global stability of interior and boundary ﬁxed …global stability of interior and boundary ﬁxed...

Differential Equations and Dynamical Systems manuscript No.(will be inserted by the editor)

Global stability of interior and boundary fixed points forLotka-Volterra systems

Stephen Baigent · Zhanyuan Hou

Received: date / Accepted: date

Abstract For permanent and partially permanent, uniformly bounded Lotka-Volterrasystems, we apply the Split Lyapunov function technique developed for competi-tive Lotka-Volterra systems to find new conditions that an interior or boundary fixedpoint of a Lotka-Volterra system with general species-species interactions is globallyasymptotically stable. Unlike previous applications of the Split Lyapunov techniqueto competitive Lotka-Volterra systems, our method does not require the existence ofa carrying simplex.

Keywords Lotka-Volterra systems · global attractors · global repellors · globalasymptotic 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 diagonalmatrix with diagonal entries α1, . . . ,αN . We consider the Lotka-Volterra equations

x = D(x)(b−Ax), (1)

where bi ∈ IR and ai j ∈ IR for i, j ∈ IN . These equations are widely used in theoreticalecology where the xi model population densities, and with such applications in mind

Stephen BaigentDepartment of Mathematics, UCL, Gower Street, London WC1E 6BT, UKTel.: +44 (0)207-679-3593Fax: 44 (0)20-7383-5519E-mail: [email protected]

Zhanyuan HouFaculty of Computing, London Metropolitan University,North Campus, 166-220 Holloway Road, London N7 8DB, UKTel.: +123-45-678910Fax: +123-45-678910E-mail: [email protected]

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 initialcondition x(0) ∈C the model makes sense.

One of the attractions of (1) to mathematicians is its simple form. Fixed pointscan be found explicitly, and there is a necessary and sufficient condition on A forall orbits to be uniformly bounded (see below). The existence and stability of fixedpoints has traditionally been a focus for ecologists, since conservation biology askswhether species sharing the same habitat can stably coexist. However, in recent yearsmany ecologists have replaced their demands for globally stable coexistence by therequirement of permanence, that is, for all initially positive densities, the evolvingdensities are bounded, and they eventually uniformly exceed some positive density.Stability and permanence are clearly closely related. Indeed, if an interior fixed pointis globally stable, so that it attracts all orbits starting in C0, then the system is per-manent, but there are permanent 3 species competitive systems with attracting limitcycles on the carrying simplex which are obviously not globally stable.

In the present paper, we are concerned with global stability of interior and bound-ary fixed points of (1). We introduce the notion of partial permanence in order toexpress permanence amongst a proper subset of populations, as is necessary for asystem where a boundary fixed point is globally attracting. Our approach is to furtherextend the Split Lyapunov function method that E.C. Zeeman and M. L. Zeeman in-troduced in [16] (see also [15]) to study the global attraction of an interior fixed pointof the totally competitive subclass (A,b > 0) of (1). Recently in [9] we have extendedthis split Liapunov function method to apply also to boundary fixed points, again forthe totally competitive case. In both [16,9] general results were obtained by first prov-ing convergence to fixed points on the carrying simplex Σ , an (N−1)−dimensionalattracting invariant manifold where the asymptotic dynamics of (1) (with A,b > 0) isknown to occur [2]. Here we prove convergence for classes of Lotka-Volterra systemsthat are not necessarily totally competitive, so that A may have entries of different signor zeroes. To the best of our knowledge, the existence of carrying simplices outsideof 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 providean alternative, more direct, proof of convergence using the split Lyapunov functionmethod. However, the price to be paid for relaxing the requirement that b and Ahave positive elements is that we will have to demonstrate uniform boundedness andpermanence or partial permanence for (1), whereas for the totally competitive casewhere 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 bedeveloped in the next section, although some of the results on permanence are newand important in their own right.

Recall that solutions of a system are called uniformly bounded if there is an M > 0such 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 −Ais a B-matrix. This condition is also necessary for uniform boundedness of solutionsof (1) for evey b ∈ IRN .

One of the many equivalent characterisations of B-matrices is: −A is a B-matrix ifand 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= 0there 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 condi-tions 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

x′i = xi fi(x), i ∈ IN , x ∈C, (2)

where fi ∈C1(IRN , IR). System (2) is called permanent if there is a compact set K⊂C0

such that for each x0 ∈C0, the solution of (2) starting at x0 satisfies x(t,x0) ∈ K forall sufficiently large t (which implies uniform boundedness). There are some well-known 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) ispermanent if there exists a p ∈C0 such that

pT (b−Ax)> 0 for all fixed points x on ∂C =C \C0. (3)

As well as ensuring that all interior trajectories are bounded, and bounded awayfrom 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 maybecome unbounded on the boundary ∂C. Thus the success of our method relies uponensuring that where necessary, an appropriate subset of components of N exceed apositive number for sufficiently large time. We will call this partial permanence. Therest 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

C0I = {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, ifthere are δ > 0 and M > δ such that

∀x0 ∈C with (∀i ∈ J,x0i > 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 sub-system of (2) on ∩i∈Iπi. Thus, in particular, J-permanence of (1) implies the existenceof a unique fixed point p in C0

I .


Lemma 1 Assume that the solutions of (2) are uniformly bounded. Assume also that,for a fixed i ∈ IN ,

∀x0 ∈ πi, ∃T (x0)> 0 such that∫ T (x0)

0fi(x(t,x0))dt > 0. (6)

Then there are δ > 0 and M > δ such that

∀x0 ∈C with x0i > 0, ∀ large t, δ ≤ xi(t,x0)≤M. (7)

Proof The first step is to show that there is a positively invariant set S such that allorbits lie in S after some finite time. The assumption of uniform boundedness ensuresthe 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 S be the closure of S. For each x0 ∈ S, thereis a time t(x0) after which the forward trajectory x(t,x0) through x0 ∈ S remains inS; that is, for each x0 ∈ S, there is a t(x0) ≥ 0 such that x(t,x0) ∈ S for all t > t(x0).Let S1 = {x(t,x0) : x0 ∈ S, t ≥ t(x0)} and let S1 be the closure of S1. Then S1 ⊂ S,S1 is compact, and S1 is positively inviarant. We claim that S1 is positively invariant.For if not, then there are some x1 ∈ S1 and some t1 > 0 such that x(t1,x1) 6∈ S1. Bycontinuous dependence of the solution on initial values, for all x0 ∈ S1 close enoughto x1 we would have x(t1,x0) 6∈ S1, a contradiction to the definition of S1. Then the setS = {x(t,x0) : x0 ∈ S, t ∈ [0,τ]} is compact, positively invariant and every solution of(2) will enter S in a finite time. Similarly, Si = S∩πi is compact, positively invariantand every solution in πi will enter Si in a finite time. So we need only prove (7) forx0 ∈ S.

By assumption (6), for each x0 ∈ Si, there is a T (x0)> 0 and a number h(x0)> 0satisfying

1T (x0)

∫ T (x0)

0fi(x(t,x0))dt > h(x0).

By continuity, there is an open neighbourhood U(x0) of x0 in S such that

∀y ∈U(x0),1

T (x0)

∫ T (x0)

0fi(x(t,y))dt > h(x0). (9)

Since Si is compact and {U(x0) : x0 ∈ Si} is an open cover of Si, so that Si has a finiteopen cover {U(xk) : xk ∈ Si,k ∈ Im}. Let

h = min{h(x1), . . . ,h(xm)}, (10)T0 = min{T (x1), . . . ,T (xm)}, (11)T1 = max{T (x1), . . . ,T (xm)}. (12)

Note that the set S = S\∪k∈ImU(xk) is also compact so that

δ0 = inf{xi : x ∈ S}> 0.


Now comes the key step, utilising (6) to show that solutions in the open cover∪k∈ImU(xk),but away from πi (i.e. satisfying x0

i > 0) cannot remain there for all time.We claim that

∀x0 ∈ ∪k∈ImU(xk)\ Si, ∃t1(x0)> 0 such that x(t1(x0),x0) ∈ S. (13)

Suppose this is not true. Then there is a k ∈ Im and an x0 ∈U(xk) \ Si such thatx(t,x0) 6∈ S for all t ≥ 0, so that since S \ Si is positively invariant, we have x(t,x0) ∈∪ j∈ImU(x j)\ Si for all t > 0. It then follows from (9)-(11) that

∫ T (xk)

0fi(x(t,x0))dt > T (xk)h(xk)≥ T0h > 0.

Since x(t,x0) is confined to ∪ j∈ImU(x j) \ Si for all t > 0, it must be the case thatx(T (xk),x0) ∈U(xs) for some s ∈ Im. Hence by (9)-(11) again,

∫ T (xk)+T (xs)

0fi(x(t,x0))dt > T0h+

∫ T (xs)

0fi(x(t,x(T (xk),x0)))dt > 2T0h.

Repeating the above process, we see that there is an increasing sequence of times trsuch that

∫ tr0 fi(x(s,x0))ds→+∞ as r→ ∞, so that

xi(tr,x0) = x0i exp

(∫ tr

0fi(x(s,x0))ds

)→+∞

as r→+∞. This contradicts the uniform boundedness of orbits and therefore showsour claim (13).

Now fix a number δ1 ∈ (0,δ0) and let

R = {x(t,x0) : x0 ∈ S with x0i ≥ δ1, t ∈ [0,T1]}. (14)

Then, as S \ Si is positively invariant, R⊂ S \ Si. Since R is the image of the solutionmap on [0,T1]×{x0 ∈ S : x0

i ≥ δ1}, which is compact, the continuity of x in (t,x0)implies the compactness of R. For each x0 ∈ S with x0

i = δ1, we have x0 ∈U(xk)\ Sifor some k ∈ Im so

xi(T (xk),x0) = x0i exp

(∫ T (xk)

0fi(x(t,x0))dt

)> δ1eT0h > δ1.

This (combined with the fact that T (xk)≤ T1) shows that the set R is forward invari-ant. By (13), each forward orbit x(t,x0) through x0 ∈ S \ Si will enter and stay in Rfor 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 asystem.


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-dimensionalrow (column) vector u (v) identical to a 1×N (N×1) matrix so that uv is regarded asthe 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 limt→+∞

x j(t) = 0 or liminft→+∞

x j(t)> 0,

(c) Aix < bi holds for each fixed point x ∈ 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)> 0satisfying ∫ T (x0)

0[bi−Aix(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 forany T (x0)> 0.

Suppose then that x0 6= 0. By assumption (b) we have the partition IN = J1 ∪ J2with i ∈ J1 such that

∀ j ∈ J1, limt→+∞

x j(t,x0) = 0; ∀k ∈ J2, liminft→+∞

xk(t,x0)> 0 (16)

(though J2 might be empty). By (16) and assumption (a), we can choose a sequence{tk} ⊂ (0,+∞) satisfying tk→+∞ as k→+∞,

∀ j ∈ J1, limk→+∞

1tk

∫ tk

0x j(t,x0)dt = 0 = y j, (17)

∀ j ∈ J2, limk→+∞

1tk

∫ tk

0x j(t,x0)dt = y j ∈ (0,+∞). (18)

For j ∈ J2 and k ≥ 1, from (1) we obtain

lnx j(tk,x0)− lnx0j

tk= b j−A j

1tk

∫ tk

0x(t,x0)dt. (19)

The left-hand side of (19) vanishes as k→+∞ due to the boundedness of lnx j(tk,x0)when j ∈ J2. Then (17)–(19) lead to

∀ j ∈ J2, b j−A jy = 0. (20)

This shows that y ∈ πi is a fixed point of (1). Thus, since bi−Aiy > 0 by (c), we have

1tk

∫ tk

0[bi−Aix(t,x0)]dt > 0

for large enough k so (15) holds for T (x0) = tk.


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 dimen-sional systems. In particular, condition (b) holds if every solution in πi converges toa fixed point (see [7], [8]). This will be demonstrated by examples later.

The following results on (partial) permanence are immediate consequences ofLemmas 1 and 2.

Theorem 3 Let J ⊂ IN with J 6= /0. Assume that the solution of (2) are uniformlybounded and that

∀ j ∈ J, ∀x0 ∈ π j, ∃T > 0 such that∫ T

0f 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∈Jπ j satisfies

limt→+∞

xk(t) = 0 or liminft→+∞

xk(t)> 0,

(c) ∀ j ∈ J, ∀ fixed point x ∈ π j, A j x < 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 competitivewith bi > 0, ai j ≥ 0 and aii > 0 for all i, j ∈ IN , by induction on N it can be shownthat condition (c) implies conditions (a) and (b). Thus, in that case, condition (c)alone implies permanence of (1). Indeed, condition (c) implies permanence of allsubsystems of (1) and their small perturbations (see [13, Corollary 3.1]). Furtherextension 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 criteriafor (1) to be globally asymptotically stable at a fixed point, which may be in theinterior 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−Aix), i ∈ IN . Explicitly, for i, j ∈ IN ,

∂Fi

∂x j

∣∣∣∣x=p

=

{bi−Ai p− piaii, i = j

−piai j, i 6= j.

Since pi > 0 implies bi−Ai p = 0, we have, for the ith row of B,

Bi =

{piAi, if pi > 0,(0 · · · 0Ai p−bi 0 · · · 0), if pi = 0. (21)

When p is an interior fixed point we obtain B = D(p)A.


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 αT 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

x =−B(x− p)−D(x− p)A(x− p).

By assumption, αT B = λαT . Now let ψ = αT (x− p). Then differentiating, and set-ting y = x− p,

ψ = αT x =−α

T By−αT D(y)Ay =−λψ− yT D(α)Ay. (23)

Let us also define V (x) = ∏Ni=1 xpiαi

i . Then

V = VN

∑i=1

piαixi

xi

= VN

∑i=1

piαi(bi−Aix) =−VN

∑i=1

piαiAi(x− p)

= −V αT B(x− p) =−λV α

T (x− p) =−λψV.

Summarising we find that,

ψ = −λψ− yT D(α)Ay, (24)V = −λψV. (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 andψ(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 ψ < 0at t = 0 and hence ψ(x(t,x0)) < 0 for all t > 0. Thus for t > 0 we have V > 0 andhence V (x(t,x0)) is increasing with positive time along the orbit. Since (1) is assumedpermanent, 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 non-empty, connected and invariant. Thus we have ω(y0)⊂ V−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 alls ∈ ω(y0) and by (26) and (24) y = 0 is the only invariant subset of ψ−1(0). Henceω(y0) = {0} and the trajectory x(t,x0) converges to p as t→+∞.


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 first case. Otherwise,ψ(x(t,x0))> 0 for all t > 0 so we would have V (t)=V (0)exp

(−λ

∫ t0 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 monotoni-cally, V → 0, so that ψ(y(tk,y0))→ 0 as k→ +∞ we have w ∈ ψ−1(0) and henceω(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 witha radius r by Br(p). We need to show that for any given ε > 0 there is a δ > 0 suchthat 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 = (α−11 ,0, . . . ,0)T , ψ

−1(0) = span{Wc2, . . . ,WcN}.

Then W is invertible. Let zT = (ψ, zT ) with z ∈ IRN−1 and y = x− p =Wz. We haveψ = (1,0, . . . ,0)z = αT (x− p) and, by (24),

ψ =−(λ +Θ(z))ψ− zT M0z, (27)

where Θ is linear with Θ(0) = 0 and M0 is an (N − 1)× (N − 1) real symmetricmatrix. Then condition (22) implies the positive definiteness of M0. From (27) wesee the existence of ε0 > 0 such that x0 ∈ Bε0(p) and ψ(x0) > 0 imply ψ < 0 att = 0.

For each l > 0, the set Sl = {x∈C : V (x) = l} defines an (N−1)-dimensional sur-face 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 V (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) thereis a t1(x0)< 0 such that ψ(x(t,x0))> 0 and x(t,x0) ∈Bε/2(p) for t ∈ (t1(x0),0) butx(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 posi-tively 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−Ax) > 0 for each boundary fixed point x. Suppose that there is aunique interior fixed point p and that B = D(p)A has a positive left eigenvector α .Then taking q = α , qT (b−Ax) = αT (b−Ax) = αT A(p− x) = −λψ(x). Hence thepermanence condition becomes that we require ψ(x)< 0 at all boundary fixed pointsx. This is certainly true if the carrying simplex Σ is convex (α is then the normal toΣ at p and all boundary fixed points lie below the tangent plane to Σ at p).


Remark 5 As discussed above, if A is a P−matrix then all orbits are uniformly boundedand there is a unique fixed point. On the other hand, any system (1) with locally sta-ble interior fixed point must have a D−stable matrix −A, since scaling the axes doesnot change the local stability of the interior fixed point, and so A must then be aP0−matrix, that is, a matrix whose principal minors are non-negative (see, for exam-ple, Theorem 2.5.8 in [4]).

Remark 6 If α can be chosen to be positive then we do not need to prove permanencefor the theorem proof to work, but then we must prove existence of an interior fixedpoint, 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 thusdo not need to assume permanence. It can be shown that if A > 0 all trajectories areuniformly bounded. However, we still need to show existence of an interior fixedpoint.

For a boundary fixed point p of (1) to be stable in C, it is necessary that it issaturated (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 C0I is saturated.

(c) The matrix D(p)A has an eigenvalue λ > 0 and an associated left eigenvectorα ∈ 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 αT y = 0. (28)

Then (1) at p is globally asymptotically stable.

Proof We first rewrite (1) as

x =−B(x− p)−D(x− p)A(x− p),

where B is given by (21). Now let ψ = αT (x− p) and y = x− p. Then ψ = αT x =−αT By−αT D(y)Ay. As yi = xi for i ∈ I, by condition (c) we can further write

ψ =−λψ−∑i∈I

αi(Ai p−bi)xi− yT D(α)Ay. (29)

With V (x) = ∏i∈J xpiαii , we have

V = V ∑i∈J

piαixi

xi

= V ∑i∈J

piαi(bi−Aix) =−V ∑i∈J

piαiAi(x− p)

= −λV αT (x− p) =−λψV.


Thus, instead of (24) and (25) in the proof of Theorem 5, we obtain

ψ = −λψ−∑i∈I

αi(Ai p−bi)xi− yT D(α)Ay, (30)

V = −λψV. (31)

Since p is saturated by condition (b), and αi > 0 for all i∈ I by condition (c), for x∈Cthe term ∑i∈I αi(Ai p− bi)xi is always nonnegative. Then, replacing permanence bypartial 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 arerestricted to C only. Referring to the proof of Theorem 5, we note that (30) implies

ψ ≤−(λ +Θ(z))ψ− zT M0z (32)

instead of (27), the set Vl is bounded by Sl , ψ−1(0) and ∪i∈Iπi, and the open subsetof Bε/2(p)∩C relative to C bounded “below” by Sl , “surrounding” by Γ and ∪i∈Iπi,and “above” by the upper boundary of Bε/2(p) satisfying ψ > 0 contains p and ispositively invariant. Then the stability follows.

3.1 Checking the positive definite condition (28)

In what follows, for a given square matrix M, MS = M +MT . Let Wc2, . . . ,WcN beN− 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)Ss≥ 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 interms of D(α) and A. For simplicity we choose

[Wc2, . . . ,WcN ] = D(α)−1

1 0

−1. . .. . . 1

0 −1

(34)

and denote by Mi j the submatrix of M obtained by deleting the ith row and the jthcolumn of M for any square matrix M, then

U = (AD(α)−1)S11 +(AD(α)−1)S

NN− (AD(α)−1)S1N− (AD(α)−1)S

N1. (35)

Thus in checking the condition (22) we simply check that U is positive definite.


4 Examples and discussion

Example 1

Consider system (1) with

A =

5 −1 14 3 1−1 − 1

2 2

, b =13

254626

. (36)

Note that −A is a B-matrix so, by Theorem 1, the solutions of (1) with (36) areuniformly bounded. The boundary fixed points xT are: (0,0,0), ( 5

3 ,0,0), (0,469 ,0),

(0,0, 133 ), ( 121

57 , 13057 ,0), ( 8

11 ,0,15533 ), (0, 132

39 , 20239 ). If q = (1,1,1)T ∈C0, we can check

that qT (b−Ax) > 0 for all x. Thus, by Theorem 2, (1) is permanent. The systemhas a unique interior fixed point p = (1,2, 16

3 )T . The matrix D(p)A has a positiveeigenvalue λ = 8 and an associated left eigenvector αT = (48,4,−21). Then, from(35) we have

(AD(α)−1)S =

524 − 1

6 −23336

− 16

32 − 29

168− 23

336 −29168 −

421

, U =

( 4924 −

8548

− 8548

13984

).

Since U is positive definite, the condition (22) is satisfied. By Theorem 5, (1) with(36) at p is globally asymptotically stable.

Example 2


A =

1 0 −θ

1 1 01 1 1

, θ = 9, 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 thematrix −DA is stable) so −D(p)A is stable and the system at p is locally asymptoti-cally 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 thatp is globally stable. An application of Theorem 5 to (1) does not yield conclusiveresults: D(p)A has a positive eigenvalue λ = 0.38732 with a real left eigenvectorαT = (0.0056859,−0.3613,1) that yields a matrix U with real eigenvalues of op-posite 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 isa global attractor. Note that this example provides an interesting attracting compactinvariant 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


02

46

8x

0

1

2

3

y

0.00.51.01.52.0

z

Fig. 1 Phase plot for Example 2 when θ = 9. The plot suggests that the interior steady state is globallyattracting 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 curvesfrom the two axial fixed points spiralling in towards to interior fixed point. But thetwo 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 giveus a clue that the geometric shape of M may be complicated; part of it joining p toM∩π2 may look like a spiralling cone with p as a vertex. This indicates that M doesnot have a tangent plane at p, which might be the essential reason for the failure ofthe application of Theorem 5.

Example 3


A =

2 0 11 3 23 − 1

2 4

, b =

271

. (38)

Clearly,−A is a B-matrix so Theorem 1 implies the uniform boundedness of solutionsof (1) with (38). There are five fixed points in π1∪π2: pT

0 = (0,0,0), pT1 = (1,0,0),

pT2 = (0, 7

3 ,0), pT3 = (0,0, 1

4 ), pT4 = (0,2, 1

2 ), 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 analysison 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 toJ = {1,2}. Thus, it has a unique fixed point p = (1,2,0)T in C0

{3}. As A3 p > b3, p issaturated. The matrix D(p)A has a positive eigenvalue λ = 6 and an associated left


01

23

4x

0

1

2

3

y

0.00.5

1.0

1.5

2.0

z

Fig. 2 The global stability of Example 2 when θ = 5. In this case the unique interior fixed point canbe shown to be globally attracting on C0 by the Volterra-Lyapunov theorem, since (DA)S > 0 when D =diag(1,3,10).

eigenvector αT = (2,4,3). By (35) we have

(AD(α)−1)S =

2 12

116

12

32

1324

116

1324

83

, U =

( 52 − 55

24− 55

243712

).

It can be checked that U is positive definite so the condition (22) holds. By Theorem6, (1) with (38) at p is globally asymptotically stable.

Example 4


A =

2 0 1 11 3 2 43 − 1

2 4 12 1

212 2

, b =

2714

. (39)

Again, −A is a B-matrix so the solutions of (1) with (39) are uniformly boundedby 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 convergesto a fixed point. Moreover, adding 0 as the fourth component to each of the fixedpoints pi(0 ≤ i ≤ 4) and p in Example 3, we see that A4 p < b4 and A4 pi < b4 forall 0≤ i≤ 4. By Theorem 4 and Remark 2, (1) with (39) is partially permanent withrespect to J = {4}. The unique fixed point in C0

I (I = {1,2,3}) is x∗ = (0,0,0,2)T .


As Aix∗ ≥ bi for all i ∈ I, x∗ is saturated. The matrix D(x∗)A has a positive eigenvalueλ = 4 and an associated left eigenvector αT = (2, 1

2 ,12 ,2). By (35) we have

(AD(α)−1)S =

2 1

272

32

12 12 3 372 3 16 3

232 3 3

2 2

, U =

13 −12 2−12 22 − 29

22 − 29

2 15

.

Since∣∣ 13 −12−12 22

∣∣ > 0 and detU = 4.75 > 0, U is positive definite so the condition (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 competitiveLotka-Voletrra systems can be used without reference to the carrying simplex, and hasa fairly general application to both competitive and non-competitive Lotka-Volterrasystems. Unlike the competitive case, however, we have to additionally show per-manence or partial permanence of the dynamics. Our results can be successfully ap-plied to a wide range of Lotka-Volterra systems, but there remain cases where thetheory does not give conclusive results as shown by example 2. For 3-species com-petitive systems, there are examples found by Zeeman and Zeeman where the systemis known to be globally convergent to an interior fixed point on C0, but the Split Lia-punov method fails. As indicated by the carrying simplex plots in [16], this appearsto be when the carrying simplex has negative curvature. It would be interesting toinvestigate when an attracting invariant manifold analogous to the carrying simplexexists for non-competitive systems, and how the Gaussian curvature at fixed pointsrelates to their stability. Further work is required to determine whether the Split Lya-punov method can be further adapted to deal with indeterminate cases, or whether anew approach is necessary.

References

1. AHMAD, S & LAZER, A.C. Average growth and total permanence in a competitive Lotka-Volterrasystem, Ann. Mat. 185, S47–S67, (2006).

2. HIRSCH, M. W. Systems of differential equations that are competitive or cooperative III: competingspecies, Nonlinearity, 1, 51–71 (1988).

3. HOFBAUER, J. & SIGMUND, K. The Theory of Evolution and Dynamical Systems, Cambridge Uni-versity Press, New York (1998).

4. HORN, R. A. & JOHNSON, C. R. Matrix Analysis, Cambridge University Press., Cambridge (1985).5. Z. Hou, Global attractor in autonomous competitive Lotka-Volterra systems, Proceedings of American

Mathematical Society 127, 3633-3642 (1999).6. Z. Hou, Global attractor in competitive Lotka-Volterra systems, Math. Nachr. 282, No. 7, 995–1008

(2009).7. Z. Hou, Vanishing components in autonomous competitive Lotka-Volterra systems, J. Math. Anal.

Appl. 359, 302–310 (2009).8. Z. Hou, Oscillations and limit cycles in competitive Lotka-Volterra systems with delays. Nonlinear

Analysis: Theory, Method and Applications 75 (2012) 358-370.


9. HOU, Z. & BAIGENT, S. Fixed point Global Attractors and Repellors in Competitive Lotka-VolterraSystems. Dynamical Systems, 26, Issue 4, 367–390 (2011).

10. HOU, Z. On permanence of all subsystems of competitive Lotka-Volterra systems with delays, Non-linear Analysis: RWA 11, 4285–4301 (2010).

11. JANSEN, W. A permanence theorem for replicator and Lotka-Volterra systems, J. Math. Biol., 25,411–422 (1987).

12. LIANG, X. & JIANG, J. The dynamical behaviour of type-K competitive Kolmogorov systems andits application to three-dimensional type-K competitive LotkaVolterra systems. Nonlinearity 16, 785801(2003).

13. MIERCZYNSKI, J. & SCHREIBER, S. J. Kolmogorov vector fields with robustly permanent subsys-tems, J. Math. Anal. Appl. 267, 329–337 (2002).

14. TAKEUCHI, Y. Global Dynamical Properties of Lotka-Volterra Systems. World Scientific (1996).15. TINEO, A. May Leonard Systems. Nonlinear Analysis: Real World Applications, 9, 1612–1618

(2007).16. ZEEMAN, E. C. & ZEEMAN, M. L. From local to global behavior in competitive Lotka-Volterra

systems, Trans. Amer. Math. Soc. 355, 713–734 (2003) .

global stability of interior and boundary ﬁxed …global stability of interior and boundary ﬁxed...

Documents