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Vol. 00, No. 00, XX 2011, 1–28
RESEARCH ARTICLE
Fixed Point Global Attractors and Repellors
in Competitive Lotka-Volterra Systems
Zhanyuan Houa∗ and Stephen Baigentb
a
Faculty of Computing, London Metropolitan University,
North Campus, 166-220 Holloway Road, London N7 8DB, UK; b Department of
Mathematics, UCL, Gower Street, London WC1E 6BT, UK
(Received 00 Month 200x; final version received 00 Month 200x)
Zeeman and Zeeman (2003, From local to global behavior in competitive
Lotka-Volterra systems, Trans. Amer. Math. Soc. 355, 713–734) show that
if a strongly competitive Lotka-Volterra system (i) has a unique interior fixed
point p and (ii) the carrying simplex Σ lies below (above) the strongly balanced tangent plane to Σ at p then the system has no periodic orbits and p is
a global attractor (repellor) relative to Σ. Condition (ii) is then translated into
the definiteness of a certain quadratic function on the tangent plane, which is
equivalent to the definiteness of an (N − 1) × (N − 1) real symmetric matrix
that can be computed. Here we adapt these methods to show that the above
conclusions are still true without the assumption (i). Hence, our results apply
to globally attracting or repelling fixed points on the boundary, as well as in
the interior, of RN
+ . Moreover, the algebraic condition for global attraction also
implies global asymptotic stability of the fixed point. We also show that the
global attraction or repulsion holds not just relative to Σ, but also relative to
the interior of the first quadrant.
Keywords: Lotka-Volterra systems; global attractors; global repellors; global
asymptotic stability.
∗ Corresponding
author. Email: z.hou@londonmet.ac.uk
ISSN: 1745-9737 print/ISSN 1745-9745 online
c 2011 Taylor & Francis
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MCS/CCS/AMS Classification/CR Category numbers: 2000 MSC:
34D05, 34D20, 34C11, 92D25
1.
Introduction
Consider the Lotka-Volterra system
x0i = xi (bi − Ai x),
i ∈ IN ,
(1)
where, for any positive integer m, Im = {1, 2, . . . , m}, and for a fixed positive integer
N and each i ∈ IN , Ai = (ai1 · · · aiN ) is the ith row of an N × N interaction matrix
A = (aij ) and the bi and aij are positive constants. Under these assumptions we will
refer to (1) as a strongly competitive system (i.e. there are no zero interaction terms). In
the remainder of the paper we only work with the strongly competitive form of (1). If,
for z ∈ RN , D(z) denotes the diagonal matrix diag[z1 , . . . , zN ] then (1) may be rewritten
as
ẋ = D(x)(b − Ax).
(2)
Since (1) or (2) models the population dynamics of a community of N species competing
for resources with xi (t) denoting the population size of the ith species at time t, as usual,
N
we restrict the study of (1) to the invariant sets RN
+ = {x ∈ R : ∀i ∈ IN , xi ≥ 0}, its
N
N
N
N
interior intRN
+ = {x ∈ R+ : ∀i ∈ IN , xi > 0} and its boundary ∂R+ = R+ \ intR+ .
Recall that for x ∈ C, C ⊆ RN , the basin of attraction A(x) in C of x under a flow
φt is defined to be the union of all subsets of C of the form V ∩ C, where V ⊆ RN is an
open neighbourhood of x, such that
lim diam φt (V ∩ C) = 0,
t→+∞
(3)
where diam (Z) = supx,y∈Z |x − y|. The basin of repulsion B(x) in C of the point x ∈ C
is defined analogously by taking the limit t → −∞ in (3). For the system (1) the origin
N
O repels trajectories and its basin of repulsion in RN
+ , B(O), is open (relative to R+ ) and
bounded, we have the following well-known result about the so-called carrying simplex
Σ = B(O) \ B(O) (see [12], [14] or [3]).
Theorem 1.1 Hirsch If system (1) is strongly competitive, then every trajectory in RN
+\
{O} is asymptotic to one in Σ; Σ is a balanced Lipschitz submanifold homeomorphic to
the closed unit simplex in RN
+ via radial projection, and intΣ is strongly balanced.
N
(A set S is said to be balanced if both u − v 6∈ intRN
+ and v − u 6∈ intR+ hold, and
N
strongly balanced if both u − v 6∈ RN
+ and v − u 6∈ R+ hold, for any distinct u, v ∈ S.)
From Theorem 1.1 it is clear that the global dynamics of (1) is completely determined by the dynamics on Σ. All limit sets, and in particular fixed points, belong to Σ.
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We shall say fixed point p ∈ Σ is a global attractor (repellor ) relative to Σ if for any
x0 ∈ Σ such that x0i > 0 whenever pi > 0, for any i ∈ IN , then limt→+∞ x(t, x0 ) = p
(limt→−∞ x(t, x0 ) = p). In this paper, when calling a fixed point p a global attractor
(repellor) we mean that p is a global attractor (repellor) relative to Σ.
Remark 1 Notice that these definitions are only concerned with trajectories on Σ. If
0
p ∈ Σ is a global attractor relative to Σ and x0 ∈ RN
+ \ Σ such that xi > 0 whenever
pi > 0 for any i ∈ IN , we cannot guarantee limt→+∞ x(t, x0 ) = p although ω(x0 ) ⊂ Σ
by Theorem 1.1. Thus, the notion of a global attractor here is different from a global
attractor in RN
+ (see, e.g. [7] and the references therein for the latter). However, we shall
see in sections 3 and 4 that a global attractor relative to Σ is also a global attractor in
RN
+ under certain conditions.
N
If an (N − 1)-dimensional plane L in RN
+ does not contain the origin O, then R+ has
+
−
−
N
the partition L ∪ L ∪ L with O ∈ L . A point p ∈ R+ is said to be below (on or above)
L if p ∈ L− (L or L+ ). A set S is said to be below (on or above) L if every point in S is
so.
Zeeman and Zeeman [14] have investigated the global dynamics on Σ by using the
methods of split Liapunov functions (see [11] for further exploration). Under the assumption that (1) has a unique fixed point p in intRN
+ , they have shown that (1) has
no periodic orbits and p is a global attractor (repellor) relative to Σ if Σ \ {p} is below
(above) the strongly balanced tangent plane Tp Σ of Σ at p. Further, they have translated this condition to the definiteness of a quadratic form on Tp Σ, which is shown to be
equivalent to the definiteness of a real symmetric (N − 1) × (N − 1) matrix.
The question is whether the methods and the conclusions are still valid if p ∈ ∂Σ =
N
Σ ∩ ∂RN
+ is a boundary fixed point and (1) has no fixed point in intR+ . The aim of
this paper is to supplement [14] so that the methods and techniques used there are still
effective to determine whether a fixed point p ∈ Σ is a global attractor (repellor), no
matter whether p is in intΣ = Σ ∩ intRN
+ or ∂Σ. We shall see that, for any fixed point
p ∈ ∂Σ, Σ may or may not have a strongly balanced tangent plane at p. However, we can
always find a suitable strongly balanced plane T̃p with p ∈ T̃p as a replacement of the
tangent plane. Then the method of split Liapunov functions can be used to show that p
is a global attractor (repellor) on Σ if Σ \ {p} is below (above) T̃p and the fixed point p
is saturated (saturated backwards in time).
Our presentation is as follows. As in [14], the strongly balanced tangent plane at
a given fixed point, but now also at boundary fixed points, plays a central role in our
application of the split Liapunov function method. We therefore start in section 2 by using
the Frobenius-Perron theorem to determine suitable conditions for a strongly balanced
tangent plane to exist at interior or boundary fixed points. When a strongly balanced
tangent plane fails to exist, we show that there is always a specific trajectory γ that is
tangent to a certain line at p. In section 3 we show how to use the strongly balanced
tangent plane, when it exists, to construct a suitable Liapunov function, and how to
utilise this Liapunov function to identify globally attracting or repelling fixed points p.
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A key point, as in [14], is that the carrying simplex Σ, minus the point p, must lie above
or below a plane through p. In addition, an attracting fixed point must be saturated for
(1) and a repelling fixed point must be saturated for (1) when time is reversed. These
latter conditions are automatically satisfied for interior attractors and repellors, but
are additional requirements for boundary fixed points. For the case where the strongly
balanced tangent plane fails to exist, we show how to construct a strongly balanced
alternative. In section 3.1 we show how to apply our results to cases where Σ is known
to be convex or concave. In the case where Σ is convex, the ruling out of the trajectory γ
mentioned above implies that both interior and boundary points have strongly balanced
tangent planes. In section 4 we establish algebraic conditions for Σ \ {p} to lie above or
below the strongly balanced plane determined in section 3. These conditions are similar
to the definiteness of a quadratic form on the tangent space found in [14]. We also show
that any attracting interior or boundary point is asymptotically stable. Next, and again
following [14], we translate these definiteness conditions to definiteness of a computable
matrix of one dimension less than the quadratic form. Finally in section 5 we give a
number of examples to illustrate how our results may be applied.
N
For convenience in what follows we will set C = RN
+ , ∂C the boundary of C in R and
0
C = C \ ∂C. For any subset I ( IN and S ⊂ C, define SI = {x ∈ S : xi = 0, ∀i ∈ I}.
Then the meaning of each of CI , CI0 , Σ0 = Σ ∩ C 0 , ΣI and Σ0I is clear.
2.
Existence of a strongly balanced tangent plane Tp Σ
As mentioned above, for any fixed point of p, we look for a plane Π containing p for
which all trajectories starting from Π either stay on Π or evolve above or below it. When
there is a strongly balanced tangent plane Tp Σ to Σ at p, Π can be chosen to be Tp Σ.
Later we will show how to choose Π when such a tangent plane does not exist. Here we
begin by finding conditions for a tangent plane at p to exist.
To this end we look at the linearisation of (1) at the fixed point p (which may be in
the interior C 0 or on the boundary of C), namely
y 0 = −By
(4)
where y = x − p and B = −DF (p) with Fi (x) = xi (bi − Ai x), i ∈ IN . Explicitly, for
i, j ∈ IN ,
(
∂Fi bi − Ai p − pi aii , i = j
=
∂xj x=p
−pi aij , i 6= j.
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Since pi > 0 implies bi − Ai p = 0, we have, for the ith row of B,
(
Bi =
p i Ai ,
if pi > 0,
(0 · · · 0 Ai p − bi 0 · · · 0), if pi = 0.
If p is an interior fixed point then B has positive elements and the Frobenius-Perron
theorem for non-negative irreducible matrices immediately tells us that B has a positive
(dominant) eigenvalue λ0 with associated positive left and right eigenvectors. However,
for a boundary fixed point p ∈ ∂Σ, B may have negative diagonal elements if Ai p < bi
for some i ∈ IN such that pi = 0, and the Frobenius-Perron theorem does not apply
directly to B. Nevertheless, we will see in lemma 2.2 that a positive left eigenvector of
B may still exist.
The next lemma concerning the existence of a strongly balanced tangent plane to Σ at
a given fixed point of (1) is relevant to the case where B has a positive left eigenvector
associated with a positive eigenvalue λ0 .
If p ∈ Σ is a fixed point of (1) (boundary or interior), we will say that Σ has a tangent
plane L at p if Tp Σ = L, where Tp Σ is the tangent space of the manifold Σ at p.
Lemma 2.1 Existence of a strongly balanced tangent plane For a fixed point p ∈ Σ
(interior or boundary) of (1), assume that B in (4) has a simple positive eigenvalue λ0
with a corresponding right eigenvector β ∈ C and a left eigenvector α ∈ C 0 , and such
that every other eigenvalue λ of B satisfies Reλ < λ0 .
Then Σ has a strongly balanced tangent plane Tp Σ = L1 at p where
L1 = {x ∈ RN : αT (x − p) = 0}.
(5)
Proof. For any y 0 ∈ RN , the solution of (4) with y(0) = y 0 satisfies
(αT y(t))0 = −αT By(t) = −λ0 αT y(t)
so that
αT y(t) = αT y 0 e−λ0 t ,
t ∈ R.
With y = x − p we see from (5) that y(t) ∈ L1 for all t ∈ R if and only if y 0 ∈ L1 . Thus,
L1 is an (N − 1)-dimensional subset of C that is invariant under (4) and which contains
the origin (where y = x − p = 0). That L1 is strongly balanced follows from α ∈ C 0 . As
y(t) = βe−λ0 t is a solution of (4), the line
L0 = {x ∈ RN : x − p = rβ, r ∈ R}
(6)
is a one-dimensional subspace of C also passing through the origin x − p = 0 and which
is invariant under (4). By β ∈ C and α ∈ C 0 we have αT β > 0 and so L0 is transverse to
L1 at x − p = 0. Let Λ1 be the set of eigenvalues of B not equal to λ0 . Since λ0 is simple
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and every λ ∈ Λ1 satisfies Reλ < λ0 , the nonlinear system (1) has a one-dimensional
local invariant manifold Γ0 ⊂ CI (βi = 0 if and only if i ∈ I) tangent to L0 at p, and
an (N − 1)-dimensional local invariant manifold Γ1 ⊂ C tangent to L1 at p (see [2]).
Such a Γ1 , in general, may not be unique. However, by Reλ < λ0 for all λ ∈ Λ1 , any
solution in Γ0 tends to p faster than any other solutions not in Γ0 as t → +∞. Hence,
any (N − 1)-dimensional local invariant manifold which L0 is transverse to at p must be
tangent to L1 at p. In particular, Σ itself is (N − 1)-dimensional invariant manifold, and
L0 is transverse to Σ at p because Σ is balanced, and therefore Tp Σ = L1 .
Before turning to the case where a strongly balanced tangent plane does not exist,
we examine via the Frobenius-Perron theorem, the possible existence of positive left
eigenvectors of B, which, as we see from above, will serve as normals to Σ at fixed
points.
If p ∈ Σ0 then B = D(p)A is a positive matrix. If p ∈ Σ0J for some non-empty J ( IN ,
since we can always relabel populations if necessary, without loss of generality we may
assume that J = IN \ Ik for some k ∈ IN −1 . Thus B can always be written in block
matrix form as
B=
B11 B12
0 B22
!
,
(7)
where

B11

p1 a11 · · · p1 a1k


=  · · · · · · · · ·  > 0,
pk ak1 · · · pk akk

B12

p1 a1(k+1) · · · p1 a1N


=  ···
· · · · · ·  > 0,
pk ak(k+1) · · · pk akN
B22 = diag[Ak+1 p − bk+1 , . . . , AN p − bN ].
(8)
(9)
Notice that since det(B − λI) = det(B11 − λI) det(B22 − λI) the spectrum of B equals
the spectrum of B11 plus the eigenvalues {Ai p − bi }N
i=k+1 of B22 .
0
In the case of p ∈ Σ , since pi aij > 0 for all i, j ∈ IN , B = B11 = D(p)A is a positive
matrix and so by the Perron-Frobenius theorem [5, Theorem 8.4.4], B has an eigenvalue
λ0 > 0 such that (i) λ0 is a simple root, (ii) there are α, β ∈ C 0 satisfying Bβ = λ0 β and
αT B = λ0 αT , and (iii) every other eigenvalue λ of B satisfies |λ| < λ0 (and therefore
Reλ < λ0 ). Thus by Lemma 2.1, Σ has a strongly balanced tangent plane Tp Σ = L1
given by (5).
In the case of p ∈ ∂Σ, we may not apply the Frobenius-Perron theorem to B itself, since
some diagonal elements of B22 may be negative. However, applying the Perron-Frobenius
theorem to the positive k × k matrix B11 , we see that B11 has an eigenvalue λ0 > 0 with
properties similar to (i)–(iii). Thus, there is a positive vector (α1 , . . . , αk )T ∈ intRk+
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satisfying
(α1 . . . αk )B11 = λ0 (α1 . . . αk ).
(10)
As remarked above, λ0 is also an eigenvalue of B. Let α ∈ C be a left eigenvector of B
corresponding to λ0 . Then α satisfies (10) and the additional equation
(α1 . . . αk )B12 + (αk+1 . . . αN )B22 = λ0 (αk+1 . . . αN ).
(11)
Equation (11) is equivalent to
(αk+1 . . . αN )(λ0 I − diag[Ak+1 p − bk+1 , . . . , AN p − bN ])
= (α1 . . . αk )B12 .
(12)
Thus, if we take the αi > 0 for i ∈ Ik satisfying (10), then by (12) we have α ∈ C 0 if and
only if
Aj p − bj < λ0 ,
∀j ∈ J
(13)
(which is equivalent to saying that λ0 > 0 is the dominant eigenvalue of B), and αj for
j ∈ J is computed from
k
αj =
X
1
αi pi aij > 0,
λ0 − (Aj p − bj )
∀j ∈ J.
(14)
i=1
Moreover, as λ0 is also an eigenvalue of B, there is a β ∈ CJ0 such that Bβ = λ0 β.
Hence, by Lemma 2.1, if (13) holds then B has a positive left eigenvector α and Σ has
a strongly balanced tangent plane Tp Σ = L1 with normal direction α given by (5), (10)
and (14).
This proves the following lemma.
Lemma 2.2 Let p ∈ Σ be a fixed point (interior or boundary) of (1). If B is given by
(7)–(9) and (13) holds, then Σ has a strongly balanced tangent plane Tp Σ = L1 given by
(5), where α ∈ C 0 is found from (10) and (14).
We now turn to the more tricky case p ∈ ∂Σ when (13) does not hold, and B may have
no positive left eigenvector (although there is always a non-negative left eigenvector). In
this case there is at least one j ∈ J such that Aj p − bj ≥ λ0 .
Lemma 2.3 Assume that p ∈ ∂Σ is a fixed point with B in (4) given by (7)–(9). Let
λ0 > 0 be the largest real eigenvalue of B11 and let β ∈ CJ0 be a right eigenvector of B
corresponding to λ0 . If Aj p − bj ≥ λ0 for some j ∈ J then (1) has a trajectory γ ⊂ Σ0J\{j}
tangent to L0 given by (6) at p.
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Proof. We consider separately the two cases (a) Aj p − bj > λ0 > 0 and (b) Aj p − bj =
λ0 > 0.
Since CJ and CJ\{j} are flow-invariant we may assume that J = {j}, and by reordering
populations we may assume that j = N .
(a) Assume AN p − bN > λ0 . Then AN p − bN is the largest real eigenvalue of B. Let
v ∈ C be a right eigenvector of B corresponding to AN p − bN and denote by ṽ ∈ RN −1
the vector composed by the first N − 1 components of v. Then, from Bv = (AN p − bN )v
we have
B11 ṽ + (a1N . . . a(N −1)N )T vN = (AN p − bN )ṽ
so
[(AN p − bN )I − B11 ]ṽ = (a1N . . . a(N −1)N )T vN .
(15)
Since every eigenvalue λ of B11 satisfies Reλ < λ0 if λ 6= λ0 , every eigenvalue of (AN p −
bN )I − B11 has a positive real part. Since the matrix (AN p − bN )I − B11 has nonpositive
off-diagonal entries, by [1, Theorem 3] it is an M -matrix so every entry of its inverse
matrix is nonnegative. Hence, if we take vN = 1 then it follows from (15) that
−1
ṽ = [(AN p − bN )I − B11 ]−1 (a1N . . . a(N −1)N )T ∈ intRN
+
so v ∈ C 0 . Since Σ is balanced, the line along v at p must be transverse to Σ.
Now the spectrum Λ(B) of B has a partition Λ(B) = Λ0 ∪Λ1 with Λ0 = {λ0 , AN p−bN }
and Λ1 = Λ(B) \ Λ0 . Corresponding to Λ0 , (4) has a two-dimensional invariant subspace
L00 = {k1 β + k2 v : k1 , k2 ∈ R}
and (1) has a two-dimensional local invariant manifold Γ00 ⊂ C tangent to L00 at y =
x − p = 0. Since bN − AN p < −λ0 < 0, the solutions of (4) in L00 are dominated by the
e−λ0 t term as t → +∞, so each trajectory of (4) in L00 , except the line along v, is tangent
to L0 at y = x − p = 0. Similarly, only two trajectories of (1) in Γ00 are tangent to the line
along v at p and the rest are all tangent to L0 at p. Since L0 is transverse to Σ{N } at p
and the line along v at p is transverse to Σ, neither Γ00 ⊂ Σ nor Σ ⊂ Γ00 is true. As Γ00 is
two-dimensional and Σ is (N −1)-dimensional, Γ00 ∩Σ must be a one-dimensional invariant
manifold, and hence there is a neighbourhood of p in Γ00 ∩ Σ which is homeomorphic to
the real line, which by invariance means that Γ00 ∩ Σ is a reparameterisation of part of a
trajectory of (1) containing p and such that the trajectory is tangent to L0 at p.
(b) Suppose AN p − bN = λ0 . Then λ0 is an eigenvalue of B with algebraic multiplicity
2. We first show that the space of right eigenvectors of B corresponding to λ0 is onedimensional. By assumption and Perron-Frobenius theorem, there is (α1 · · · αN −1 )T ∈
−1
intRN
such that (α1 · · · αN −1 )B11 = λ0 (α1 · · · αN −1 ). Let u be a left eigenvector of
+
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B corresponding to λ0 . Then
(u1 · · · uN −1 )B11 = λ0 (u1 , · · · uN −1 ),
(16)
(u1 · · · uN −1 )(a1N · · · a(N −1)N )T + (AN p − bN )uN = λ0 uN .
(17)
Since λ0 is a simple eigenvalue of B11 , (16) implies that (u1 · · · uN −1 ) = δ(α1 · · · αN −1 )
for some δ ∈ R. As AN p − bN = λ0 , from (17) we have
δ(α1 · · · αN −1 )(a1N · · · a(N −1)N )T = 0.
This, together with
(α1 · · · αN −1 )(a1N · · · a(N −1)N )T = α1 a1N + · · · + αN −1 a(N −1)N > 0,
gives δ = 0. Thus, the space of left eigenvectors u = (0 · · · 0 uN )T of B corresponding to
λ0 is one-dimensional and so is the space of right eigenvectors.
As β satisfies −Bβ = −λ0 β, λ0 is a simple eigenvalue of B11 and λ0 is an eigenvalue
of B with algebraic multiplicity 2, there is a generalised eigenvector v satisfying −Bv =
−λ0 v+β. Then, with the partition Λ(B) = Λ1 ∪{λ0 }, (4) has a two-dimensional invariant
subspace L000 = {k1 v + k2 β : k1 , k2 ∈ R}, which is generated by linear combinations of the
two solutions βe−λ0 t and (βt + v)e−λ0 t . Correspondingly, (1) has a two-dimensional local
invariant manifold Γ000 tangent to L000 at y = x − p = 0. Clearly, every trajectory of (4)
in L000 is tangent to L0 given by (6) at y = 0 and so are the trajectories of (1) in Γ000 . By
the same argument as that in part (a), Σ ∩ Γ000 is at least one-dimensional and contains
a trajectory γ ⊂ Σ0 tangent to L0 at p.
Combining lemmas 2.1, 2.2 and 2.3 we obtain
Theorem 2.4 For a fixed point p ∈ Σ0J of (1), where J ( IN , let B11 be the submatrix
of B obtained by deleting the jth row and column of B for all j ∈ J. Then the following
conclusions hold.
(i) B has an eigenvalue λ0 > 0, which is a simple and the largest real eigenvalue of
B11 , and a β ∈ CJ0 such that Bβ = λ0 β.
(ii) If either J = ∅ or Aj p − bj < λ0 for all j ∈ J, then there is an α ∈ C 0 such that
αT B = λ0 αT and Σ has a strongly balanced tangent plane Tp Σ = L1 at p given by
L1 = {x ∈ RN : αT (x − p) = 0}.
(iii) For any j ∈ J, if Aj p − bj ≥ λ0 then there is a trajectory γ ⊂ Σ0J\{j} tangent to the
line along β at p.
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Geometric condition for p to be a global attractor or repellor
We now demonstrate how to use the strongly balanced tangent plane Tp Σ through an
interior or boundary fixed point p ∈ Σ, when it exists, to build a Liapunov function for
(1). When the strongly balanced tangent plane does not exist we show how to construct
a replacement strongly balanced plane.
Let πi be the ith coordinate plane and γi the ith nullcline plane defined by
πi = {x ∈ C : xi = 0},
γi = {x ∈ C : Ai x = bi },
i ∈ IN ,
(18)
i ∈ IN .
(19)
It turns out, as shown in the next lemma that global attraction to (repulsion from)
an interior or boundary fixed point requires that a certain subset of the nullcline planes
must not lie above (below) that fixed point in C.
First recall that a fixed point p of (1) is said to be saturated (see, for example, [4]) if
pi = 0 ⇒ bi − Ai p ≤ 0, i ∈ IN .
(20)
Conversely, we will say that a fixed point q is reverse-saturated if the opposite holds:
pi = 0 ⇒ bi − Ai p ≥ 0, i ∈ IN .
(21)
If a fixed point p ∈ CJ0 is saturated then p lies on or above all nullcline planes γj for
which j ∈ J. Dynamically saturation means that points in a sufficiently small neighbourhood of p are attracted to xj = 0 for j ∈ J.
It is known (for example, Theorem 3.2.5 in [9]) that when a fixed point of (1) is stable,
it is necessarily saturated. We also have
Lemma 3.1 If p is an attractor (repellor) of (1), then p is an isolated saturated (reversesaturated) fixed point in C.
Proof. It is obvious that p is an isolated fixed point since p cannot attract (repel)
any other fixed point. Suppose, for some J ( IN , that p ∈ ΣJ is an attractor and
is below γj for some j ∈ J, so that Aj p < bj . By attractiveness, for x0 ∈ Σ0 close
to p, we have limt→+∞ x(t, x0 ) = p. Thus there is a T > 0 such that, for all t ≥ T ,
|Aj x(t, x0 ) − Aj p| < 21 (bj − Aj p) so
Z
t
(bj − Aj x(s, x0 ))ds
xj (t, x0 ) = xj (T, x0 ) exp
T
1
> xj (T, x0 )e 2 (bj −Aj p)(t−T ) .
This leads to xj (t, x0 ) → +∞ as t → +∞, a contradiction to the boundedness of Σ.
Hence, p must be on or above γj for all j ∈ J, i.e. p is saturated. If p is a repellor, by an
argument similar to the above we can show that p is on or below γj for all j ∈ J.
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Now we turn to sufficient conditions for a fixed point p to be global attractor or repellor.
We first construct a strongly balanced plane that passes through p such that all trajectories move above or below the plane. As mentioned above, if there is a strongly balanced
tangent plane to Σ at p then this is the plane that we choose. Now we demonstrate how
to define the plane when there is no suitable strongly balanced tangent plane.
Assume that p ∈ Σ0J for some J ( IN is an interior or boundary fixed point of (1). As
shown in the proof of lemma 2.2 (see equation (10)) there are αi > 0 for all i ∈ IN \ J
and λ0 > 0 such that
∀j ∈ IN \ J,
X
αi pi aij = λ0 αj .
(22)
X
1
αi pi aij .
λ0 − (Aj p − bj )
(23)
1 X
αi pi aij .
λ0
(24)
i∈IN \J
For j ∈ J such that Aj p − bj < λ0 we define
αj =
i∈IN \J
Let
α̂j =
i∈IN \J
Then we define a normal vector α̃ ∈ C 0 for the required plane via
(
α̃j =
αj if j ∈ IN \ J,
α̂j if j ∈ J and Aj p − bj ≥ λ0 ,
α̃j ∈ [αj , α̂j ] or [α̂j , αj ] if j ∈ J and Aj p − bj < λ0 .
(25)
(26)
The required plane is then defined by
Tp0 = {x ∈ C : α̃T (x − p) = 0}.
(27)
Note that we are now slightly abusing our previous terminology by using ‘plane’ to refer
to the bounded (N − 1)-dimensional closed convex set formed by the intersection with
C of the plane passing through p with normal α̃.
We now show how to construct the split Liapunov function for interior and boundary
fixed points.
Theorem 3.2 Assume that p ∈ Σ0J for some J ( IN is an isolated saturated fixed point
of (1). If there are αi > 0 for i ∈ IN \ J and λ0 > 0 satisfying (22) and Σ \ {p} is below
Tp0 then p is a global attractor relative to C.
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Proof. Consider the function
xαi i pi ,
Y
V (x) =
x ∈ C,
(28)
i∈IN \J
where the αi > 0 satisfy (22). Then
V̇ (x)|(1) = V (x)
X
αi pi (bi − Ai x)
i∈IN \J
= V (x)
X
αi pi Ai (p − x)
i∈IN \J
X
X
= V (x)
αi pi ai1 · · ·
αi pi aiN (p − x).
i∈IN \J
i∈IN \J
From (22) and (25) we obtain
X
∀j ∈ IN \ J,
αi pi aij = λ0 αj = λ0 α̃j .
(29)
i∈IN \J
For j ∈ J with Aj p − bj ≥ λ0 , it follows from (24) and (25) that
X
αi pi aij = λ0 α̂j = λ0 α̃j .
(30)
i∈IN \J
For j ∈ J with Aj p − bj < λ0 , we have Aj p − bj ≥ 0 since p is saturated. Thus, from
(23), (24) and (26), we have α̃j ∈ [α̂j , αj ] so
X
αi pi aij = λ0 α̂j ≤ λ0 α̃j .
(31)
i∈IN \J
Then substitution of (29)–(31) into V̇ (x)|(1) results in
V̇ (x)|(1)
X
X
= V (x)
λ0 αj (pj − xj ) +
λ0 α̂j (pj − xj )
j∈IN \J
j∈J
X
X
λ0 α̂j (−xj )
= V (x)
λ0 αj (pj − xj ) +
j∈IN \J
j∈J
≥ V (x)λ0 α̃T (p − x).
Hence, for x below Tp0 , V (x) > 0 implies V̇ (x)|(1) > 0. For any x0 ∈ Σ \ {p} with x0i > 0
for i ∈ IN \ J and x0j ≥ 0 for j ∈ J, we have x(t, x0 ) ∈ Σ \ {p} and V (x(t, x0 )) > 0 for
all t ∈ R. Since Σ \ {p} is below Tp0 by assumption, V (x(t, x0 )) is strictly increasing as t
increases. This shows that x(t, x0 ) is not a periodic solution. By the boundedness of Σ,
there is a c > 0 such that limt→+∞ V (x(t, x0 )) = c. Thus, V (q) = c for all q ∈ ω(x0 ).
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If ω(x0 ) 6= {p}, then there is a q 0 ∈ ω(x0 ) \ {p} ⊂ Σ \ {p} so V̇ (x(t, q 0 ))|(1) > 0 and
V (x(t, q 0 )) > V (q 0 ) = c for t > 0, a contradiction to V (q) = c for all q ∈ ω(x0 ).
Therefore, we must have limt→+∞ x(t, x0 ) = p for all x0 ∈ Σ \ {p} provided x0i > 0 for
all i ∈ IN \ J. This shows that p is a global attractor relative to Σ.
Next, we show that p is also a global attractor relative to C. For any x0 ∈ C \ Σ with
x0i > 0 for i ∈ IN \ J and x0j ≥ 0 for j ∈ J, Theorem 1.1 ensures that ω(x0 ) ⊂ Σ. If
x(t, x0 ) stays on or above Tp0 for all t ≥ 0, then we must have ω(x0 ) = {p}. If there is
a T ≥ 0 such that x(t, x0 ) is below Tp0 for all t ≥ T , then the same reasoning used in
the previous paragraph leads to limt→+∞ x(t, x0 ) = p. Otherwise, there are increasing
sequences {tn }, {sn } ⊂ R+ such that for all integer n ≥ 1, tn < sn ≤ tn+1 , x(tn , x0 ) ∈
Tp0 , x(sn , x0 ) ∈ Tp0 , x(t, x0 ) is below Tp0 for t ∈ (tn , sn ), and x(t, x0 ) is on or above
Tp0 for t ∈ [sn , tn+1 ]. Since p is the only point of Σ on or above Tp0 , restricting t to
0
0
∪∞
n=1 [sn , tn+1 ] we must have limt→+∞ x(t, x ) = p. Thus, limn→∞ V (x(tn , x )) = V (p) =
limn→∞ V (x(sn , x0 )). Note that V (x(t, x0 )) is increasing for t ∈ (tn , sn ) since x(t, x0 ) is
0
below Tp0 . Hence, restricting t to ∪∞
n=1 [tn , sn ], we have limt→+∞ V (x(t, x )) = V (p) so
limt→+∞ x(t, x0 ) = p. Therefore, p is also a global attractor in C.
If we know that every eigenvalue of B in (4) has a positive real part, namely, −B is
stable, then (1) at p is stable in RN thus also stable relative to C. Then, from Theorem
3.2 we have the following corollary.
Corollary 3.3 Under the assumptions of Theorem 3.2, if the matrix −B is stable then
(1) at p is globally asymptotically stable relative to C.
Similarly we have
Theorem 3.4 Assume that p ∈ Σ0J for some J ( IN is an isolated reverse-saturated
fixed point of (1). If there are αi > 0 for i ∈ IN \ J and λ0 > 0 satisfying (22) and Σ \ {p}
is above Tp0 then p is a global repellor.
Proof. The proof of Theorem 3.2 is still valid up to (29). Since p is on or below γj for
all j ∈ J, we have Aj p − bj ≤ 0 for all j ∈ J. Thus, by (23)–(26), we have
∀j ∈ J, α̃j ∈ [αj , α̂j ],
X
αi pi aij = λ0 α̂j ≥ λ0 α̃j .
(32)
i∈IN \J
Then, from (29) and (32), we obtain
V̇ (x)|(1) ≤ V (x)λ0 α̃T (p − x).
Hence, for x above Tp0 , V (x) > 0 implies V̇ (x)|(1) < 0. By an argument similar to that
used in the proof of Theorem 3.2, we obtain limt→−∞ x(t, x0 ) = p for any x0 ∈ Σ \ {p}
as long as x0i > 0 for all i ∈ IN \ J. This shows that p is a global repellor.
Remark 1 The examples in section 5 illustrate the flexibility in choice of α̃ afforded in
equation (26). Indeed, some examples show that in some cases choosing α̃j = min{αj , α̂j }
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may not lead to conclusive results from Theorem 3.2 and Theorem 4.1 (see later), whereas
choosing α̃j in the interior of the interval does lead to a conclusive result.
3.1.
Σ is concave or convex
Recall that B(O) is the basin of repulsion of the origin in RN
+ . We call Σ convex (concave)
(see [14]) if for any distinct u, v ∈ Σ, su + (1 − s)v ∈ B(O) (B(∞)) holds for all s ∈ (0, 1).
In [14] the authors show that when Σ is convex (concave) and contains an interior fixed
point p, then Σ \ {p} lies below (above) Tp Σ and p is globally attracting (repelling). We
now demonstrate how to extend this result to cover also the case when p is a boundary
fixed point. As might be expected, when there is a strongly balanced tangent plane Tp Σ
to Σ at p, Σ \ {p} lies below (above) Tp Σ when Σ is convex (concave).
Corollary 3.5 Assume that p ∈ Σ0J for some J ( IN is an isolated saturated fixed
point of (1). If Σ is convex then p is a global attractor relative to C.
Proof. By Theorem 2.4 (i), B11 has a simple largest real eigenvalue λ0 > 0, λ0 is an
eigenvalue of B and there is a β ∈ CJ0 such that Bβ = λ0 β. Since Σ is balanced and
convex and Σ0 is strongly balanced (and so is the restriction of Σ0 to every subsystem),
there is no trajectory on Σ tangent at p to the line L0 defined by (6). By Theorem 2.4
(iii), we have either J = ∅ or Aj p − bj < λ0 for all j ∈ J. So, by Theorem 2.4 (ii), there
is an α ∈ C 0 such that αT B = λ0 αT and Σ has a tangent plane Tp Σ = L1 at p given by
(5). Moreover, the convexity of Σ implies that Σ \ {p} is below Tp Σ. Note from the proof
of Theorem 2.4 that α is determined by (22) and (23). By saturation Aj p − bj ≥ 0 for all
j ∈ J, by (22)–(27) we have α̃j ≤ αj for all j ∈ J and α̃j = αj for all j ∈ IN \ J. Thus,
∀x ∈ Tp Σ,
α̃T (x − p) =
X
α̃j xj +
j∈J
≤
X
j∈J
X
αj (xj − pj )
j∈IN \J
αj xj +
X
αj (xj − pj )
j∈IN \J
= αT (x − p) = 0
so each point of Tp Σ is on or below Tp0 . Therefore, Σ \ {p} is below Tp0 . By Theorem 3.2,
p is a global attractor.
Corollary 3.6 Assume that p ∈ Σ0J for some J ( IN is an isolated reverse-saturated
fixed point of (1). If Σ is concave then p is a global repellor.
Proof. Since Aj p − bj ≤ 0 for all j ∈ J, by Theorem 2.4 Σ has a strongly balanced
tangent plane Tp Σ at p. As α̃j = αj for all j ∈ IN \ J and α̃j ≥ αj for all j ∈ J, for each
x ∈ Tp Σ, we have α̃T (x − p) ≥ αT (x − p) = 0 so x is on or above Tp0 . As the concavity
of Σ implies that Σ \ {p} is above Tp Σ, Σ \ {p} is also above Tp0 . By Theorem 3.4, p is a
global repellor.
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Remark 2 When p ∈ Σ0 = Σ0J with J = ∅, Tp0 defined by (27) coincides with Tp Σ. Thus,
the combination of Theorems 3.2 and 3.4 is consistent with [14, Theorem 4.3] and the
combination of Corollaries 3.5 and 3.6 is consistent with [14, Corollary 4.5].
Remark 3 When p ∈ Σ0J for some J ( IN , we have seen from (22)–(27) that there is
always a plane Tp0 through p no matter whether p ∈ Σ0 or p ∈ ∂Σ and no matter whether
Tp Σ exists or not. Moreover, when J 6= ∅ and λ0 > Aj p − bj 6= 0 for some j ∈ J, there is
a freedom of choice for α̃j . This will be demonstrated by Example 3 in the last section.
4.
Algebraic conditions for p to be a global attractor or repellor
From Theorems 3.2 and 3.4 we see that a fixed point p ∈ Σ is a global attractor (repellor)
if Σ \ {p} is below (above) Tp0 . Since this condition cannot be checked directly due to the
unknown Σ \ {p}, we need to find checkable conditions to ensure this. For Tp0 defined by
(27), we extend it from C to RN by letting
T̃p = {x ∈ RN : α̃T (x − p) = 0}.
(33)
We say that the quadratic form (x − p)T D(α̃)A(x − p) is positive (negative) definite on
T̃p if
∀x ∈ T̃p \ {p},
(x − p)T D(α̃)A(x − p) > 0 (< 0).
(34)
Theorem 4.1 Assume that p ∈ Σ0J for some J ( IN is an isolated fixed point of (1)
and p is saturated (reverse-saturated). If α̃ is the positive vector constructed in Theorem
3.2 and (x − p)T D(α̃)A(x − p) is positive (negative) definite on T̃p given by (33) then p
is a global attractor (repellor). Moreover, p is globally asymptotically stable relative to C
if p is a global attractor.
Proof. By Theorems 3.2 and 3.4, we need only show that Σ \ {p} is below (above) Tp0 .
Let W be an N × N matrix such that the columns of W , Wcj (1 ≤ j ≤ N ), satisfy
Wc1 = (α̃1−1 0 · · · 0)T ,
T̃p = span{Wc2 , . . . , WcN }.
(35)
Then α̃T W = (1 0 · · · 0) so Wc2 , . . . , WcN are linearly independent and W is invertible.
Let x − p = W z. Then α̃T (x − p) = α̃T W z = z1 and
z10 |(1)
T 0
= α̃ x |(1) =
N
X
α̃i xi (bi − Ai x)
i=1
= −α̃T B(x − p) − (x − p)T D(α̃)A(x − p).
(This last expression is just the Taylor expansion of α̃T x0 |(1) about x = p.) When p is
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an interior fixed point, α̃ = α is a positive left eigenvector of B and so the linear term
α̃T B(x − p) = λ0 αT (x − p) = 0 on z1 = 0 and z10 is one-signed by virtue of (34). When p
belongs to the boundary of Σ, we use the definition of α̃ as follows. For j ∈ IN \ J, from
(22) and (25) we have
N
X
X
α̃i bij =
i=1
αi pi aij = λ0 αj = λ0 α̃j .
(36)
i∈IN \J
For j ∈ J with Aj p − bj ≥ λ0 , (24) and (25) result in
N
X
X
α̃i bij =
i=1
αi pi aij + α̂j (Aj p − bj ) = λ0 α̃j + εj ,
(37)
i∈IN \J
where εj = α̂j (Aj p − bj ) ≥ λ0 α̂j > 0. (Note that this case does not occur if p is on or
below γj .)
For j ∈ J with Aj p − bj < λ0 , as a consequence of (23), (24) and (26) we have
N
X
i=1
α̃i bij =
X
αi pi aij + α̃j (Aj p − bj )
i∈IN \J
= [λ0 − (Aj p − bj )]αj + α̃j [(Aj p − bj ) − λ0 ] + λ0 α̃j
= λ0 α̃j + εj ,
(38)
where εj = [λ0 − (Aj p − bj )](αj − α̃j ) ≥ 0 (= 0 or ≤ 0) if p is above (on or below) γj .
Substituting (36)–(38) into z10 |(1) and letting
Q(z) = z T W T D(α̃)AW z = z1 Θ(z) + Q1 (z2 , . . . , zN ),
(39)
where Θ is a linear function of z with Θ(0) = 0, we obtain, setting z̃ = (z2 , . . . , zN )T ,
z10 |(1) = −λ0 z1 −
X
εj xj (z) − Q(z)
j∈J
= −(λ0 + Θ(z))z1 −
X
εj xj (z) − Q1 (z̃).
(40)
j∈J
As x = p and T̃p are transformed to z = 0 and π10 = {z ∈ RN : z1 = 0} respectively, the
assumption (34) becomes
∀z ∈ π10 \ {0}, Q1 (z̃) > 0 (< 0).
(41)
Thus, Q1 is a positive (negative) definite quadratic form of z̃ = (z2 , . . . , zN )T .
Let C 0 = {W −1 (x−p) : x ∈ C} and π̃1 = {W −1 (x−p) : x ∈ Tp0 }. Thus, Tp0 = T̃p ∩C and
π̃1 = π10 ∩ C 0 . Note that xj (z) ≥ 0 for all j ∈ J and z ∈ C 0 . Then, from the assumption
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and the positive (negative) definiteness of Q1 , we obtain
∀z ∈ C 0 \ {0},
X
εj xj (z) + Q1 (z̃) > 0 (< 0).
(42)
j∈J
It then follows from this and (40) that
∀x ∈ Tp0 \ {p},
(α̃T x)0 |(1) < 0 (> 0).
By continuity, there is an open set S0 of C with Tp0 \ {p} ⊂ S0 such that
∀x ∈ S0 ,
(α̃T x)0 |(1) < 0 (> 0).
(43)
This shows that Tp0 \ {p} ⊂ C \ B(O) (C \ B(∞)). Thus, the set
Σ \ {p} = (B(O) \ B(O)) \ {p} = (B(∞) \ B(∞)) \ {p}
is below (above) Tp0 . By Theorem 3.2 (3.4), p is a global attractor (repellor).
Next we show that p is stable in C so that it is globally asymptotically stable. For any
r > 0, let Br (p) denote the open ball in RN with centre p and radius r > 0. Recall that
p ∈ C is stable in C if for every ε > 0 there is a ρ > 0 such that x0 ∈ C ∩ Bρ (p) implies
that x(t, x0 ) ∈ C ∩ Bε (p) for all t ≥ 0.
From (40) and (42), there is an r1 > 0 such that z ∈ Br1 (0) implies |Θ(z)| < 12 λ0 so
that
∀z ∈ Br1 (0) with z1 > 0, (α̃T x)0 |(1) = z10 |(1) < 0.
This, together with (43), shows the existence of δ1 > α̃T p satisfying, with L(α̃, δ) = {x ∈
C : α̃T x = δ},
∀δ ∈ [α̃T p, δ1 ], ∀x ∈ L(α̃, δ) \ {p}, (α̃T x)0 |(1) < 0.
(44)
For the function V (x) defined by (28) and for each ` > 0, S` = {x ∈ C : V (x) = `}
defines an (N − 1)-dimensional surface in C parallel to the xj -axis for all j ∈ J. For the
Q
particular value `∗ = i∈IN \J pαi i pi , from (28) we see that x ∈ S`∗ if and only if
X
αi pi (log(xi ) − log(pi )) = 0.
i∈IN \J
Now − log is a strictly convex function on R+ and hence for xi 6= pi and i ∈ IN \ J,
(− log(xi )) − (− log(pi )) >
1
−
pi
(xi − pi )
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which upon rearrangement gives
(xi − pi ) > pi (log(xi ) − log(pi )).
Thus, since pi = 0 for i ∈ J, we obtain
α̃T (x − p) ≥
X
αi (xi − pi )
i∈IN \J
>
X
αi pi (log(xi ) − log(pi )) = 0,
i∈IN \J
so x is above Tp0 if x ∈ S`∗ \ {p}. Thus, S`∗ ∩ Tp0 = {p}. Moreover, for each ` ∈ (0, `∗ ), the
set
V` = {x ∈ C : V (x) ≥ `, α̃T (p − x) ≥ 0}
is bounded below by S` and above by Tp0 and has an (N − 2)-dimensional edge S` ∩ Tp0 .
For a fixed ε > 0, there is an ` ∈ (0, `∗ ) such that V` ⊂ Bε (p). Then, from (44), there
is a δ0 ∈ (α̃T p, δ1 ) such that
∀x0 ∈ S` ∩ Tp0 , ∃t1 (x0 ) < 0 satisfying
x(t1 , x0 ) ∈ L(α̃, δ0 ) and ∀t ∈ [t1 , 0], x(t, x0 ) ∈ Bε (p).
Let
S1 = {x(t, x0 ) : x0 ∈ S` ∩ Tp0 , t ∈ [t1 (x0 ), 0]}.
Then S1 is an (N − 1)-dimensional surface consisting of the trajectory segments of (1)
connecting S` to L(α̃, δ0 ) within Bε (p). Hence, the region bounded by S` , S1 and L(α̃, δ0 )
is positively invariant and a subset of C ∩ Bε (p) and has p as an interior point in C.
Therefore, we can choose a small ρ > 0 so that C ∩ Bρ (p) is contained in this region.
Then x0 ∈ C ∩ Bρ (p) implies x(t, x0 ) ∈ C ∩ Bε (p) for all t ≥ 0 so p is stable.
Remark 1 Notice that the definiteness condition (34) is sufficient but not necessary for
global stability of a fixed point of (1). This will be demonstrated by examples in section
5.
In general, if S is an open connected set of RN and p ∈ S is a fixed point of x0 = f (x)
in S, then stability of the matrix Df (p) implies stability of x0 = f (x) at p and existence
of an eigenvalue of Df (p) with a positive real part implies instability of the system at
p. Thus, if p is a global attractor of the system in S and Df (p) is stable then x0 = f (x)
at p is globally asymptotically stable in S. Conversely, if x0 = f (x) at p is globally
asymptotically stable, Df (p) may not necessarily be stable but all of its eigenvalues have
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non-positive real parts. However, for some particular systems under certain conditions,
global asymptotic stability of the system at p does imply stability of Df (p). We shall see
that, under the assumptions of Theorem 4.1, system (1) with p ∈ C 0 is such an example.
Lemma 4.2 Assume that p ∈ Σ0J for some J ( IN is an isolated fixed point of (1), p is
saturated (reverse-saturated), and (x − p)T D(α̃)A(x − p) is positive (negative) definite
on T̃p . Let B11 be the submatrix of B in (4) and let λ0 > 0 be the largest real eigenvalue
of B11 as described in Theorem 2.4. Then every eigenvalue of B11 other than λ0 has a
positive (negative) real part and every eigenvalue of B has a non-negative (non-positive)
real part.
Proof. We follow the notation used earlier in section 2. Thus we suppose that the
populations are labeled such that B takes the block form in (7)–(9) so that B22 is a
diagonal matrix with N −k (1 ≤ k ≤ N ) entries Ak+1 p−bk+1 , . . . , AN p−bN . By saturation
these entries are all non-negative (non-positive), so that B has N − k non-negative (nonpositive) eigenvalues. The remaining k eigenvalues are those of the k × k positive matrix
B11 . As in section 2 there is a positive left eigenvector α of B11 corresponding to λ0 > 0,
and λ0 is also an eigenvalue of B. Now α̃ is the k-dimensional α extended to an N dimensional positive vector. Let β be a right eigenvector of B11 associated with any
eigenvalue µ 6= λ0 of B11 . Then α and β are orthogonal. Now extend β to β̃ by adding
N − k zeroes to β. Then α̃T β̃ = 0, so that β̃ lies in T̃p . Since β̃, µ may be complex, we
write β = φ + iψ, β̃ = φ̃ + iψ̃ and µ = σ + iω. (If µ is real then ω = 0 and ψ = 0.) The
eigenvector equation for µ, β then yields
B11 φ = σφ − ωψ,
B11 ψ = σψ + ωφ.
Thus we have
D(α)A11 φ = σD(α/p)φ − ωD(α/p)ψ,
D(α)A11 ψ = σD(α/p)ψ + ωD(α/p)φ.
where D(α/p) is the k × k diagonal matrix with entries αi /pi , i = 1, . . . , k. Now note
that φ̃, ψ̃ ∈ T̃p and hence, by (34), φ̃T D(α̃)Aφ̃ ≥ 0 (≤ 0) and ψ̃ T D(α̃)Aψ̃ ≥ 0 (≤ 0) with
strict inequality holding in at least one of these. Thus we find
φ̃T D(α̃)Aφ̃ = φT D(α)A11 φ = σφT D(α/p)φ − ωφT D(α/p)ψ ≥ 0 (≤ 0),
ψ̃ T D(α̃)Aψ̃ = ψ T D(α)A11 ψ = σψ T D(α/p)ψ + ωψ T D(α/p)φ ≥ 0 (≤ 0).
Adding these last two equations yields σ(φT D(α/p)φ + ψ T D(α/p)ψ) ≥ 0 (≤ 0) with
equality if and only if φ = ψ = 0. Since φ + iψ is an eigenvector, at least one of φ, ψ
must be non-zero and so σ > 0 (< 0). Hence if µ 6= λ0 is an eigenvalue of B11 then its
real part is positive (negative).
Then the result below immediately follows from Lemma 4.2.
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Corollary 4.3 If p ∈ C 0 and the global asymptotic stability of (1) at p relative to C
is achieved by the application of Theorem 4.1, then the matrix −B in (4) is stable.
Open Problem 1 Assume that p ∈ Σ0J for some J ( IN is an isolated fixed point of
(1) and p is saturated (reverse-saturated). If every eigenvalue of B other than λ0 has a
positive (negative) real part, is (x − p)T D(α̃)A(x − p) positive (negative) definite on T̃p ?
If so, then we obtain a stronger but convenient sufficient condition for global asymptotic
stability.
Remark 2 When p is a unique fixed point of (1) in C 0 , Theorem 4.1 without the stability
part coincides with [14, Theorem 5.3]. While the method of induction on N is employed
in the proof of Theorem 5.3, a different approach has been used here for the proof of
Theorem 4.1, though the same technique of transformation of [14, Lemma 5.2] has been
adopted. As an attracting fixed point does not have to be stable in general, the method
of our proof has an added value that the algebraic condition for global attraction also
ensures global asymptotic stability.
From the proof of Theorem 4.1 we see that (x − p)T D(α̃)A(x − p) is positive (negative)
definite on T̃p if and only if Q1 (z̃) defined by (39) is a positive (negative) definite quadratic
form. Following [14] we define
1
M s = (M + M T )
2
for any square matrix M . Then
Q(z) = z T W T D(α̃)AW z
T
= (z1 Wc1
+ z̃ T [Wc2 , . . . , WcN ]T )D(α̃)A(z1 Wc1 + [Wc2 , . . . , WcN ]z̃)
T
= z1 Wc1
(D(α̃)Az + AT D(α̃)[Wc2 , . . . , WcN ]z̃)
+z̃ T [Wc2 , . . . , WcN ]T (D(α̃)A)S [Wc2 , . . . , WcN ]z̃.
From (39) we obtain
Q1 (z2 , . . . , zN ) = z̃ T [Wc2 , . . . , WcN ]T (D(α̃)A)S [Wc2 , . . . , WcN ]z̃.
Thus, Q1 is positive (negative) definite if and only if the (N − 1) × (N − 1) symmetric
matrix
U = [Wc2 , . . . , WcN ]T (D(α̃)A)S [Wc2 , . . . , WcN ]
(45)
is positive (negative) definite.
Since the definiteness of U is independent of the choice of basis {Wc2 , . . . , WcN }, we
are at liberty to choose the basis so that U has a simple explicit expression in terms of
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D(α̃) and A. For example, if we choose

1




−1  −1
[Wc2 , . . . , WcN ] = D(α̃)

0
..
.
..
.
0




1 
−1
(46)
and denote by Mij the submatrix of M obtained by deleting the ith row and the jth
column of M for any square matrix M , then
U = (AD(α̃)−1 )S11 + (AD(α̃)−1 )SN N − (AD(α̃)−1 )S1N − (AD(α̃)−1 )SN 1 .
(47)
Alternatively, if we choose

1
..



 0
−1
−1 
[Wc2 , . . . , WcN ] = D(α̃)

0
.
···




1 
−1
(48)
and write
(AD(α̃)−1 )S = (dij )N ×N ,
(49)
U = (dij + dN N − diN − dN j )(N −1)×(N −1) .
(50)
then
Note that U given by (50) is obtained from (AD(α̃)−1 )S as follows: subtract the N th
column from each column of (AD(α̃)−1 )S to obtain U 0 , subtract the N th row from each
row of U 0 to obtain U 1 , and then delete the N th row and the N th column of U 1 to
obtain U . In general, if we choose

1
0
0 ···

..

.

 0
1 0 ···

−1 
[Wc2 , . . . , WcN ] = D(α̃)  −1 · · · −1 −1 · · ·

 0 ··· 0 1


..

.
0 ··· 0 0
0




0 


−1  ,

0 



(51)
1
where every entry on the i0 th row is −1, we have
U = (dij + di0 i0 − dii0 − di0 j )i,j∈IN \{i0 } .
(52)
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Summarising the above, we have proved the following theorem.
Theorem 4.4 Assume that p 6= 0 is a saturated (reverse-saturated) fixed point of (1). If
U given by (47) or (50) or (52) or (45) is positive (negative) definite, then p is a global
attractor (repellor). Moreover, the global attraction of p implies the global asymptotic
stability of p relative to C.
Remark 3 When p ∈ Σ0 , Theorem 4.4 without the stability part with U given by (47) is
consistent with [14, Theorem 6.7]. However, some other forms of U such as (50) or (52)
might be more convenient for some concrete examples. This will be demonstrated in the
next section.
5.
Some examples and further remarks
In this section, we utilise results on fixed point global attractors (repellors) from [6–8]
and present a few examples as an application of Theorem 4.4.
Example 1 Consider system (1) with

ε
1

A=
1
δ
1
ε
1
δ
1
1
ε
δ

1
1

,
1
ρ
 
1
1
 
b =  .
1
1
(53)
It is shown in [6] that if
0 < ε < 1,
0 < ρ < 1,
δ ≥ 1,
(54)
then (1) with (53) has an interior fixed point x∗ given by
x∗1 = x∗2 = x∗3 =
3δ − (2 + ε)
1−ρ
, x∗4 =
3δ − ρ(2 + ε)
3δ − ρ(2 + ε)
(55)
and x∗ is a global repellor. Obviously, it is not a trivial task to find the positive eigenvalue
λ0 and a corresponding eigenvector α ∈ intR4+ of D(x∗ )A for this example in particular,
and for N -dimensional systems (1) in general when N is large. The advantage of the
method given in [6] is that it does not require the computation of the interior fixed point
x∗ , the eigenvalue λ0 > 0 and a corresponding left eigenvector α ∈ C 0 of D(x∗ )A. The
disadvantage is that it cannot be applied directly to a boundary fixed point p ∈ ∂Σ.
We shall see that Theorem 4.4 can be applied to (1) with (53) for some boundary fixed
points.
Conclusion 5.1 The boundary fixed point p = (2 + ε)−1 (1, 1, 1, 0)T is a global repellor if
0 < ε < 1,
2
δ< ,
3
ρ<
ε+2
.
2(2 + ε) − 3δ
(56)
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Proof. We have

ε
1 
1

B(p) =

2 + ε 1
0

1
1

1
1

,

ε
1
0 3δ − 2 − ε
1
ε
1
0
ε−1 3δ−ε−2
which has spectrum {1, ε−1
2+ε , 2+ε , 2+ε }. The eigenvalue λ = 1 has a positive left eigen3
3δ
vector (1, 1, 1, 2(2+ε)−3δ
). Moreover, A4 p − b4 = 2+ε
− 1 < 0 so p lies below γ4 . The
matrix


ε 1 1
1 

B11 =
1 ε 1
2+ε
1 1 ε
has an eigenvalue λ0 = 1 with a left eigenvector (α1 , α2 , α3 ) = (1, 1, 1). We have λ0 −
(A4 p − b4 ) = 1 − 3δ/(2 + ε) + 1 > 0, so that by (23), (24) and (26)
3
X
1
3
α4 =
> 0,
αi pi ai4 =
λ0 − (A4 p − b4 )
2(2 + ε) − 3δ
i=1
α̂4 =
3
1 X
3
αi pi ai4 =
.
λ0
2+ε
i=1
3
Under the conditions (56), α4 ≤ α̂4 holds. Let us choose α̃ = (1, 1, 1, 2(2+ε)−3δ
)T . Then



(AD(α̃)−1 )S = 

2+ε
3
2+ε
3
2+ε
3
2+ε 2+ε 2+ε ρ(2ε−3δ+4)
3
3
3
3
ε
1
1
1
ε
1
1
1
ε



.

(57)
Using (50) we obtain


U =
ε+ρ(2(2+ε)−3δ)−4 ρ(2(2+ε)−3δ)−2ε−1 ρ(2(2+ε)−3δ)−2ε−1
3
3
3
ρ(2(2+ε)−3δ)−2ε−1 ε+ρ(2(2+ε)−3δ)−4 ρ(2(2+ε)−3δ)−2ε−1
3
3
3
ρ(2(2+ε)−3δ)−2ε−1 ρ(2(2+ε)−3δ)−2ε−1 ε+ρ(2(2+ε)−3δ)−4
3
3
3
Under the inequalities (56), the leading principal minors of −U are
1
2
`1 = [4 − ρ(4 + 2ε − 3δ) − ε] > (1 − ε) > 0,
3
3
1
1
`2 = (ε − 1)(2ρ(4 + 2ε − 3δ) − ε − 5) > (1 − ε) > 0,
3
3
`3 = det(−U ) = −(1 − ε)2 ((4 + 2ε) − 3δ)ρ − ε − 2) > 0.


.
(58)
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So U is negative definite and p is a global repellor.
Conclusion 5.2 The boundary fixed point q = (0, 0, 0, ρ−1 )T is a global repellor of (1)
with (53) if
0 < ε < 1,
(2ρ − 1)(2 + ε)
< δ.
3ρ
1 ≤ ρ,
(59)
Proof. As ρ ≥ 1, the boundary fixed point q is on or below γi for all i ∈ I3 . The
linearised system at q is (4) with

A1 q − b 1
0
0

0
A2 q − b2
0

B=

0
0
A3 q − b 3
δ/ρ
δ/ρ
δ/ρ
  −1

0
ρ −1
0
0
0

0 
0
ρ−1 − 1
0
0
 

=
.
0  
0
0
ρ−1 − 1 0 
ρ/ρ
δ/ρ
δ/ρ
δ/ρ 1
Then B has a spectrum {1, ρ1 − 1, ρ1 − 1, ρ1 − 1} so λ0 = 1. Also λ0 − (Ai q − bi ) =
2 − 1/ρ > 0 for i = 1, 2, 3. Hence, again there is some flexibility in choosing α̃. We choose
T
α̃ = (1, 1, 1, 2ρ−1
δ ) , which is a positive left eigenvector associated with λ0 = 1. Then



(AD(α̃)−1 )S = 

δρ
2ρ−1
δρ
2ρ−1
δρ
2ρ−1
δρ
δρ
δρ
δρ
2ρ−1 2ρ−1 2ρ−1 2ρ−1

2ρ−δρ−1
2ρ−δρ−1
2ρ−1
2ρ−1
ε(2ρ−1)−δρ 2ρ−δρ−1
2ρ−1
2ρ−1
2ρ−δρ−1 ε(2ρ−1)−δρ
2ρ−1
2ρ−1

ε
1
1
1
ε
1
1
1
ε


.

(60)
By (50) or the note following it we have
 ε(2ρ−1)−δρ

U =
2ρ−1
2ρ−δρ−1
2ρ−1
2ρ−δρ−1
2ρ−1

.
(61)
The leading principal minors of −U are
`1 =
δρ − ε(2ρ − 1)
,
2ρ − 1
`2 =
(1 − ε)
[2δρ − (2ρ − 1)(1 + ε)],
2ρ − 1
`3 = det(−U ) =
(1 − ε)2
[3δρ − (ε + 2)(1ρ − 1)].
2ρ − 1
It can be checked that condition (59) ensures the positive definiteness of −U . By Theorem 4.4, q is a global repellor.
Note that we have a choice of α̃i ∈ [1, 2ρ−1
ρ ] for i ∈ I3 . If we do not take α̃i = 1 then we
cannot prove the conclusion. Indeed, when 0 < ε < 1 and ρ ≥ 1, among all the choices
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of α̃i , (2ρ − 1)(2 + ε)/(3ρ) is the smallest possible lower bound of δ in order for q to be
a global repellor.
Conclusion 5.3 System (1) with (53) is globally asymptotically stable at the boundary
fixed point p = (2 + ε)−1 (1, 1, 1, 0)T if
ε > 1,
2+ε
ρ(2 + ε)
≤δ<
.
3
3
ρ > 1,
(62)
Proof. From the proof of Conclusion 1 we have λ0 = 1 with (α1 , α2 , α3 ) = (1, 1, 1).
From (62) we have A4 p − b4 = 3δ/(2 + ε) − 1 ≥ 0 so p is on or above γ4 . Note that α̃ in
the proof of Conclusion 1 may not be positive any more. Even if it is positive for some
values of ε and δ, we cannot guarantee the positive definiteness of U given by (58). But
3 T
we can always take α̃4 = α̂4 and α̃ = (1, 1, 1, 2+ε
) . Then (AD(α̃)−1 )S is given by (57)
with the replacements of

ε+

U = 1 +
1+
2+ε
3
by
(2+ε)(ρ−1)
3
(2+ε)(ρ−1)
3
(2+ε)(ρ−1)
3
2+ε+3δ
6
and
−δ 1+
−δ ε+
−δ 1+
ρ(2ε−3δ+4)
3
(2+ε)(ρ−1)
3
(2+ε)(ρ−1)
3
(2+ε)(ρ−1)
3
by
ρ(2+ε)
3
−δ 1+
−δ 1+
−δ ε+
and, by (50), we obtain
(2+ε)(ρ−1)
3
(2+ε)(ρ−1)
3
(2+ε)(ρ−1)
3

−δ

− δ.
−δ
Then the positive definiteness of U follows from (62). By Theorem 4.4, (1) with (53) is
globally asymptotically stable at p.
Conclusion 5.4 System (1) with (53) is globally asymptotically stable at the boundary
fixed point q = (0, 0, 0, ρ−1 )T if
ε > 1,
1
< ρ ≤ 1,
2
0<δ<
(2 + ε)(2ρ − 1)
.
3ρ
(63)
Proof. Since 12 < ρ ≤ 1, q is on or above γi for all i ∈ I3 . From the proof of Conclusion
2 we see that U given by (61) is positive definite under (63). Then the conclusion follows
from Theorem 4.4.
Example 2 Consider system (1) with

2
1

A=
0
2
5
3
1
2
12
25
2
5

1 12
1 45 

,
2 2
2
5 3
 
1
1
 
b =  .
1
1
(64)
5 T
It is shown in [8] that the fixed point p = (0, 110
59 , 0, 59 ) is a global attractor. But we
cannot apply Theorem 4.4 to (1) with (64) as a31 = 0.
Open Problem 2 Further investigation is needed to explore the possibility of extending the method of this paper to (1) with aij ≥ 0, aii > 0 for all i, j ∈ IN .
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Example 3 Consider system (1) with

1 ε δ


A = ε 1 δ,
δ δ 1

 
1
 
b = 1,
1
(65)
where 0 < ε < 1 and 0 < δ < 1.
Conclusion 5.5 If ε = 0.8 and δ = 0.948, then the conditions of Theorem 4.4 are satisfied for (1) with (65) and the boundary fixed point p = (1 + ε)−1 (1, 1, 0)T is globally
asymptotically stable. However, Σ is not convex.
Proof. It is shown in [13, counterexample 6.1] that Σ is not convex if
δ >2−
4
.
3+ε
(66)
As ε = 0.8 and δ = 0.948 satisfy (66), Σ is not convex. That p is above γ3 follows from
A3 p − b3 = 2δ/(1 + ε) − 1 = 0.096/1.8 > 0. Note that the matrix
B11
1
=
1+ε
1ε
ε1
!
has an eigenvalue λ0 = 1 and a left eigenvector (α1 , α2 ) = (1, 1). By (23) and (24),
1
2δ
(α1 p1 a13 + α2 p2 a23 ) =
,
λ0
1+ε
δ
1
(α1 p1 a13 + α2 p2 a23 ) =
.
α3 =
λ0 − (A3 p − b3 )
1+ε−δ
α̂3 =
Then, by (26), we take α̃3 =
δ
ρ
∈ [α̂3 , α3 ] with 1 + ε − δ ≤ ρ ≤ (1 + ε)/2. Thus,


1
ε
(δ + ρ)/2


(AD(α̃)−1 )S = 
ε
1
(δ + ρ)/2 
(δ + ρ)/2 (δ + ρ)/2
ρ/δ
and
U=
1 + ρ/δ − ρ − δ ε + ρ/δ − ρ − δ
ε + ρ/δ − ρ − δ 1 + ρ/δ − ρ − δ
!
.
(67)
For ε = 0.8 and δ = 0.948, we have 1 + ρ(δ −1 − 1) − δ > 0 and, with ρ = (1 + ε)/2 = 0.9,
det(U ) = (1 − ε)(1 + ε + 2ρ(δ −1 − 1) − 2δ) > 0. Therefore, U is positive definite and the
δ
2δ
> 1+ε
,
conclusion follows. Notice also that if we take ρ = 1 + ε − δ, so that α̃3 = 1+ε−δ
i.e. the top end of the interval for α̃3 , we obtain det U = −0.00051 < 0, so not all values
of α̃3 ∈ [α̂3 , α3 ] give conclusive results.
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27
Conclusion 5.6 If (66) holds then Σ is not convex but the boundary fixed point p = (1 +
ε)−1 (1, 1, 0)T is a global attractor of (1) with (65). However, for ε = 0.8 and 0.95 ≤ δ < 1,
the positive definite condition of Theorem 4.4 on U is not met.
Proof. That Σ is not convex follows from the proof of Conclusion 5. Since (66) implies
2δ/(1 + ε) > 2 − δ > 1 so (1 + ε)/2 < δ, we have (A1 + A2 )/2 < A3 . Then, by [8, Theorem
2], there is a δ0 > 0 such that x0 ∈ intR3+ implies x3 (x0 , t) = o(e−δ0 t ) as t → +∞. By [8,
Theorem 4], p is a global attractor. When ε = 0.8, δ = 0.95 and ρ = (1 + ε)/2 = 0.9, we
have
1 + ε + 2ρ(δ −1 − 1) − 2δ = 2(0.9 − δ 2 )/δ = −0.005/δ < 0
so U given by (67) is not positive definite.
6.
Closing comments
We have demonstrated that the Split Liapunov method of [14] can be extended to detect
global attractors and repellors of totally competitive Lotka-Volterra systems. A key step
in our method is to find a strongly balanced tangent or replacement plane through the
examined fixed point on which all trajectories remain or cross in the same direction.
We have clarified how convexity or concavity of the carrying simplex, plus saturation or
reverse-saturation leads to global stability or repulsion.
Regarding the relationship between any two of the following three:
(i) the convexity (concavity) of Σ,
(ii) the global attraction (repulsion) of a fixed point p ∈ Σ,
(iii) the positive (negative) definite condition on U of Theorem 4.4.
We see from Corollaries 3.5 and 3.6 and Theorem 4.4 that both (i) and (iii) are sufficient conditions for (ii). Example 3 demonstrates that neither (i) nor (iii) is a necessary
condition of (ii). Conclusion 5 also shows that (iii) does not necessarily imply (i). An
interesting open question is whether (i) necessarily implies (iii).
Open Problem 3 Sufficient conditions are needed for the carrying simplex Σ to be
convex (concave).
References
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