Differential Equations and Dynamical Systems manuscript No.
(will be inserted by the editor)
Global stability of interior and boundary fixed points for
Lotka-Volterra systems
Stephen Baigent · Zhanyuan Hou
Received: date / Accepted: date
Abstract For permanent and partially permanent, uniformly bounded Lotka-Volterra
systems, we apply the Split Lyapunov function technique developed for competitive Lotka-Volterra systems to find new conditions that an interior or boundary fixed
point of a Lotka-Volterra system with general species-species interactions is globally
asymptotically stable. Unlike previous applications of the Split Lyapunov technique
to competitive Lotka-Volterra systems, our method does not require the existence of
a carrying simplex.
Keywords Lotka-Volterra systems · global attractors · global repellors · global
asymptotic stability
Mathematics Subject Classification (2000) 34D05 · 34D20 · 34C11 · 92D25
1 Introduction
For N ≥ 1 integer let IN = {1, . . . , N} and denote by D(α) = diag[α] the diagonal
matrix with diagonal entries α1 , . . . , αN . We consider the Lotka-Volterra equations
ẋ = D(x)(b − Ax),
(1)
where bi ∈ IR and ai j ∈ IR for i, j ∈ IN . These equations are widely used in theoretical
ecology where the xi model population densities, and with such applications in mind
Stephen Baigent
Department of Mathematics, UCL, Gower Street, London WC1E 6BT, UK
Tel.: +44 (0)207-679-3593
Fax: 44 (0)20-7383-5519
E-mail: s.baigent@ucl.ac.uk
Zhanyuan Hou
Faculty of Computing, London Metropolitan University,
North Campus, 166-220 Holloway Road, London N7 8DB, UK
Tel.: +123-45-678910
Fax: +123-45-678910
E-mail: z.hou@londonmet.ac.uk
2
Stephen Baigent, Zhanyuan Hou
we are only interested in solutions to (1) that lie in the positive cone C = IRN+ . As
can be easily seen, both C and its interior C0 are invariant for (1), so for any initial
condition x(0) ∈ C the model makes sense.
One of the attractions of (1) to mathematicians is its simple form. Fixed points
can be found explicitly, and there is a necessary and sufficient condition on A for
all orbits to be uniformly bounded (see below). The existence and stability of fixed
points has traditionally been a focus for ecologists, since conservation biology asks
whether species sharing the same habitat can stably coexist. However, in recent years
many ecologists have replaced their demands for globally stable coexistence by the
requirement of permanence, that is, for all initially positive densities, the evolving
densities are bounded, and they eventually uniformly exceed some positive density.
Stability and permanence are clearly closely related. Indeed, if an interior fixed point
is globally stable, so that it attracts all orbits starting in C0 , then the system is permanent, but there are permanent 3 species competitive systems with attracting limit
cycles on the carrying simplex which are obviously not globally stable.
In the present paper, we are concerned with global stability of interior and boundary fixed points of (1). We introduce the notion of partial permanence in order to
express permanence amongst a proper subset of populations, as is necessary for a
system where a boundary fixed point is globally attracting. Our approach is to further
extend the Split Lyapunov function method that E.C. Zeeman and M. L. Zeeman introduced in [16] (see also [15]) to study the global attraction of an interior fixed point
of the totally competitive subclass (A, b > 0) of (1). Recently in [9] we have extended
this split Liapunov function method to apply also to boundary fixed points, again for
the totally competitive case. In both [16, 9] general results were obtained by first proving convergence to fixed points on the carrying simplex Σ , an (N − 1)−dimensional
attracting invariant manifold where the asymptotic dynamics of (1) (with A, b > 0) is
known to occur [2]. Here we prove convergence for classes of Lotka-Volterra systems
that are not necessarily totally competitive, so that A may have entries of different sign
or zeroes. To the best of our knowledge, the existence of carrying simplices outside
of the subclass of totally competitive systems [2] or type-K competitive systems [12]
is an open problem, so we circumscribe the use of a carrying simplex and provide
an alternative, more direct, proof of convergence using the split Lyapunov function
method. However, the price to be paid for relaxing the requirement that b and A
have positive elements is that we will have to demonstrate uniform boundedness and
permanence or partial permanence for (1), whereas for the totally competitive case
where b, A > 0 they are guaranteed.
2 Permanence and boundedness of solutions
This section serves as a preparation for the global asymptotic stability results to be
developed in the next section, although some of the results on permanence are new
and important in their own right.
Recall that solutions of a system are called uniformly bounded if there is an M > 0
such that every solution satisfies |xi (t)| ≤ M for all i ∈ IN and large enough t.
Global stability of interior and boundary fixed points for Lotka-Volterra systems
3
Theorem 1 [3, Theorem 15.2.1] Solutions of (1) in C are uniformly bounded if −A
is a B-matrix. This condition is also necessary for uniform boundedness of solutions
of (1) for evey b ∈ IRN .
One of the many equivalent characterisations of B-matrices is: −A is a B-matrix if
and only if for all x ≥ 0 with x 6= 0 there is an index i such that xi > 0 and (Ax)i > 0
[3, Theorem 15.2.4]. Hence, recalling that A is a P-matrix if and only if for all x 6= 0
there exists an index i such that xi (Ax)i > 0, we see that the following is obvious.
Corollary 1 The solutions of (1) in C are uniformly bounded if either of the conditions below is met:
(i) A is a P-matrix,
(ii) ∀i, j ∈ IN , ai j ≥ 0 and aii > 0.
System (1) is a special case of the Kolmogorov system
xi0 = xi fi (x),
i ∈ IN ,
x ∈ C,
(2)
∈ C1 (IRN , IR). System (2) is called permanent if there is a compact set K
where fi
⊂ C0
0
0
0
0
such that for each x ∈ C , the solution of (2) starting at x satisfies x(t, x ) ∈ K for
all sufficiently large t (which implies uniform boundedness). There are some wellknown conditions on A, b that ensure permanence. The following due to Jansen [11]
is one of them.
Theorem 2 Assume that the solutions of (1) in C are uniformly bounded. Then (1) is
permanent if there exists a p ∈ C0 such that
pT (b − Ax̂) > 0 for all fixed points x̂ on ∂C = C \C0 .
(3)
As well as ensuring that all interior trajectories are bounded, and bounded away
from the boundary, permanence also ensures that there is a unique interior fixed point
[11].
The stability results of the next section will utilise a Lyapunov function that may
become unbounded on the boundary ∂C. Thus the success of our method relies upon
ensuring that where necessary, an appropriate subset of components of N exceed a
positive number for sufficiently large time. We will call this partial permanence. The
rest of this section is reserved for partial permanence of (1) and (2).
Let J be a nonempty subset of IN with I = IN \ J and
CI0 = {x ∈ C : ∀i ∈ I, xi = 0, ∀ j ∈ J, x j > 0}.
We say that (2) is partially permanent with respect to J, or J-permanent for short, if
there are δ > 0 and M > δ such that
∀x0 ∈ C with (∀i ∈ J, xi0 > 0), ∀ j ∈ J, ∀ large t, δ ≤ x j (t, x0 ) ≤ M.
(4)
Obviously, partial permanence with respect to J = IN is permanence. Let
πi = {x ∈ C : xi = 0},
i ∈ IN .
(5)
We observe that J-permanence of (2) implies permanence of the |J|-dimensional subsystem of (2) on ∩i∈I πi . Thus, in particular, J-permanence of (1) implies the existence
of a unique fixed point p in CI0 .
4
Stephen Baigent, Zhanyuan Hou
Lemma 1 Assume that the solutions of (2) are uniformly bounded. Assume also that,
for a fixed i ∈ IN ,
0
0
∀x ∈ πi , ∃T (x ) > 0 such that
Z T (x0 )
0
fi (x(t, x0 ))dt > 0.
(6)
Then there are δ > 0 and M > δ such that
∀x0 ∈ C with xi0 > 0, ∀ large t, δ ≤ xi (t, x0 ) ≤ M.
(7)
Proof The first step is to show that there is a positively invariant set Ŝ such that all
orbits lie in Ŝ after some finite time. The assumption of uniform boundedness ensures
the existence of M > 0 such that the solutions of (2) satisfy
∀x0 ∈ C, ∀ j ∈ IN , ∀ large t, 0 ≤ x j (t, x0 ) < M.
(8)
Let S = {x ∈ C : ∀ j ∈ IN , x j < M} and let S̄ be the closure of S. For each x0 ∈ S̄, there
is a time t(x0 ) after which the forward trajectory x(t, x0 ) through x0 ∈ S̄ remains in
S; that is, for each x0 ∈ S̄, there is a t(x0 ) ≥ 0 such that x(t, x0 ) ∈ S for all t > t(x0 ).
Let S1 = {x(t, x0 ) : x0 ∈ S̄,t ≥ t(x0 )} and let S̄1 be the closure of S1 . Then S̄1 ⊂ S̄,
S̄1 is compact, and S1 is positively inviarant. We claim that S̄1 is positively invariant.
For if not, then there are some x1 ∈ S̄1 and some t1 > 0 such that x(t1 , x1 ) 6∈ S̄1 . By
continuous dependence of the solution on initial values, for all x0 ∈ S̄1 close enough
to x1 we would have x(t1 , x0 ) 6∈ S̄1 , a contradiction to the definition of S1 . Then the set
Ŝ = {x(t, x0 ) : x0 ∈ S̄,t ∈ [0, τ]} is compact, positively invariant and every solution of
(2) will enter Ŝ in a finite time. Similarly, Ŝi = Ŝ ∩ πi is compact, positively invariant
and every solution in πi will enter Ŝi in a finite time. So we need only prove (7) for
x0 ∈ Ŝ.
By assumption (6), for each x0 ∈ Ŝi , there is a T (x0 ) > 0 and a number h(x0 ) > 0
satisfying
Z T (x0 )
1
fi (x(t, x0 ))dt > h(x0 ).
T (x0 ) 0
By continuity, there is an open neighbourhood U(x0 ) of x0 in Ŝ such that
∀y ∈ U(x0 ),
1
T (x0 )
Z T (x0 )
0
fi (x(t, y))dt > h(x0 ).
(9)
Since Ŝi is compact and {U(x0 ) : x0 ∈ Ŝi } is an open cover of Ŝi , so that Ŝi has a finite
open cover {U(xk ) : xk ∈ Ŝi , k ∈ Im }. Let
h = min{h(x1 ), . . . , h(xm )},
(10)
1
m
(11)
1
m
(12)
T0 = min{T (x ), . . . , T (x )},
T1 = max{T (x ), . . . , T (x )}.
Note that the set S̃ = Ŝ \ ∪k∈Im U(xk ) is also compact so that
δ0 = inf{xi : x ∈ S̃} > 0.
Global stability of interior and boundary fixed points for Lotka-Volterra systems
5
Now comes the key step, utilising (6) to show that solutions in the open cover ∪k∈Im U(xk ),
but away from πi (i.e. satisfying xi0 > 0) cannot remain there for all time.
We claim that
∀x0 ∈ ∪k∈Im U(xk ) \ Ŝi , ∃t1 (x0 ) > 0 such that x(t1 (x0 ), x0 ) ∈ S̃.
(13)
Suppose this is not true. Then there is a k ∈ Im and an x0 ∈ U(xk ) \ Ŝi such that
x(t, x0 ) 6∈ S̃ for all t ≥ 0, so that since Ŝ \ Ŝi is positively invariant, we have x(t, x0 ) ∈
∪ j∈Im U(x j ) \ Ŝi for all t > 0. It then follows from (9)-(11) that
Z T (xk )
0
fi (x(t, x0 ))dt > T (xk )h(xk ) ≥ T0 h > 0.
Since x(t, x0 ) is confined to ∪ j∈Im U(x j ) \ Ŝi for all t > 0, it must be the case that
x(T (xk ), x0 ) ∈ U(xs ) for some s ∈ Im . Hence by (9)-(11) again,
Z T (xk )+T (xs )
0
Z T (xs )
0
fi (x(t, x ))dt > T0 h +
0
fi (x(t, x(T (xk ), x0 )))dt > 2T0 h.
RepeatingR the above process, we see that there is an increasing sequence of times tr
such that 0tr fi (x(s, x0 ))ds → +∞ as r → ∞, so that
Z t
r
0
0
0
xi (tr , x ) = xi exp
fi (x(s, x ))ds → +∞
0
as r → +∞. This contradicts the uniform boundedness of orbits and therefore shows
our claim (13).
Now fix a number δ1 ∈ (0, δ0 ) and let
R = {x(t, x0 ) : x0 ∈ Ŝ with xi0 ≥ δ1 , t ∈ [0, T1 ]}.
(14)
Then, as Ŝ \ Ŝi is positively invariant, R ⊂ Ŝ \ Ŝi . Since R is the image of the solution
map on [0, T1 ] × {x0 ∈ Ŝ : xi0 ≥ δ1 }, which is compact, the continuity of x in (t, x0 )
implies the compactness of R. For each x0 ∈ Ŝ with xi0 = δ1 , we have x0 ∈ U(xk ) \ Ŝi
for some k ∈ Im so
Z T (xk )
xi (T (xk ), x0 ) = xi0 exp
fi (x(t, x0 ))dt > δ1 eT0 h > δ1 .
0
This (combined with the fact that T (xk ) ≤ T1 ) shows that the set R is forward invariant. By (13), each forward orbit x(t, x0 ) through x0 ∈ Ŝ \ Ŝi will enter and stay in R
for all large t. Since the compactness of R implies
δ = inf{xi : x ∈ R} > 0,
this δ meets the requirement of (7).
Remark 1 Lemma 1 is an adaptation of Theorem 12.2.1 in [3] for permanence of a
system.
6
Stephen Baigent, Zhanyuan Hou
Note that (1) is a special case of (2) when f = ( f1 , . . . , fN )T is affine. Lemma 1
can then be simplified as follows for (1). In this paper, we view an N-dimensional
row (column) vector u (v) identical to a 1 × N (N × 1) matrix so that uv is regarded as
the product of two matrices rather than the dot multiplication of two vectors.
Lemma 2 Assume that
(a) the solutions of (1) are uniformly bounded,
(b) for a fixed i ∈ IN , every solution of (1) in πi satisfies
∀ j ∈ IN \ {i}, either lim x j (t) = 0 or lim inf x j (t) > 0,
t→+∞
t→+∞
(c) Ai x̂ < bi holds for each fixed point x̂ ∈ πi .
Then there are δ > 0 and M > δ such that (7) holds.
Proof For each x0 ∈ πi , by Lemma 1 we need only show the existence of T (x0 ) > 0
satisfying
Z T (x0 )
0
[bi − Ai x(t, x0 )]dt > 0.
(15)
If x0 = 0 then x(t, x0 ) ≡ 0 for t ≥ 0 and condition (c) implies bi > 0, so (15) holds for
any T (x0 ) > 0.
Suppose then that x0 6= 0. By assumption (b) we have the partition IN = J1 ∪ J2
with i ∈ J1 such that
∀ j ∈ J1 , lim x j (t, x0 ) = 0; ∀k ∈ J2 , lim inf xk (t, x0 ) > 0
t→+∞
t→+∞
(16)
(though J2 might be empty). By (16) and assumption (a), we can choose a sequence
{tk } ⊂ (0, +∞) satisfying tk → +∞ as k → +∞,
∀ j ∈ J1 ,
∀ j ∈ J2 ,
1 tk
x j (t, x0 )dt = 0 = y j ,
k→+∞ tk 0
Z
1 tk
lim
x j (t, x0 )dt = y j ∈ (0, +∞).
k→+∞ tk 0
Z
lim
(17)
(18)
For j ∈ J2 and k ≥ 1, from (1) we obtain
ln x j (tk , x0 ) − ln x0j
tk
= bj −Aj
1
tk
Z tk
x(t, x0 )dt.
(19)
0
The left-hand side of (19) vanishes as k → +∞ due to the boundedness of ln x j (tk , x0 )
when j ∈ J2 . Then (17)–(19) lead to
∀ j ∈ J2 , b j − A j y = 0.
(20)
This shows that y ∈ πi is a fixed point of (1). Thus, since bi − Ai y > 0 by (c), we have
1
tk
Z tk
0
[bi − Ai x(t, x0 )]dt > 0
for large enough k so (15) holds for T (x0 ) = tk .
Global stability of interior and boundary fixed points for Lotka-Volterra systems
7
Remark 2 The condition (a) of Lemma 2 is automatically satisfied if for all i, j ∈ IN ,
aii > 0 and ai j ≥ 0 (see [5] or [6]). Condition (b) is easily checked for lower dimensional systems. In particular, condition (b) holds if every solution in πi converges to
a fixed point (see [7], [8]). This will be demonstrated by examples later.
The following results on (partial) permanence are immediate consequences of
Lemmas 1 and 2.
Theorem 3 Let J ⊂ IN with J 6= 0.
/ Assume that the solution of (2) are uniformly
bounded and that
∀ j ∈ J, ∀x0 ∈ π j , ∃T > 0 such that
Z T
0
f j (x(t, x0 ))dt > 0.
Then (2) is J−permanent. In particular, (2) is permanent if J = IN .
Theorem 4 Let J ⊂ IN with J 6= 0.
/ Assume that
(a) the solutions of (1) are uniformly bounded,
(b) each nonzero component xk of every solution in ∪ j∈J π j satisfies
lim xk (t) = 0 or lim inf xk (t) > 0,
t→+∞
t→+∞
(c) ∀ j ∈ J, ∀ fixed point x̂ ∈ π j , A j x̂ < b j .
Then (1) is J−permanent. In particular, (1) is permanent if J = IN .
Remark 3 Consider the particular case J = IN of Theorem 4. When (1) is competitive
with bi > 0, ai j ≥ 0 and aii > 0 for all i, j ∈ IN , by induction on N it can be shown
that condition (c) implies conditions (a) and (b). Thus, in that case, condition (c)
alone implies permanence of (1). Indeed, condition (c) implies permanence of all
subsystems of (1) and their small perturbations (see [13, Corollary 3.1]). Further
extension of the result to nonautonomous and delayed systems can be found in [1]
and [10].
3 Application of the Split Lyapunov function method
In this section, we apply the split Lyapunov function method to (1) to obtain criteria
for (1) to be globally asymptotically stable at a fixed point, which may be in the
interior C0 or the boundary ∂C.
In the following, for p an interior or boundary fixed point of (1) we set B =
−DF(p) where Fi (x) = xi (bi − Ai x), i ∈ IN . Explicitly, for i, j ∈ IN ,
∂ Fi
bi − Ai p − pi aii , i = j
=
−pi ai j , i 6= j.
∂ x j x=p
Since pi > 0 implies bi − Ai p = 0, we have, for the ith row of B,
pi Ai ,
if pi > 0,
Bi =
(0 · · · 0 Ai p − bi 0 · · · 0), if pi = 0.
When p is an interior fixed point we obtain B = D(p)A.
(21)
8
Stephen Baigent, Zhanyuan Hou
Theorem 5 Suppose that for the system (1) the following conditions hold:
1. The system is permanent, so that there is a unique interior fixed point p;
2. There is a left eigenvector α ∈ IRN of D(p)A associated with an eigenvalue λ > 0
and αi 6= 0 for all i ∈ IN ;
3. The following holds:
yT D(α)Ay > 0 for all non-zero y ∈ IRN such that α T y = 0.
(22)
Then the system (1) at p is globally asymptotically stable.
Proof Our assumption means that the unique interior fixed point p satisfies Ap = b.
With B = D(p)A the Jacobian matrix at p is −B. Notice that (1) can be rewritten as
ẋ = −B(x − p) − D(x − p)A(x − p).
By assumption, α T B = λ α T . Now let ψ = α T (x − p). Then differentiating, and setting y = x − p,
ψ̇ = α T ẋ = −α T By − α T D(y)Ay = −λ ψ − yT D(α)Ay.
(23)
Let us also define V (x) = ∏Ni=1 xipi αi . Then
N
V̇ = V
ẋi
∑ pi αi xi
i=1
N
=V
N
∑ pi αi (bi − Ai x) = −V ∑ pi αi Ai (x − p)
i=1
i=1
= −V α T B(x − p) = −λV α T (x − p) = −λ ψV.
Summarising we find that,
ψ̇ = −λ ψ − yT D(α)Ay,
(24)
V̇ = −λ ψV.
(25)
Then the assumption (22) becomes
yT D(α)Ay > 0 for y ∈ ψ −1 (0) \ {0}.
(26)
Now suppose that x0 ∈ C0 \ {p} is given. We consider the two cases ψ(x0 ) ≤ 0 and
ψ(x0 ) > 0 separately. Let y0 = x0 − p and y(t, y0 ) = x(t, x0 ) − p.
First suppose that ψ(x0 ) ≤ 0. Then since x0 6= p, by (24) and (26) we have ψ̇ < 0
at t = 0 and hence ψ(x(t, x0 )) < 0 for all t > 0. Thus for t > 0 we have V̇ > 0 and
hence V (x(t, x0 )) is increasing with positive time along the orbit. Since (1) is assumed
permanent, V (x(t, x0 )) is bounded and V (x(t, x0 )) ≥ ε for some ε > 0 and all t ≥ 0.
Let ω(y0 ) denote the omega limit set of y0 = x0 − p; by compactness ω(y0 ) is nonempty, connected and invariant. Thus we have ω(y0 ) ⊂ V̇ −1 (0) = V −1 (0) ∪ ψ −1 (0).
But ω(y0 ) is a connected invariant set, and lies within C0 so that V (s) > 0 for all
s ∈ ω(y0 ) and by (26) and (24) y = 0 is the only invariant subset of ψ −1 (0). Hence
ω(y0 ) = {0} and the trajectory x(t, x0 ) converges to p as t → +∞.
Global stability of interior and boundary fixed points for Lotka-Volterra systems
9
Next we suppose ψ(x0 ) > 0. If there is a T > 0 such that ψ(x(t, x0 )) > 0 for 0 ≤
t < T but ψ(x(T, x0 )) = 0, then limt→+∞ x(t, x0 ) = p from the firstR case. Otherwise,
ψ(x(t, x0 )) > 0 for all t > 0 so we would have V (t) = V (0) exp −λ 0t ψ(x(s, x0 )) ds →
c ≥ 0 as t → +∞. If c = 0 this would contradict permanence. For c > 0, let w ∈ ω(y0 ).
Then there exists tk → +∞ such that y(tk , y0 ) → w. Since V (t) → c > 0 monotonically, V̇ → 0, so that ψ(y(tk , y0 )) → 0 as k → +∞ we have w ∈ ψ −1 (0) and hence
ω(y0 ) ⊂ ψ −1 (0). Arguing as above again we obtain x(t, x0 ) → p as t → +∞.
Finally, we prove the stability of (1) at p. Denote the open ball centered at p with
a radius r by Br (p). We need to show that for any given ε > 0 there is a δ > 0 such
that x0 ∈ Bδ (p) implies x(t, x0 ) ∈ Bε (p) for all t ≥ 0.
Let W be an N × N matrix such that the columns of W , Wc j for j ∈ IN , satisfy
Wc1 = (α1−1 , 0, . . . , 0)T , ψ −1 (0) = span{Wc2 , . . . ,WcN }.
Then W is invertible. Let zT = (ψ, z̃T ) with z̃ ∈ IRN−1 and y = x − p = W z. We have
ψ = (1, 0, . . . , 0)z = α T (x − p) and, by (24),
ψ̇ = −(λ +Θ (z))ψ − z̃T M0 z̃,
(27)
where Θ is linear with Θ (0) = 0 and M0 is an (N − 1) × (N − 1) real symmetric
matrix. Then condition (22) implies the positive definiteness of M0 . From (27) we
see the existence of ε0 > 0 such that x0 ∈ Bε0 (p) and ψ(x0 ) > 0 imply ψ̇ < 0 at
t = 0.
For each l > 0, the set Sl = {x ∈ C : V (x) = l} defines an (N − 1)-dimensional surface and, by the definition of V , l ∗ = V (p) > 0. For each y0 = x0 − p ∈ ψ −1 (0) \ {0},
since ψ(x(t, x0 )) < 0 by (24) and V̇ (x(t, x0 )) > 0 by (25) for t > 0 so that V (x(t, x0 )) ↑
l ∗ as t → +∞, we have V (x0 ) < l ∗ so Sl ∗ ∩ ψ −1 (0) = {x = p}.
Then, for any fixed ε ∈ (0, ε0 ), there is an l ∈ (0, l ∗ ) such that Vl = {x ∈ C : V (x) ≥
l, ψ(x) ≤ 0} ⊂ Bε/2 (p). The set Vl is bounded by Sl and ψ −1 (0) with an (N − 2)dimensional edge Sl ∩ ψ −1 (0). Since p 6∈ Sl ∩ ψ −1 (0), for each x0 ∈ Sl ∩ ψ −1 (0) there
is a t1 (x0 ) < 0 such that ψ(x(t, x0 )) > 0 and x(t, x0 ) ∈ Bε/2 (p) for t ∈ (t1 (x0 ), 0) but
x(t1 (x0 ), x0 ) 6∈ Bε/2 (p). Let
Γ = {x(t, x0 ) : x0 ∈ Sl ∩ ψ −1 (0),t ∈ [t1 (x0 ), 0]}.
Then the open subset of Bε/2 (p) bounded “below” by Sl , “surrounding” by Γ and
“above” by the upper boundary of Bε/2 (p) satisfying ψ > 0 contains p and is positively invariant. Therefore, it contains Bδ (p) for a small δ ∈ (0, ε/2) and x0 ∈ Bδ (p)
implies x(t, x0 ) ∈ Bε (p) for all t ≥ 0.
Remark 4 The sufficient condition for permanence (3) is that there exists q ∈ C0
such that qT (b − Ax̂) > 0 for each boundary fixed point x̂. Suppose that there is a
unique interior fixed point p and that B = D(p)A has a positive left eigenvector α.
Then taking q = α, qT (b − Ax̂) = α T (b − Ax̂) = α T A(p − x̂) = −λ ψ(x̂). Hence the
permanence condition becomes that we require ψ(x̂) < 0 at all boundary fixed points
x̂. This is certainly true if the carrying simplex Σ is convex (α is then the normal to
Σ at p and all boundary fixed points lie below the tangent plane to Σ at p).
10
Stephen Baigent, Zhanyuan Hou
Remark 5 As discussed above, if A is a P−matrix then all orbits are uniformly bounded
and there is a unique fixed point. On the other hand, any system (1) with locally stable interior fixed point must have a D−stable matrix −A, since scaling the axes does
not change the local stability of the interior fixed point, and so A must then be a
P0 −matrix, that is, a matrix whose principal minors are non-negative (see, for example, Theorem 2.5.8 in [4]).
Remark 6 If α can be chosen to be positive then we do not need to prove permanence
for the theorem proof to work, but then we must prove existence of an interior fixed
point, and still show that all trajectories are uniformly bounded.
Remark 7 If A > 0 then as shown in [16], assuming that an interior fixed point exists,
the Perron-Frobenius theorem tells us that α can be chosen to be positive. We thus
do not need to assume permanence. It can be shown that if A > 0 all trajectories are
uniformly bounded. However, we still need to show existence of an interior fixed
point.
For a boundary fixed point p of (1) to be stable in C, it is necessary that it is
saturated (see [14, Theorem 3.2.5]), i.e.
∀i ∈ IN , pi = 0 =⇒ bi − Ai p ≤ 0.
Theorem 6 Suppose (1) satisfies the following conditions:
(a) The solutions of (1) are uniformly bounded.
(b) For a proper subset J ⊂ IN with I = IN \ J, (1) is J−permanent and the unique
fixed point p in CI0 is saturated.
(c) The matrix D(p)A has an eigenvalue λ > 0 and an associated left eigenvector
α ∈ IRN such that α j 6= 0 for all j ∈ J and αi > 0 for all i ∈ I.
(d) The following holds:
yT D(α)Ay > 0 for all non-zero y ∈ IRN such that α T y = 0.
(28)
Then (1) at p is globally asymptotically stable.
Proof We first rewrite (1) as
ẋ = −B(x − p) − D(x − p)A(x − p),
where B is given by (21). Now let ψ = α T (x − p) and y = x − p. Then ψ̇ = α T ẋ =
−α T By − α T D(y)Ay. As yi = xi for i ∈ I, by condition (c) we can further write
ψ̇ = −λ ψ − ∑ αi (Ai p − bi )xi − yT D(α)Ay.
i∈I
With V (x) = ∏i∈J xipi αi , we have
V̇ = V ∑ pi αi
i∈J
ẋi
xi
= V ∑ pi αi (bi − Ai x) = −V ∑ pi αi Ai (x − p)
i∈J
i∈J
= −λV α T (x − p) = −λ ψV.
(29)
Global stability of interior and boundary fixed points for Lotka-Volterra systems
11
Thus, instead of (24) and (25) in the proof of Theorem 5, we obtain
ψ̇ = −λ ψ − ∑ αi (Ai p − bi )xi − yT D(α)Ay,
(30)
V̇ = −λ ψV.
(31)
i∈I
Since p is saturated by condition (b), and αi > 0 for all i ∈ I by condition (c), for x ∈ C
the term ∑i∈I αi (Ai p − bi )xi is always nonnegative. Then, replacing permanence by
partial permanence in the proof of Theorem 5, we obtain the global attraction of p.
To show the stability of (1) at p in C, since p ∈ ∂C, the balls centered at p are
restricted to C only. Referring to the proof of Theorem 5, we note that (30) implies
ψ̇ ≤ −(λ +Θ (z))ψ − z̃T M0 z̃
(32)
instead of (27), the set Vl is bounded by Sl , ψ −1 (0) and ∪i∈I πi , and the open subset
of Bε/2 (p) ∩C relative to C bounded “below” by Sl , “surrounding” by Γ and ∪i∈I πi ,
and “above” by the upper boundary of Bε/2 (p) satisfying ψ > 0 contains p and is
positively invariant. Then the stability follows.
3.1 Checking the positive definite condition (28)
In what follows, for a given square matrix M, M S = M + M T . Let Wc2 , . . . ,WcN be
N − 1 column vectors that span ψ −1 (0). As was shown in [16, 9], for any α ∈ IRN
with αi 6= 0 for all i ∈ IN , sT (D(α)A)S s ≥ 0 for all s ∈ ψ −1 (0) \ {0} if and only if the
(N − 1) × (N − 1) symmetric matrix
U = [Wc2 , . . . ,WcN ]T (D(α)A)S [Wc2 , . . . ,WcN ]
(33)
is positive 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 D(α) and A. For simplicity we choose

1




−1  −1
[Wc2 , . . . ,WcN ] = D(α)
0
0
..
.
..
.





1 
−1
(34)
and denote by Mi j the submatrix of M obtained by deleting the ith row and the jth
column of M for any square matrix M, then
U = (AD(α)−1 )S11 + (AD(α)−1 )SNN − (AD(α)−1 )S1N − (AD(α)−1 )SN1 .
Thus in checking the condition (22) we simply check that U is positive definite.
(35)
12
Stephen Baigent, Zhanyuan Hou
4 Examples and discussion
Example 1
Consider system (1) with


5 −1 1
A =  4 3 1 ,
−1 − 12 2
 
25
1 
46 .
b=
3
26
(36)
Note that −A is a B-matrix so, by Theorem 1, the solutions of (1) with (36) are
uniformly bounded. The boundary fixed points x̂T are: (0, 0, 0), ( 53 , 0, 0), (0, 46
9 , 0),
121 130
8
155
132 202
T ∈ C0 , we can check
),
(
,
,
0),
(
,
0,
),
(0,
,
).
If
q
=
(1,
1,
1)
(0, 0, 13
3
57 57
11
33
39 39
that qT (b − Ax̂) > 0 for all x̂. Thus, by Theorem 2, (1) is permanent. The system
T
has a unique interior fixed point p = (1, 2, 16
3 ) . The matrix D(p)A has a positive
eigenvalue λ = 8 and an associated left eigenvector α T = (48, 4, −21). Then, from
(35) we have

 5
23
49
− 16 − 336
24
− 85
−1 S
1
3
29 
24
48

.
(AD(α) ) =
−6
− 168 , U =
139
2
− 85
23
29
4
48 84
− 336
− 168
− 21
Since U is positive definite, the condition (22) is satisfied. By Theorem 5, (1) with
(36) at p is globally asymptotically stable.
Example 2
Consider system (1) with


1 0 −θ
A =  1 1 0  , θ = 9,
11 1
b = (1, 3, 19/6)T .
(37)
There is an interior fixed point pT = (5/2, 1/2, 1/6) and the system is permanent
(via checking (3)). The matrix −A is D−stable (that is for every diagonal D > 0 the
matrix −DA is stable) so −D(p)A is stable and the system at p is locally asymptotically stable. In [3, Conjecture 15.6.7] the authors suggest that for (1) with a (unique)
interior fixed point p, D−stability of −A is a necessary and sufficient condition that
p is globally stable. An application of Theorem 5 to (1) does not yield conclusive
results: D(p)A has a positive eigenvalue λ = 0.38732 with a real left eigenvector
α T = (0.0056859, −0.3613, 1) that yields a matrix U with real eigenvalues of opposite sign. Moreover there is no positive 3 × 3 diagonal matrix D such that (DA)S
is positive definite, so that the Volterra-Lyapunov Theorem (e.g. [16, Theorem 7.1])
does not apply either. Thus, it needs further investigation to determine whether p is
a global attractor. Note that this example provides an interesting attracting compact
invariant set M, which separates R(0) and R(∞), the basin of repulsion of 0 and of ∞,
parallel to the carrying simplex for a totally competitive system. Since the subsystem
Global stability of interior and boundary fixed points for Lotka-Volterra systems
x
13
8
6
4
2
0
3
2.0
2
1.5
1.0
y
0.5
0.0
1
z
0
Fig. 1 Phase plot for Example 2 when θ = 9. The plot suggests that the interior steady state is globally
attracting on C0 , but neither Theorem 5 nor the Volterra-Lyapunov theorem provide a proof.
for (x1 , x3 ) has an interior globally attracting focus, M ∩ π2 consists of two curves
from the two axial fixed points spiralling in towards to interior fixed point. But the
two subsystems for (x1 , x2 ) and (x2 , x3 ) both have an interior globally attracting node,
so both M ∩ π3 and M ∩ π1 are convex curves joining the three fixed points. This give
us a clue that the geometric shape of M may be complicated; part of it joining p to
M ∩ π2 may look like a spiralling cone with p as a vertex. This indicates that M does
not have a tangent plane at p, which might be the essential reason for the failure of
the application of Theorem 5.
Example 3
Consider system (1) with

2 0 1
A =  1 3 2 ,
3 − 21 4

 
2
b =  7 .
1
(38)
Clearly, −A is a B-matrix so Theorem 1 implies the uniform boundedness of solutions
of (1) with (38). There are five fixed points in π1 ∪ π2 : pT0 = (0, 0, 0), pT1 = (1, 0, 0),
pT2 = (0, 37 , 0), pT3 = (0, 0, 41 ), pT4 = (0, 2, 21 ), and they satisfy A1 p0 < b1 , A1 p2 < b1 ,
A1 p3 < b1 , A1 p4 < b1 , A2 p0 < b2 , A2 p1 < b2 and A2 p3 < b2 . A phase portrait analysis
on π1 and π2 shows that every solution in π1 ∪ π2 converges to a fixed point. Then,
by Theorem 4 and Remark 2, (1) with (38) is partially permanent with respect to
0 . As A p > b , p is
J = {1, 2}. Thus, it has a unique fixed point p = (1, 2, 0)T in C{3}
3
3
saturated. The matrix D(p)A has a positive eigenvalue λ = 6 and an associated left
14
Stephen Baigent, Zhanyuan Hou
x
4
3
3
2
1
2
0
2.0
y
1.5
1
z 1.0
0.5
0.0
0
Fig. 2 The global stability of Example 2 when θ = 5. In this case the unique interior fixed point can
be shown to be globally attracting on C0 by the Volterra-Lyapunov theorem, since (DA)S > 0 when D =
diag(1, 3, 10).
eigenvector α T = (2, 4, 3). By (35) we have


2 12 11
6
(AD(α)−1 )S =  1 3 13  ,
2 2 24
11 13 8
6 24 3
U=
5
2
55
− 24
− 55
24
37
12
.
It can be checked that U is positive definite so the condition (22) holds. By Theorem
6, (1) with (38) at p is globally asymptotically stable.
Example 4
Consider system (1) with


2 0 11
1 3 2 4

A=
 3 −1 4 1 ,
2
2 21 12 2
 
2
7

b=
 1 .
4
(39)
Again, −A is a B-matrix so the solutions of (1) with (39) are uniformly bounded
by Theorem 1. The subsystem of (1) with (39) on π4 is the system (1) with (38).
Thus, from Example 3 we know that every solution of (1) with (39) in π4 converges
to a fixed point. Moreover, adding 0 as the fourth component to each of the fixed
points pi (0 ≤ i ≤ 4) and p in Example 3, we see that A4 p < b4 and A4 pi < b4 for
all 0 ≤ i ≤ 4. By Theorem 4 and Remark 2, (1) with (39) is partially permanent with
respect to J = {4}. The unique fixed point in CI0 (I = {1, 2, 3}) is x∗ = (0, 0, 0, 2)T .
Global stability of interior and boundary fixed points for Lotka-Volterra systems
15
As Ai x∗ ≥ bi for all i ∈ I, x∗ is saturated. The matrix D(x∗ )A has a positive eigenvalue
λ = 4 and an associated left eigenvector α T = (2, 12 , 12 , 2). By (35) we have

2
1
2
7 3
2 2

 1 12 3 3 
2

(AD(α)−1 )S = 
 7 3 16 3  ,
2
2
3
3
2 3 2 2


13 −12 2
.
U =  −12 22 − 29
2
29
2 − 2 15
13 −12
> 0 and detU = 4.75 > 0, U is positive definite so the condition (22)
Since −12
22
holds. By Theorem 6, (1) with (39) at x∗ is globally asymptotically stable.
5 Conclusions and discussion
We have shown that the Split Lyapunov method introduced in [16] for competitive
Lotka-Voletrra systems can be used without reference to the carrying simplex, and has
a fairly general application to both competitive and non-competitive Lotka-Volterra
systems. Unlike the competitive case, however, we have to additionally show permanence or partial permanence of the dynamics. Our results can be successfully applied to a wide range of Lotka-Volterra systems, but there remain cases where the
theory does not give conclusive results as shown by example 2. For 3-species competitive systems, there are examples found by Zeeman and Zeeman where the system
is known to be globally convergent to an interior fixed point on C0 , but the Split Liapunov method fails. As indicated by the carrying simplex plots in [16], this appears
to be when the carrying simplex has negative curvature. It would be interesting to
investigate when an attracting invariant manifold analogous to the carrying simplex
exists for non-competitive systems, and how the Gaussian curvature at fixed points
relates to their stability. Further work is required to determine whether the Split Lyapunov method can be further adapted to deal with indeterminate cases, or whether a
new approach is necessary.
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