v_{2} + δ^{2}u_{2}v_{1}^{2})
+ xd_{21}yd_{23}u_{2}v_{2}δ(u_{1}v_{3} + u_{3}v_{1})(u_{1}v_{3} − u_{3}v_{1})^{2}) ≥ 0,

where d_{12}/d_{21} = d′_{12}/d′_{21} = x and d_{23}/d_{32} = d′_{23}/d′_{32} = y. The other coefficients D_{61}, D_{62}, D_{63} can also be shown to be positive. Similarly, the leading principal minor of size i (i = 1, …, 5) can be shown to have sign (−1)^{i}.

Once we know that any positive equilibrium is asymptotically stable, the rest follows in the same way as in the proof of Theorem 2.1. □

Remark. One can also use Maple to show that Theorem 3.1 holds for the case of coordinated migration

d_{ij}/d_{ji} = d′_{ij}/d′_{ji} = κ_{ij}, 1 ≤ i < j ≤ 3  (3.4)

provided additionally

κ_{12}κ_{23} = κ_{13}(≠ 0)  (3.5)

holds. This covers in particular the case of symmetric migration patterns (i. e. d_{ij} = d_{ji} and d′_{ij} = d′_{ji} for all 1 ≤ i, j ≤ 3).

Condition (3.1) appears to be sharp. If it is violated then D_{60} is an indefinite expression. The following example of cyclic migration shows that the above global stability results do not hold for arbitrary migration patterns.

Take as migration rates d_{11} = −200, d_{22} = d_{33} = −100, d_{12} = d_{23} = d_{31} = 0, d_{21} = d_{32} = 1, d_{13} = 50 for both species. As interaction coefficients we choose α = 1.001, β = γ = δ = 1. If we take the basic growth rates as r_{1} = 52.001, r_{2} = 102.502, r_{3} = 310009/3000, s_{1} = 152, s_{2} = 102, s_{3} = 103 then u = (1, 2, 3), v = (1, 1, 1) will be a positive equilibrium point. Using Maple one can verify that the determinant of the Jacobian at this equilibrium is −192597329057/750000000. Since this is negative, this equilibrium is unstable. Hence (see last remark in section 1) there will be at least two stable equilibria in this system.

4. Conclusion

The results in sections 2 and 3 suggest the following Conjecture: Under assumptions (1.2), (1.3) and ε = αδ −βγ > 0, the n–patch system (1.1) is globally stable, provided the dispersal matrices D and D′ are symmetric.

We note that Theorems 2.1 and 3.1 hold also for two mutualists, i.e. in (1.3) β and γ are negative. The assumption ε = αδ − βγ > 0 guarantees (besides stability) the boundedness of orbits, i.e. the dissipativity of system (1.1). The only difference in the proof is to use the usual ordering in IR^{2n} instead of (1.7).

For predator–prey systems (i.e. β > 0, γ < 0) (1.1) does not generate a monotone dynamical system. Also, the signs of the minors of the Jacobian do not determine the stability of the equilibrium E^{∗}. For that Routh–Hurwitz conditions are needed. One can expect stable limit cycles to exist in (1.1) for certain parameter values. However, using similar Maple calculations, one can still show that the determinant of the Jacobian is positive (at least for two or three patches). Therefore, by the index theorem of [Ho], there is a unique saturated equilibrium for (1.1) and hence at most one interior equilibrium.

The method of analysis in this paper is similar to the one in [HS] on competition in the gradostat. Also the results are related: In the case of two patches or two vessels there is global stability. In the case of three patches or three vessels there is a difference, however. For the gradostat, multiple stable equilibria are possible both in the case of linearly connected vessels and in the case of a cyclic connection, as shown in [HS]. The reason for this more complicated behaviour in the gradostat seems to be due to the nonlinear nature of the nutrient uptake functions (i.e. Monod type) in contrast to the linear Lotka–Volterra interactions which we discuss here.

Remark: It was pointed out to us by M. Shirvani of University of Alberta that the problem of deciding if a multivariate polynomial f(x_{1}, …, x_{n}) with real coefficients is positive for x_{1}, …, x_{n} > 0 can be reformulated as one involving a polynomial equation by considering the polynomial g(y_{1}, …, y_{n}, y_{n+1}) = f(y_{1}^{2}, …, y_{n}^{2})y_{n+1}^{2} + 1. Then g(y_{1}^{∗}, …, y_{n+1}^{∗}) = 0 for some y_{1}^{∗}, …, y_{n+1}^{∗} ∈ IR if and only if f(x_{1}^{∗}, …, x_{n}^{∗}) < 0 for some x_{1}^{∗}, …, x_{n}^{∗} ≥ 0. The question of whether g = 0 has a real root can be decided (at least in principle) by means of Tarski’s theorem (c.f. Theorem 5.8 on p. 335 of [J]). This involves implementing Tarski’s method in Maple and we have not pursued this approach further here.

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