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The Coexistence of a Community of Species with Limited Competition
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
217, 233]245 Ž1998.
AY975711
The Coexistence of a Community of Species with Limited Competition Tu Caifeng and Jiang Jifa* Department of Mathematics, The Uni¨ ersity of Science and Technology of China, Hefei, China Submitted by Hal L. Smith Received March 17, 1997
This paper is a study of a system modeling a biological community of species with limited competition. The community consists of two competing subcommunities, all species of which cooperate, and some species of one subcommunity can invade the steady state of another subcommunity, whereas others cannot. Sufficient conditions are given that all species can coexist. For Lotka]Volterra systems we can improve this result by showing that there is a unique, globally asymptotically stable steady state. Q 1998 Academic Press
1. INTRODUCTION Consider the model of a biological community of species,
˙x i s x i f i Ž x 1 , x 2 , . . . , x n . ,
1 F i F n,
x i G 0,
Ž S.
in which x i represents the population density of the ith species and f i Ž x . represents the per capita growth rate of the ith species. Assume that the species community can be divided into two groups, I s 1, . . . , k 4 , J s k q 1, . . . , n4 , 0 F k F n, satisfying the requirement that the species in group I and J cooperate, and that species i and species j compete if i g I and j g J. This kind of system is called a system with limited competition. Mathematically, this situation can be described in terms of the Jacobian of f s Ž f 1 , . . . , f n . in the following manner: Df Ž x . s
ž
A yC
yB , D
/
Ž 1.1.
* Project supported by the National Natural Science Foundation of China. 233 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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where A and D have nonnegative off-diagonal elements and B G 0, C G 0. It is assumed that f is a continuously differentiable function n . defined on some open set containing the nonnegative orthant Rq If L is a nonempty subset of N s 1, 2, . . . , n4 , then the set Hq Ls x g n Rq : x p s 0, for p f L4 is an invariant set for ŽS.. The subsystem
˙x l s x l f l Ž x . ,
l g L,
x g Hq L
Ž SL .
will play an important role in this paper. In w1x, Smith extended the theories for cooperative systems developed by Hirsch w2, 3x to systems Ž S . and presented sufficient conditions for the persistence of all species. His essential conditions are that Ž SI . possesses a nyk positive steady state x 01 that is unstable to Rq , and Ž S J . possesses a 2 k positive steady state x 0 that is unstable to Rq , that is x 01 can be invaded by each species j g J and x 02 can be invaded by each species i g I. In mathematical language, this condition can be expressed as f j Ž x 01 , 0. ) 0 for all j g J and f i Ž0, x 02 . ) 0 for all i g I. This implies that f Ž0. ) 0 and the system Ž S . is nonobligate. But as mentioned by Smith w1, p. 870x, in the general case where positive steady states of Ž SI . and Ž S J . exist, we might imagine that some of the species j g J can invade the steady state for Ž SI . whereas others cannot, and similarly for the positive sready state for Ž S J .. Few of results of Smith w1x apply to this situation. Therefore, Smith’s results in w1x cannot be applied to those systems like the following system:
˙x s diag Ž x . Ž r q Mx . ,
4 x g Rq ,
r g R4 ,
Ž 1.2.
where y18 12 Ms y3 y3
12 y36 y1 y1
y1 y1 y12 6
y6 y6 18 y24
0
and
y6 108 rs . 60 y6
0
In this example, the subsystem Ž SI . has a unique positive steady state x 01 s Ž 157 , 267 . with f 3 Ž x 01 , 0. ) 0, f 4Ž x 01 , 0. - 0, and the subsystem Ž S J . has a unique positive steady state x 20 s Ž 375 , 58 . with f 1Ž0, x 02 . - 0, f 2 Ž0, x 02 . ) 0, where f Ž x . s Ž f 1 Ž x . , f 2 Ž x . , f 3 Ž x . , f 4 Ž x . . s r q Mx. This system does not satisfy the conditions in w1, Theorem 3.6x, but it has a globally asymptotically stable positive steady state Ž1, 3, 6, 1. in the interior of the nonnegative orthant. In this paper we shall pay attention to this situation. Using the idea of Smith w1x, we shall present some sufficient conditions such that all species can persist uniformly. These results generalize Theorem 3.6 of Smith w1x. Applying our general results, we can sharpen the result with the
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Lotka]Volterra system,
˙x s diag Ž x . Ž r q Mx . ,
n x g Rq ,
r g Rn,
and obtain sufficient conditions such that the two groups I and J can coexist in a globally asymptotically stable steady state in the case where r is not positive.
2. THE MAIN RESULTS In this section we will agree on some notation and establish some conventions that will be employed throughout this work. In addition, the main result will be stated. n Let Rq s x g R n : x i G 0, 1 F i F n4 denote the nonnegative orthant n n and Rq s x g Rq : x i ) 0, 1 F i F n4 denote its interior. In this paper one of the important notations is a cone. Recall that a cone K in R n is a nonempty closed subset of R n with the properties Rq K ; K ,
KqK;K
and
K l Ž yK . s 0 4 ,
where Rq is the set of nonnegative real numbers. If the partial order relation is generated by a cone K, we write x Fk y Ž x -K y . whenever y y x g K Ž y y x g Int K .. If x, y g R n and K is a cone in R n, we let n w x, y x K s z g R n : x FK z FK y4 . We drop the K if the cone is Rq . Fix k ny k . 1 k 2 Ž 1 F k F n. Then K s Rq = yRq is a cone. Let x g R , x g R nyk ; it is convenient to write x s Ž x 1, x 2 . g R n. Then x FK y, where x s Ž x 1, x 2 ., y s Ž y 1, y 2 . implies x 1 F y 1 , x 2 G y 2 . k ny k .. k If M is an n = n matrix such that Ž M q l I .Ž Rq = ŽyRq ; Rq = ny k . ŽyRq Ž for sufficiently large l, then M must be of the structure 1.1.. We call such a matrix M a type K matrix in Smith’s meaning Žsee w1x., and call such a system ŽS. a type K monotone system if Df Ž x . has a type K structure. Let N s 1, 2, . . . , n4 , I s 1, 2, . . . , k 4 , and J s k q 1, . . . , n4 , where n is the dimension of our Euclidean space R n. If L, P are nonempty sets of N such that L > I and P > J, then L s N y L and P s N y P denote their complementary sets in N. Hereafter, u, u, and u 0 , etc., will always aL represent vectors in aL-dimensional Euclidean space Rq ; ¨ , ¨ , and ¨ 0 , aL etc. in aL-dimensional Euclidean space Pq ; w, w, and w 0 , etc., aP aP in Rq ; and z, z, and z 0 , etc., in Rq . x 1 , x 2 will always represent k nyk n vectors in Rq and Rq , respectively. Then x s Ž x 1, x 2 . g Rq , f Ž x. s 1 2 1 2 Ž f 1Ž x , x ., f 2 Ž x , x ... Without loss of generality, we may assume L s 1, 2, . . . , k, k q 1, . . . , l 4 , k - l - n and P s n y p q 1, . . . , k q 1, . . . n4 , p ) n y k. Let x s Ž u, ¨ .
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and f Ž x . s Ž f LŽ u, ¨ ., f LŽ u, ¨ ... Then we can rewrite the system Ž S . as
½
u ˙ s diag Ž u . f LŽ u, ¨ . ¨˙ s diag Ž ¨ . f L Ž u, ¨ .
,
aL u g Rq ,
aL ¨ g Rq .
Ž S1 .
Let x s Ž z, w . and f Ž x . s Ž f P Ž z, w ., f P Ž z, w ... Then we can rewrite the system Ž S . as
½
˙z s diag Ž z . f P Ž z, w . , w ˙ s diag Ž w . f P Ž z, w .
aP z g Rq ,
aP w g Rq .
Ž S2 .
Set ¨ s 0 and z s 0 in Ž S1 . and Ž S2 ., respectively; we obtain two subsystems, u ˙ s diag Ž u . f LŽ u, 0 . ,
aL , u g Rq
Ž SL .
aP w g Rq .
Ž SP .
and w ˙ s diag Ž w . f P Ž 0, w . ,
Because Df Ž u, ¨ . and Df Ž z, w . are type K matrices as in Ž1.1., Df LŽ u, 0. is a type K 1 submatrix of Df Ž u, ¨ ., and Df P Ž0, w . is a type K 2 submatrix of k lyk . ny k . Df Ž z, w ., where K 1 s Rq = ŽyRq and K 2 s Rqpq kyn = ŽyRq . We write f t Ž x . for the unqiue solution x Ž t . of Ž S . satisfying x Ž0. s x, and f tL Ž u., f tP Ž w . are the unique solutions uŽ t . of Ž SL . and w Ž t . of Ž SP ., respectively, satisfying uŽ0. s u and w Ž0. s w. Ž f tL Ž u..I consists of components f tL Ž u.4i of f tL Ž u. for all i g I; Ž f tP Ž w .. J is defined similarly. n It is also necessary to give the following useful definitions. x g Rq is said to be a steady state of Ž S . if F Ž x . s 0. Denote the v-limit set of f t Ž x . as n v Ž x . s y g Rq : f t kŽ x . ª y,
for some sequence t k ª q` 4 .
The domain of attraction of a steady state x is the set of initial conditions x for which v Ž x . s x 4 . We say x is globally asymptotically stable with respect to a set W if x is asymptotically stable and W belongs to the domain of attraction of x. Throughout this paper, denote the cardinality of L and P, respectively, l nyl as l s aL, p s aP. If x s Ž u, ¨ ., then u g Rq and ¨ g Rq . If x s ny p p 1 2 1 k Ž z, w ., then z g Rq and w g Rq . If x s Ž x , x ., then x g Rq and 2 ny k x g Rq . Our main results are as follows.
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THEOREM 2.1. Assume that u 0 is a positi¨ e steady state for Ž SL ., f LŽ u 0 , 0. ) 0 and w 0 is a positi¨ e steady state for Ž S P ., f P Ž0, w 0 . ) 0. Let Ž0, w 0 . FK Ž u 0 , 0.. Then there exist positi¨ e steady states x and ˜ x for Ž S . with the following properties: Ž1. 0 - x, ˜ x F ŽŽ u 0 .I , Ž w 0 . J ., and x FK ˜ x. 0 Ž2. If x ) 0 and ˜ x FK x FK Ž u , 0., then f t Ž x . ª ˜ x as t ª q`. 0. Ž Ž . If x ) 0 and 0, w FK x FK x, then f t x ª x as t ª q`. Ž3. If x ) 0 and Ž0, w 0 . FK x FK Ž u 0 , 0., then v Ž x . ; w x, ˜ xxK . In addition, if u 0 q K 1 lies in the domain of attraction of u 0 for Ž SL ., and w y K 2 lies in the domain of attraction of w 0 for Ž SP ., then v Ž x . s x 4 for all x ) 0 with x FK x, v Ž x . s ˜ x 4 for all x ) 0 with x GK ˜ x, and v Ž x . ; w x, ˜ x x K for all x ) 0. 0
Applying Theorem 2.1, we can deduce a better result for the Lotka]Volterra system:
˙x s diag Ž x . Ž r q Mx . ,
n x g Rq ,
r g Rn,
Ž S.
under the assumption that M is stable Žthe definition can be found in section 4., and M is a type K matrix as in Ž1.1.. THEOREM 2.2. Assume that the Lotka]Volterra system Ž S . satisfies the following conditions. Ž1. There exists L ; N with L > I such that Ž S L . has a positi¨ e steady state u 0 and f LŽ u 0 , 0. ) 0. Ž2. There exists P ; N with P > J such that Ž S P . has a positi¨ e steady state w 0 and f P Ž0, w 0 . ) 0. Then Ž S . has a unique positi¨ e steady state that is globally asymptotically n stable relati¨ e to R˙q . Remark 2.1. Theorem 3.6 in w1x is the special case where L s I, P s J in Theorem 2.1. Remark 2.2. Let x 01 be a positive steady state of Ž S I .. In Theorem 2.1, the condition f LŽ u 0 , 0. ) 0 cannot guarantee that f J Ž x 01 , 0. ) 0. This is because for some j g J, it is possible that f j Ž x 01 , 0. - 0. We can know this fact from Ž1.2.. Remark 2.3. Recall that the system Ž S . is permanent if there exist constant numbers d , D ) 0 such that for every solution x Ž t . of Ž S ., d F lim inf t ªq` x Ž t . F lim sup t ªq` x Ž t . F D for all i g N. We notice that under the hypotheses of Theorem 2.1, together with its additional assumptions, the block w x, ˜ x x K attracts all solutions with positive initial data. Thus the system Ž S . is permanent.
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3. THE PROOF OF MAIN RESULT It is useful to establish some preliminary results before giving the proof of Theorem 2.1. The fundamental tool used in this paper is the generalized Kamke theorem. This theorem is extended in a natural way from cooperative systems to type K monotone systems Žsee w1, Theorem 2.4x.. THEOREM 3.1 ŽKamke Theorem.. Assume that the system Ž S . is type K monotone, and x Ž t ., y Ž t . are the solutions of Ž S . defined on a F t F b with x Ž a. FK y Ž a.. Then x Ž t . FK y Ž t . for all t g w a, b x. In w1, p. 862x, Smith extended a criterion for the monotonicity of every component of a solution for a cooperative system given by Selgrade w5x to a type K monotone system. Now we quote it here. THEOREM 3.2. Let the system Ž S . be a type K monotone system and let n f Ž x . GK 0 for some x g Rq . Then f t Ž x .4i is nondecreasing if i g I and f t Ž x .4j is nonincreasing if j g J for all t G 0 for which the solution exists. A similar result holds if f Ž x . FK 0. The existence of a steady state for Ž S . shall be related to the existence of the steady states for Ž SL . and Ž SP .. This is reflected by the following proposition. PROPOSITION 3.3. Assume that x s Ž x 1 , x 2 . is a positi¨ e steady state of l nyl nyp Ž S .. Rewrite x s Ž u, ¨ . or x s Ž z, w ., where u g Rq , ¨ g Rq , z g Rq , p w g Rq . Then Ž1. Ž x q K . l Hq L is positi¨ ely in¨ ariant and either has a nonnegati¨ e steady state or f tL Ž u. is unbounded for e¨ ery u GK 1 u and t G 0. Ž2. Ž x y K . l Hq P is positi¨ ely in¨ ariant and either has a nonnegati¨ e steady state or f tP Ž w . is unbounded for e¨ ery w FK 2 w and t G 0. Proof. We only prove Ž1., since Ž2. can be shown in a similar way. Consider the subsystem u ˙ s diag Ž u . f LŽ u, 0 . ,
l u g Rq .
Let x s Ž u, ¨ .. By the given condition, we have f LŽ u, ¨ . s 0. If i g I ; L, then f i Ž u, 0. G f i Ž u, ¨ . s 0 because f i Ž u, ¨ .r ¨ F 0. If j g J l L, then f j Ž u, 0. F f j Ž u, ¨ . s 0, because f j Ž u, ¨ .r ¨ G 0. Thus we have k lyk . f LŽ u, 0. GK 1 0, where K 1 s Rq = ŽyRq . By Theorem 3.2, f t Ž u, 0.4i is nondecreasing for i g I and f t Ž u, 0.4j is nonincreasing for j g J l L. Let Ž f t Ž u, 0..L s f tL Ž u.. Therefore either f tL Ž u.: t G 04 is unbounded or f tL Ž u.4 converges to a nonnegative steady state u 0 GK 1 u as t tends to
SPECIES WITH LIMITED COMPETITION
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infinity. Obviously, Ž u 0 , 0. GK Ž u, ¨ . s x; this implies that Ž u 0 , 0. g x q K. LŽ . LŽ . Ž u 0 , 0. g Ž x q K . l Hq Since Ž u 0 , 0. g Hq L, L . Because f t u GK 1 f t u LŽ . LŽ . holds for every u GK 1 u, f t u is unbounded for t G 0 if f t u is unbounded. Because x p s 0 for p f L, f t Ž x .4p ' 0 holds, Hq is invariant. L If x GK x, then f t Ž x . GK x for all t ) 0 by Theorem 3.1, that is, f t Ž x . g x q K for t G 0. This implies that x q K is positively invariant. Therefore, Ž x q K . l Hq L is also positively invariant. The proposition will be used in the proof of our main result. PROPOSITION 3.4.
n Let x s Ž x 1, x 2 . s Ž u, ¨ . s Ž z, w . g Rq . Then
f t Ž x . F Ž Ž f tL Ž u . . I , Ž f tP Ž w . . J .
for t G 0.
l nyl Proof. Let x s Ž u, ¨ ., where u g Rq , ¨ g Rq . Then Ž u, ¨ . FK Ž u, 0.. By Theorem 3.1,
f t Ž x . FK f t Ž u, 0 . s Ž f tL Ž u . , 0 . s Ž Ž f tL Ž u . . I , Ž f tL Ž u . . LrI , 0 .
for t G 0.
Thus for t G 0,
Ž ft Ž x . . I F Ž ftL Ž u . . I .
Ž 3.1.
ny p Similarly, let x s Ž z, w ., where z g Rq , w g Rqp . Then Ž z, w . GK Ž0, w . and
f t Ž x . GK f t Ž 0, w . s Ž 0, f tP Ž w . . s Ž 0, Ž f tP Ž w . . PrJ , Ž f tP Ž w . . J .
for t G 0.
Hence for t G 0,
Ž ft Ž x . . J F Ž ftP Ž w . . J .
Ž 3.2.
From Ž3.1. and Ž3.2., we derive that
f t Ž x . F Ž Ž f tL Ž u . . I , Ž f tP Ž w . . J .
for t G 0.
Proposition 3.3 provides some relation between a positive steady state x for Ž S . and nonnegative steady states u 0 for Ž SL . and w 0 for Ž SP .. Hence to prove the existence of a nonnegative steady state Ž S ., it is natural to suppose the existence of positive steady states u 0 for Ž SL . and w 0 for Ž SP ..
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LEMMA 3.5. Assume that w 0 is a positi¨ e steady state for Ž SP . and f P Ž0, w 0 . ) 0, and u 0 is a positi¨ e steady state for Ž S L .. Let Ž0, w 0 . FK Ž u 0 , 0.. Then there is a steady state x s Ž x 1, x 2 . for Ž S . with the following properties: Ž1. Ž0, w 0 . FK x FK Ž u 0 , 0. and x i ) 0 for i g I. Ž2. If x ) 0, Ž0, w 0 . FK x FK x, then f t Ž x . ª x as t ª q`. kqpyn ny k . Ž3. x s Ž z, w . ) 0, w GK w 0 , where K 2 s Rq = ŽyRq , then 2 v Ž x . ; x q K. Ž4. If f j Ž u 0 , 0. ) 0 for some j g L, then x j ) 0. Proof. Ž1. By the continuity of f, f P Ž0, w 0 . ) 0 implies that f P Ž z, w 0 . ) 0 for sufficiently small z ) 0, where P ; I. For i g I l P, f i Ž z, w 0 . G f i Ž0, w 0 . s 0, because f i Ž z, w .r z G 0, and for j g J, f j Ž z, w 0 . F f j Ž0, w 0 . s 0, because f j Ž z, w .r z F 0. Hence f Ž z, w 0 . GK 0 for z sufficiently small. By Theorem 3.2, for sufficiently small z ) 0, f t Ž z, w 0 .4i is nondecreasing in t G 0 for i g I and f t Ž z, w 0 .4j is nonincreasing in t G 0 for j g J. Since Ž0, w 0 . FK Ž u 0 , 0., we have Ž0, w 0 . FK Ž z, w 0 . FK Ž u 0 , 0. for sufficiently small z ) 0. By Theorem 3.1, Ž0, w 0 . FK f t Ž z, w 0 . FK Ž u 0 , 0. for all t G 0. Hence f t Ž z, w 0 . is bounded. Consequently, f t Ž z, w 0 . converges to x as t ª q`, and Ž0, w 0 . FK x FK Ž u 0 , 0.. It is clear that x 1 G Ž z, w 0 .4I ) 0. Ž2. If x ) 0, Ž0, w 0 . FK x FK x, then there exists sufficiently small z ) 0 such that Ž z, w 0 . FK x FK x and f t Ž z, w 0 . FK f t Ž x . FK x. Because f t Ž z, w 0 . ª x as t ª q` by the proof of Ž1., f t Ž x . ª x as t ª q`. Ž3. Let x s Ž z, w . ) 0. Since w GK w 0 , we fix a sufficiently small z 0 2 such that z G z 0 . Then x s Ž z, w . GK Ž z 0 , w 0 . and f t Ž z, w . GK f t Ž z 0 , w 0 . for t ) 0. We have v Ž x . GK x 4 for f t Ž z 0 , w 0 . ª x as t ª q`. This means that v Ž x . ; x q K. Ž4. Suppose that f j Ž u 0 , 0. ) 0 for some j g L. Let x s Ž u, ¨ .. If u FK u 0 1 and ¨ G 0 with ¨ j s 0, then f j Ž u, ¨ . G f j Ž u 0 , 0. ) 0 for j g L. Let x s Ž u, ¨ .. Since Ž0, w 0 . FK Ž u, ¨ . FK Ž u 0 , 0., u FK u 0 . 1 If x j s ¨ j s 0 for some j g L, then f j Ž x . s f j Ž u, ¨ . G f j Ž u 0 , 0. ) 0 for j g L. Let u s Ž z,). and u 0 s Ž z 0 , )., where ) denotes other components of u and u 0 . Then for fixed sufficiently small z ) 0, we find that f t Ž z, w 0 . ª x as t ª q`. Since f j Ž x . ) 0, f j Ž f t Ž z, w 0 .. ) 0 for large enough t. But as t ª q`, f t Ž z, w .4j ª x j s 0, which is a contradiction. As a result, x j ) 0. The proof is completed. Remark 3.1. If the condition f P Ž0, w 0 . ) 0 in Lemma 3.5 is replaced by f LŽ u 0 , 0. ) 0, there are similar conclusions. If f LŽ u 0 , 0. ) 0 and f P Ž0, w 0 . ) 0 hold symmetrically, we obtain our main result, Theorem 2.1.
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Before proving our main result, we again stress that x s Ž u, ¨ . with l nyl ny p u g Rq , ¨ g Rq and x s Ž z, w . with z g Rq , w g Rqp , if necessary. Proof of Theorem 2.1. Ž1. By Lemma 3.5, for sufficiently small z ) 0, f t Ž z, w 0 . ª x as t tends to infinity, and x 1 ) 0.
Ž 3.3.
Similarly, fix sufficiently small ¨ such that Ž0, w 0 . FK Ž u 0 , ¨ . FK Ž u 0 , 0.; then Ž0, w 0 . FK f t Ž u 0 , ¨ . FK Ž u 0 , 0. for all t G 0. Hence f t Ž u 0 , ¨ . ª ˜ x as t ª q`, Ž0, w 0 . FK ˜ x F Ž u 0 , 0., and
˜x 2 G Ž u 0 , ¨ . 4 J ) 0.
Ž 3.4.
Let z ) 0, ¨ ) 0 be small enough such that Ž z, w 0 . FK Ž u 0 , ¨ .. Then f t Ž z, w 0 . FK f t Ž u 0 , ¨ . for t ) 0. So x FK ˜ x, namely, Ž x 1, x 2 . FK Ž ˜ x 1, ˜ x 2 .. This means that
˜x 1 G x 1 ) 0,
x2 G ˜ x 2 ) 0.
Ž 3.5.
Combining Ž3.3., Ž3.4., and Ž3.5., we conclude that x ) 0 and ˜ x ) 0. By Proposition 3.4, for x s Ž u, ¨ . s Ž z, w . and t ) 0, f t Ž x . F ŽŽ f tL Ž u..I , Ž f tP Ž w .. J . holds. Then for t G 0,
f t Ž z, w 0 . F Ž Ž f tL Ž u . . I , Ž f tP Ž w 0 . . J . and
f t Ž u 0 , ¨ . F Ž Ž f tL Ž u 0 . . I , Ž f tP Ž w . . J . . By a limit process, this derives x 2 F Ž w 0 . J and ˜ x 1 F Ž u 0 .I . It follows from 1 1 0 2 2 0 Ž3.5. that x F ˜ x F Ž u .I and ˜ x F x F Ž w . J . Therefore, 0 - x, ˜ xF ŽŽ u 0 .I , Ž w 0 . J .. Ž2. It follows from Lemma 3.5 and Remark 3.1 that if x ) 0 and Ž0, w 0 . FK x FK x, then f t Ž x . ª x as t ª q`, and ˜ x FK x FK Ž u 0 , 0., ft Ž x . ª ˜ x as t ª q`. Ž3. Let x s Ž z, w . ) 0 with w GK w 0 . Then from Ž3. of Lemma 3.5, 2 v Ž x . ; x q K. Similarly, if x s Ž u, ¨ . ) 0 with u FK 1 u 0 , then v Ž x . ; ˜x y K. If x ) 0 and Ž0, w 0 . FK x FK Ž u 0 , 0., then u FK 1 u 0 by rewriting x to be x s Ž u, ¨ .. This yields
vŽ x. ; ˜ x y K.
Ž 3.6.
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On the other hand, we have
vŽ x. ; x q K
Ž 3.7.
if x is rewritten to be x s Ž z, w . with w GK 2 w . Combining Ž3.6. and Ž3.7., we get v Ž x . ; w x, ˜ xxK . In addition, for x ) 0, we choose w g w 0 y K 2 , u g u 0 q K 1 , such that 0
Ž 0, w . FK x FK Ž u, 0 . . By Theorem 3.1, Ž0, f tP Ž w .. FK f t Ž x . FK Ž f tL Ž u., 0. for t G 0. This yields Ž0, w 0 . FK v Ž x . FK Ž u 0 , 0.. Let x ) 0, y g v Ž x .. Since Ž0, w 0 . FK v Ž x . FK Ž u 0 , 0., we have yi G 0 wi ) 0 for i g P l I. We claim that yi ) 0 for all i g P ; I. If contrary, denote i 0 g P such that yi 0 s 0. Then f t kŽ x .4i 0 ª 0 for some t k ª q`. Since f P Ž0, w 0 . ) 0, by the continuity of f, f P Ž0, w 1 . ) 0 for w 1 FK 2 w 0 and w 1 ) 0. The monotonicity of f implies that f P Ž0, w . ) 0 for w GK 2 w 1 and w ) 0. Again from the continuity of f, there exists some ˆz ) 0 such that f P Ž ˆz, w . ) 0, that is, f i Ž ˆz, w . ) 0 for every i g P, w GK 2 w 1 and w ) 0. Without loss of generality, we may fix some sufficiently large k 0 such that f t k Ž x . s ˆ x s Žˆ z, w ˆ . with w ˆ GK 2 w 1 and w ˆ ) 0. It follows from 0 f i 0Ž ˆ x . ) 0 that f t Ž ˆ x .4i 0 G ˆ x i 0 ) 0 for t G 0. It implies that f tqt Ž x .4i G ˆ x i 0 ) 0 for t G 0, which contradicts f t kŽ x .4i 0 ª 0 for some k0 0 t k ª q`. This proves the claim. Similarly, we can prove y j ) 0 for all j g J. Therefore y ) 0 for every n y g v Ž x ., namely, v Ž x . ; R˙q . Later we will conclude that v Ž x . ; w x, ˜ x x K for x ) 0. In fact, for every 0. 0 Ž Ž . Ž . Ž x ) 0, 0, w FK v x FK u , 0 and v x . ) 0 Ž; y g v Ž x ., y ) 0. hold. We can fix a sufficiently large t 0 such that Ž0, w 0 . FK f t 0Ž x . FK Ž u 0 , 0. and f t 0Ž x . ) 0. Let ˆ x s f t 0Ž x .. Since ˆ x ) 0 and Ž0, w 0 . FK ˆ x FK Ž u 0 , 0., 0 there exist sufficiently small z, ¨ such that Ž z, w . FK ˆ x FK Ž u 0 , ¨ .. It 0 0 follows that f t Ž z, w . FK f t Ž ˆ x . FK f t Ž u , ¨ . by Theorem 3.1. This deduces x FK v Ž x . FK ˜ x. But v Ž ˆ x . s v Ž x .. Hence, v Ž x . ; w x, ˜ x x K for x ) 0. If x FK x, then v Ž x . FK x 4 and v Ž x . ; w x, ˜ x x K . So v Ž x . s x 4 . x GK ˜ x implies v Ž x . GK ˜ x 4 and v Ž x . ; w x, ˜ x x K . Hence v Ž x . s ˜ x 4 . The proof is completed.
4. LOTKA]VOLTERRA SYSTEMS Consider the Lotka]Volterra system
˙x s diag Ž x . Ž r q Mx . , where M is a type K matrix as in Ž1.1..
n x g Rq ,
r g Rn,
Ž S.
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One of the most interesting problems concerning the Lotka]Volterra system is whether the two groups I and J can coexist in a globally asymptotically stable steady state. This problem was studied in w1, 4x. For such system, Smith w1x gave the following result Žsee w1, Theorem 4.1x.. SMITH’S THEOREM. If Ž SI . has a positi¨ e steady state x 01 that is unstable ny k Ž to Rq that is, f J Ž x 01 , 0. ) 0. and Ž S J . has a positi¨ e steady state x 02 that is k Ž unstable to Rq that is, f I Ž0, x 02 . ) 0., then r 1 ) 0, r 2 ) 0, and the system Ž S . has a unique steady state x that is globally asymptotically stable with n respect to R˙q . In w4x, Takeuchi and Adachi proved that if M is stable, then Ž S . has a unique nonnegative steady state x with x q s 0 for q g Q and x r ) 0 for r g N y Q, which attracts all solutions with initial conditions in x g n Rq : x r ) 0, for r g N y Q4 . They also proved that x is globally asymptotin cally stable relative to x g Rq : x r ) 0, for r g N y Q4 if and only if n
rq q
Ý
mqj x j F 0
for all q g Q.
js1
Therefore, to give the conditions that guarantee that the Lotka] n Volterra system has a globally asymptotically stable steady state in R˙q , it is natural to assume that M is stable, that is, the principal minors of Mq alternate in sign as follows:
Ž y1 .
k
q a11 .. .
aq k1
... ...
a1qk .. . ) 0, q ak k
1 F k F n,
q < < where aq i j s a i j for i / j and a ii s a ii . Under this assumption, the existence of a positive steady state for the Lotka]Volterra system implies its n global stability in R˙q . Theorem 2.2 just gives the existence of such a steady state. To apply Theorem 2.1 to Theorem 2.2, we only want to prove that Theorem 2.2 satisfies the condition Ž0, w 0 . FK Ž u 0 , 0.. Indeed, we choose w in R˙qp such that wi s Ž w 0 . i for i g P l I, wj is ny p sufficiently large for j g J ; P, and z g Rq such that z i s 0 for i g l P ; I. Similarly, choose u in R˙q such that u i is sufficiently large ny l for i g I ; L, u j s Ž u 0 . j for j g L l J, and ¨ g Rq such that ¨ j s 0 for j g L ; J. Let x s Ž z, w . and y s Ž u, ¨ .; then x FK y. Thus, f t Ž x . FK f t Ž y . for t G 0, that is, Ž0, f tP Ž w .. FK Ž f tL Ž u., 0.. Since w 0 is globally asymptotically
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stable relative to R˙qp and u 0 is globally asymptotically stable relative to l R˙q , then
Ž 0, ftP Ž w . . ª Ž 0, w 0 .
as t ª q`,
Ž ftL Ž u . , 0 . ª Ž u 0 , 0.
as t ª q`.
and
Hence Ž0, w 0 . FK Ž u 0 , 0.. Smith’s result cannot be applied to the system Ž1.2. in the Introduction, but our Theorem 2.2 can. In fact, set L s 1, 2, 34 ; then the subsystem Ž SL . is u ˙ s diag Ž u . Ž r L q ML u . ,
3 u g Rq ,
Ž SL .
where ML s
ž
y18 12 y3
12 y36 y1
y1 y1 y12
/
y6 r L s 108 . 60
ž /
and
Ž SL . has a unique positive steady state u0 s
ž
1699 3387 4188 , , 979 979 979
/
and f LŽ u 0 , 0. s 10770r979 ) 0. Set P s 2, 3, 44 . The subsystem is as w ˙ s diag Ž w . Ž r P q MP w . ,
3 w g Rq ,
Ž SP .
where MP s
ž
y36 y1 y1
y1 y12 6
y6 18 y24
/
and
rP s
108 60 . y6
ž /
Ž SP . has a unique positive steady state w 0 s Ž546r211, 1434r211, 283r211. and f P Ž0, w 0 . s 2154r211 ) 0. By Theorem 2.2, ŽS. has a globally asymptotically stable positive steady state. By calculation it is Ž1, 3, 6, 1.. We conjecture that the conditions presented in Theorem 2.2 are sufficient and necessary ones for a type K Lotka]Volterra system to have a globally asymptotically stable positive steady state in the positive orthant; that is, if Ž S . has a globally asymptotically stable positive steady state, then there exist two index subsets L, P ; N that satisfy Ž1. and Ž2. of Theorem 2.2.
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REFERENCES 1. H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM. J. Appl. Math. 46 Ž1986., 856]873. 2. M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets, SIAM J. Math. Anal. 13 Ž1982., 167]179. 3. M. W. Hirsch, Systems of differential equations which are competitive or cooperative. II. Convergence almost everywhere, SIAM J. Math. Anal. 16 Ž1985., 423]439. 4. Y. Takeuchi and N. Adachi, The existence of globally stable equilibria of ecosystems of the generalized Volterra type, J. Math. Biol. 10 Ž1980., 401]415. 5. J. F. Selgrade, Asymptotic behavior of solutions to single loop positive feedback systems, J. Differential Equations 38 Ž1980., 80]103.
217, 233]245 Ž1998.
AY975711
The Coexistence of a Community of Species with Limited Competition Tu Caifeng and Jiang Jifa* Department of Mathematics, The Uni¨ ersity of Science and Technology of China, Hefei, China Submitted by Hal L. Smith Received March 17, 1997
This paper is a study of a system modeling a biological community of species with limited competition. The community consists of two competing subcommunities, all species of which cooperate, and some species of one subcommunity can invade the steady state of another subcommunity, whereas others cannot. Sufficient conditions are given that all species can coexist. For Lotka]Volterra systems we can improve this result by showing that there is a unique, globally asymptotically stable steady state. Q 1998 Academic Press
1. INTRODUCTION Consider the model of a biological community of species,
˙x i s x i f i Ž x 1 , x 2 , . . . , x n . ,
1 F i F n,
x i G 0,
Ž S.
in which x i represents the population density of the ith species and f i Ž x . represents the per capita growth rate of the ith species. Assume that the species community can be divided into two groups, I s 1, . . . , k 4 , J s k q 1, . . . , n4 , 0 F k F n, satisfying the requirement that the species in group I and J cooperate, and that species i and species j compete if i g I and j g J. This kind of system is called a system with limited competition. Mathematically, this situation can be described in terms of the Jacobian of f s Ž f 1 , . . . , f n . in the following manner: Df Ž x . s
ž
A yC
yB , D
/
Ž 1.1.
* Project supported by the National Natural Science Foundation of China. 233 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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where A and D have nonnegative off-diagonal elements and B G 0, C G 0. It is assumed that f is a continuously differentiable function n . defined on some open set containing the nonnegative orthant Rq If L is a nonempty subset of N s 1, 2, . . . , n4 , then the set Hq Ls x g n Rq : x p s 0, for p f L4 is an invariant set for ŽS.. The subsystem
˙x l s x l f l Ž x . ,
l g L,
x g Hq L
Ž SL .
will play an important role in this paper. In w1x, Smith extended the theories for cooperative systems developed by Hirsch w2, 3x to systems Ž S . and presented sufficient conditions for the persistence of all species. His essential conditions are that Ž SI . possesses a nyk positive steady state x 01 that is unstable to Rq , and Ž S J . possesses a 2 k positive steady state x 0 that is unstable to Rq , that is x 01 can be invaded by each species j g J and x 02 can be invaded by each species i g I. In mathematical language, this condition can be expressed as f j Ž x 01 , 0. ) 0 for all j g J and f i Ž0, x 02 . ) 0 for all i g I. This implies that f Ž0. ) 0 and the system Ž S . is nonobligate. But as mentioned by Smith w1, p. 870x, in the general case where positive steady states of Ž SI . and Ž S J . exist, we might imagine that some of the species j g J can invade the steady state for Ž SI . whereas others cannot, and similarly for the positive sready state for Ž S J .. Few of results of Smith w1x apply to this situation. Therefore, Smith’s results in w1x cannot be applied to those systems like the following system:
˙x s diag Ž x . Ž r q Mx . ,
4 x g Rq ,
r g R4 ,
Ž 1.2.
where y18 12 Ms y3 y3
12 y36 y1 y1
y1 y1 y12 6
y6 y6 18 y24
0
and
y6 108 rs . 60 y6
0
In this example, the subsystem Ž SI . has a unique positive steady state x 01 s Ž 157 , 267 . with f 3 Ž x 01 , 0. ) 0, f 4Ž x 01 , 0. - 0, and the subsystem Ž S J . has a unique positive steady state x 20 s Ž 375 , 58 . with f 1Ž0, x 02 . - 0, f 2 Ž0, x 02 . ) 0, where f Ž x . s Ž f 1 Ž x . , f 2 Ž x . , f 3 Ž x . , f 4 Ž x . . s r q Mx. This system does not satisfy the conditions in w1, Theorem 3.6x, but it has a globally asymptotically stable positive steady state Ž1, 3, 6, 1. in the interior of the nonnegative orthant. In this paper we shall pay attention to this situation. Using the idea of Smith w1x, we shall present some sufficient conditions such that all species can persist uniformly. These results generalize Theorem 3.6 of Smith w1x. Applying our general results, we can sharpen the result with the
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235
Lotka]Volterra system,
˙x s diag Ž x . Ž r q Mx . ,
n x g Rq ,
r g Rn,
and obtain sufficient conditions such that the two groups I and J can coexist in a globally asymptotically stable steady state in the case where r is not positive.
2. THE MAIN RESULTS In this section we will agree on some notation and establish some conventions that will be employed throughout this work. In addition, the main result will be stated. n Let Rq s x g R n : x i G 0, 1 F i F n4 denote the nonnegative orthant n n and Rq s x g Rq : x i ) 0, 1 F i F n4 denote its interior. In this paper one of the important notations is a cone. Recall that a cone K in R n is a nonempty closed subset of R n with the properties Rq K ; K ,
KqK;K
and
K l Ž yK . s 0 4 ,
where Rq is the set of nonnegative real numbers. If the partial order relation is generated by a cone K, we write x Fk y Ž x -K y . whenever y y x g K Ž y y x g Int K .. If x, y g R n and K is a cone in R n, we let n w x, y x K s z g R n : x FK z FK y4 . We drop the K if the cone is Rq . Fix k ny k . 1 k 2 Ž 1 F k F n. Then K s Rq = yRq is a cone. Let x g R , x g R nyk ; it is convenient to write x s Ž x 1, x 2 . g R n. Then x FK y, where x s Ž x 1, x 2 ., y s Ž y 1, y 2 . implies x 1 F y 1 , x 2 G y 2 . k ny k .. k If M is an n = n matrix such that Ž M q l I .Ž Rq = ŽyRq ; Rq = ny k . ŽyRq Ž for sufficiently large l, then M must be of the structure 1.1.. We call such a matrix M a type K matrix in Smith’s meaning Žsee w1x., and call such a system ŽS. a type K monotone system if Df Ž x . has a type K structure. Let N s 1, 2, . . . , n4 , I s 1, 2, . . . , k 4 , and J s k q 1, . . . , n4 , where n is the dimension of our Euclidean space R n. If L, P are nonempty sets of N such that L > I and P > J, then L s N y L and P s N y P denote their complementary sets in N. Hereafter, u, u, and u 0 , etc., will always aL represent vectors in aL-dimensional Euclidean space Rq ; ¨ , ¨ , and ¨ 0 , aL etc. in aL-dimensional Euclidean space Pq ; w, w, and w 0 , etc., aP aP in Rq ; and z, z, and z 0 , etc., in Rq . x 1 , x 2 will always represent k nyk n vectors in Rq and Rq , respectively. Then x s Ž x 1, x 2 . g Rq , f Ž x. s 1 2 1 2 Ž f 1Ž x , x ., f 2 Ž x , x ... Without loss of generality, we may assume L s 1, 2, . . . , k, k q 1, . . . , l 4 , k - l - n and P s n y p q 1, . . . , k q 1, . . . n4 , p ) n y k. Let x s Ž u, ¨ .
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and f Ž x . s Ž f LŽ u, ¨ ., f LŽ u, ¨ ... Then we can rewrite the system Ž S . as
½
u ˙ s diag Ž u . f LŽ u, ¨ . ¨˙ s diag Ž ¨ . f L Ž u, ¨ .
,
aL u g Rq ,
aL ¨ g Rq .
Ž S1 .
Let x s Ž z, w . and f Ž x . s Ž f P Ž z, w ., f P Ž z, w ... Then we can rewrite the system Ž S . as
½
˙z s diag Ž z . f P Ž z, w . , w ˙ s diag Ž w . f P Ž z, w .
aP z g Rq ,
aP w g Rq .
Ž S2 .
Set ¨ s 0 and z s 0 in Ž S1 . and Ž S2 ., respectively; we obtain two subsystems, u ˙ s diag Ž u . f LŽ u, 0 . ,
aL , u g Rq
Ž SL .
aP w g Rq .
Ž SP .
and w ˙ s diag Ž w . f P Ž 0, w . ,
Because Df Ž u, ¨ . and Df Ž z, w . are type K matrices as in Ž1.1., Df LŽ u, 0. is a type K 1 submatrix of Df Ž u, ¨ ., and Df P Ž0, w . is a type K 2 submatrix of k lyk . ny k . Df Ž z, w ., where K 1 s Rq = ŽyRq and K 2 s Rqpq kyn = ŽyRq . We write f t Ž x . for the unqiue solution x Ž t . of Ž S . satisfying x Ž0. s x, and f tL Ž u., f tP Ž w . are the unique solutions uŽ t . of Ž SL . and w Ž t . of Ž SP ., respectively, satisfying uŽ0. s u and w Ž0. s w. Ž f tL Ž u..I consists of components f tL Ž u.4i of f tL Ž u. for all i g I; Ž f tP Ž w .. J is defined similarly. n It is also necessary to give the following useful definitions. x g Rq is said to be a steady state of Ž S . if F Ž x . s 0. Denote the v-limit set of f t Ž x . as n v Ž x . s y g Rq : f t kŽ x . ª y,
for some sequence t k ª q` 4 .
The domain of attraction of a steady state x is the set of initial conditions x for which v Ž x . s x 4 . We say x is globally asymptotically stable with respect to a set W if x is asymptotically stable and W belongs to the domain of attraction of x. Throughout this paper, denote the cardinality of L and P, respectively, l nyl as l s aL, p s aP. If x s Ž u, ¨ ., then u g Rq and ¨ g Rq . If x s ny p p 1 2 1 k Ž z, w ., then z g Rq and w g Rq . If x s Ž x , x ., then x g Rq and 2 ny k x g Rq . Our main results are as follows.
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THEOREM 2.1. Assume that u 0 is a positi¨ e steady state for Ž SL ., f LŽ u 0 , 0. ) 0 and w 0 is a positi¨ e steady state for Ž S P ., f P Ž0, w 0 . ) 0. Let Ž0, w 0 . FK Ž u 0 , 0.. Then there exist positi¨ e steady states x and ˜ x for Ž S . with the following properties: Ž1. 0 - x, ˜ x F ŽŽ u 0 .I , Ž w 0 . J ., and x FK ˜ x. 0 Ž2. If x ) 0 and ˜ x FK x FK Ž u , 0., then f t Ž x . ª ˜ x as t ª q`. 0. Ž Ž . If x ) 0 and 0, w FK x FK x, then f t x ª x as t ª q`. Ž3. If x ) 0 and Ž0, w 0 . FK x FK Ž u 0 , 0., then v Ž x . ; w x, ˜ xxK . In addition, if u 0 q K 1 lies in the domain of attraction of u 0 for Ž SL ., and w y K 2 lies in the domain of attraction of w 0 for Ž SP ., then v Ž x . s x 4 for all x ) 0 with x FK x, v Ž x . s ˜ x 4 for all x ) 0 with x GK ˜ x, and v Ž x . ; w x, ˜ x x K for all x ) 0. 0
Applying Theorem 2.1, we can deduce a better result for the Lotka]Volterra system:
˙x s diag Ž x . Ž r q Mx . ,
n x g Rq ,
r g Rn,
Ž S.
under the assumption that M is stable Žthe definition can be found in section 4., and M is a type K matrix as in Ž1.1.. THEOREM 2.2. Assume that the Lotka]Volterra system Ž S . satisfies the following conditions. Ž1. There exists L ; N with L > I such that Ž S L . has a positi¨ e steady state u 0 and f LŽ u 0 , 0. ) 0. Ž2. There exists P ; N with P > J such that Ž S P . has a positi¨ e steady state w 0 and f P Ž0, w 0 . ) 0. Then Ž S . has a unique positi¨ e steady state that is globally asymptotically n stable relati¨ e to R˙q . Remark 2.1. Theorem 3.6 in w1x is the special case where L s I, P s J in Theorem 2.1. Remark 2.2. Let x 01 be a positive steady state of Ž S I .. In Theorem 2.1, the condition f LŽ u 0 , 0. ) 0 cannot guarantee that f J Ž x 01 , 0. ) 0. This is because for some j g J, it is possible that f j Ž x 01 , 0. - 0. We can know this fact from Ž1.2.. Remark 2.3. Recall that the system Ž S . is permanent if there exist constant numbers d , D ) 0 such that for every solution x Ž t . of Ž S ., d F lim inf t ªq` x Ž t . F lim sup t ªq` x Ž t . F D for all i g N. We notice that under the hypotheses of Theorem 2.1, together with its additional assumptions, the block w x, ˜ x x K attracts all solutions with positive initial data. Thus the system Ž S . is permanent.
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3. THE PROOF OF MAIN RESULT It is useful to establish some preliminary results before giving the proof of Theorem 2.1. The fundamental tool used in this paper is the generalized Kamke theorem. This theorem is extended in a natural way from cooperative systems to type K monotone systems Žsee w1, Theorem 2.4x.. THEOREM 3.1 ŽKamke Theorem.. Assume that the system Ž S . is type K monotone, and x Ž t ., y Ž t . are the solutions of Ž S . defined on a F t F b with x Ž a. FK y Ž a.. Then x Ž t . FK y Ž t . for all t g w a, b x. In w1, p. 862x, Smith extended a criterion for the monotonicity of every component of a solution for a cooperative system given by Selgrade w5x to a type K monotone system. Now we quote it here. THEOREM 3.2. Let the system Ž S . be a type K monotone system and let n f Ž x . GK 0 for some x g Rq . Then f t Ž x .4i is nondecreasing if i g I and f t Ž x .4j is nonincreasing if j g J for all t G 0 for which the solution exists. A similar result holds if f Ž x . FK 0. The existence of a steady state for Ž S . shall be related to the existence of the steady states for Ž SL . and Ž SP .. This is reflected by the following proposition. PROPOSITION 3.3. Assume that x s Ž x 1 , x 2 . is a positi¨ e steady state of l nyl nyp Ž S .. Rewrite x s Ž u, ¨ . or x s Ž z, w ., where u g Rq , ¨ g Rq , z g Rq , p w g Rq . Then Ž1. Ž x q K . l Hq L is positi¨ ely in¨ ariant and either has a nonnegati¨ e steady state or f tL Ž u. is unbounded for e¨ ery u GK 1 u and t G 0. Ž2. Ž x y K . l Hq P is positi¨ ely in¨ ariant and either has a nonnegati¨ e steady state or f tP Ž w . is unbounded for e¨ ery w FK 2 w and t G 0. Proof. We only prove Ž1., since Ž2. can be shown in a similar way. Consider the subsystem u ˙ s diag Ž u . f LŽ u, 0 . ,
l u g Rq .
Let x s Ž u, ¨ .. By the given condition, we have f LŽ u, ¨ . s 0. If i g I ; L, then f i Ž u, 0. G f i Ž u, ¨ . s 0 because f i Ž u, ¨ .r ¨ F 0. If j g J l L, then f j Ž u, 0. F f j Ž u, ¨ . s 0, because f j Ž u, ¨ .r ¨ G 0. Thus we have k lyk . f LŽ u, 0. GK 1 0, where K 1 s Rq = ŽyRq . By Theorem 3.2, f t Ž u, 0.4i is nondecreasing for i g I and f t Ž u, 0.4j is nonincreasing for j g J l L. Let Ž f t Ž u, 0..L s f tL Ž u.. Therefore either f tL Ž u.: t G 04 is unbounded or f tL Ž u.4 converges to a nonnegative steady state u 0 GK 1 u as t tends to
SPECIES WITH LIMITED COMPETITION
239
infinity. Obviously, Ž u 0 , 0. GK Ž u, ¨ . s x; this implies that Ž u 0 , 0. g x q K. LŽ . LŽ . Ž u 0 , 0. g Ž x q K . l Hq Since Ž u 0 , 0. g Hq L, L . Because f t u GK 1 f t u LŽ . LŽ . holds for every u GK 1 u, f t u is unbounded for t G 0 if f t u is unbounded. Because x p s 0 for p f L, f t Ž x .4p ' 0 holds, Hq is invariant. L If x GK x, then f t Ž x . GK x for all t ) 0 by Theorem 3.1, that is, f t Ž x . g x q K for t G 0. This implies that x q K is positively invariant. Therefore, Ž x q K . l Hq L is also positively invariant. The proposition will be used in the proof of our main result. PROPOSITION 3.4.
n Let x s Ž x 1, x 2 . s Ž u, ¨ . s Ž z, w . g Rq . Then
f t Ž x . F Ž Ž f tL Ž u . . I , Ž f tP Ž w . . J .
for t G 0.
l nyl Proof. Let x s Ž u, ¨ ., where u g Rq , ¨ g Rq . Then Ž u, ¨ . FK Ž u, 0.. By Theorem 3.1,
f t Ž x . FK f t Ž u, 0 . s Ž f tL Ž u . , 0 . s Ž Ž f tL Ž u . . I , Ž f tL Ž u . . LrI , 0 .
for t G 0.
Thus for t G 0,
Ž ft Ž x . . I F Ž ftL Ž u . . I .
Ž 3.1.
ny p Similarly, let x s Ž z, w ., where z g Rq , w g Rqp . Then Ž z, w . GK Ž0, w . and
f t Ž x . GK f t Ž 0, w . s Ž 0, f tP Ž w . . s Ž 0, Ž f tP Ž w . . PrJ , Ž f tP Ž w . . J .
for t G 0.
Hence for t G 0,
Ž ft Ž x . . J F Ž ftP Ž w . . J .
Ž 3.2.
From Ž3.1. and Ž3.2., we derive that
f t Ž x . F Ž Ž f tL Ž u . . I , Ž f tP Ž w . . J .
for t G 0.
Proposition 3.3 provides some relation between a positive steady state x for Ž S . and nonnegative steady states u 0 for Ž SL . and w 0 for Ž SP .. Hence to prove the existence of a nonnegative steady state Ž S ., it is natural to suppose the existence of positive steady states u 0 for Ž SL . and w 0 for Ž SP ..
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LEMMA 3.5. Assume that w 0 is a positi¨ e steady state for Ž SP . and f P Ž0, w 0 . ) 0, and u 0 is a positi¨ e steady state for Ž S L .. Let Ž0, w 0 . FK Ž u 0 , 0.. Then there is a steady state x s Ž x 1, x 2 . for Ž S . with the following properties: Ž1. Ž0, w 0 . FK x FK Ž u 0 , 0. and x i ) 0 for i g I. Ž2. If x ) 0, Ž0, w 0 . FK x FK x, then f t Ž x . ª x as t ª q`. kqpyn ny k . Ž3. x s Ž z, w . ) 0, w GK w 0 , where K 2 s Rq = ŽyRq , then 2 v Ž x . ; x q K. Ž4. If f j Ž u 0 , 0. ) 0 for some j g L, then x j ) 0. Proof. Ž1. By the continuity of f, f P Ž0, w 0 . ) 0 implies that f P Ž z, w 0 . ) 0 for sufficiently small z ) 0, where P ; I. For i g I l P, f i Ž z, w 0 . G f i Ž0, w 0 . s 0, because f i Ž z, w .r z G 0, and for j g J, f j Ž z, w 0 . F f j Ž0, w 0 . s 0, because f j Ž z, w .r z F 0. Hence f Ž z, w 0 . GK 0 for z sufficiently small. By Theorem 3.2, for sufficiently small z ) 0, f t Ž z, w 0 .4i is nondecreasing in t G 0 for i g I and f t Ž z, w 0 .4j is nonincreasing in t G 0 for j g J. Since Ž0, w 0 . FK Ž u 0 , 0., we have Ž0, w 0 . FK Ž z, w 0 . FK Ž u 0 , 0. for sufficiently small z ) 0. By Theorem 3.1, Ž0, w 0 . FK f t Ž z, w 0 . FK Ž u 0 , 0. for all t G 0. Hence f t Ž z, w 0 . is bounded. Consequently, f t Ž z, w 0 . converges to x as t ª q`, and Ž0, w 0 . FK x FK Ž u 0 , 0.. It is clear that x 1 G Ž z, w 0 .4I ) 0. Ž2. If x ) 0, Ž0, w 0 . FK x FK x, then there exists sufficiently small z ) 0 such that Ž z, w 0 . FK x FK x and f t Ž z, w 0 . FK f t Ž x . FK x. Because f t Ž z, w 0 . ª x as t ª q` by the proof of Ž1., f t Ž x . ª x as t ª q`. Ž3. Let x s Ž z, w . ) 0. Since w GK w 0 , we fix a sufficiently small z 0 2 such that z G z 0 . Then x s Ž z, w . GK Ž z 0 , w 0 . and f t Ž z, w . GK f t Ž z 0 , w 0 . for t ) 0. We have v Ž x . GK x 4 for f t Ž z 0 , w 0 . ª x as t ª q`. This means that v Ž x . ; x q K. Ž4. Suppose that f j Ž u 0 , 0. ) 0 for some j g L. Let x s Ž u, ¨ .. If u FK u 0 1 and ¨ G 0 with ¨ j s 0, then f j Ž u, ¨ . G f j Ž u 0 , 0. ) 0 for j g L. Let x s Ž u, ¨ .. Since Ž0, w 0 . FK Ž u, ¨ . FK Ž u 0 , 0., u FK u 0 . 1 If x j s ¨ j s 0 for some j g L, then f j Ž x . s f j Ž u, ¨ . G f j Ž u 0 , 0. ) 0 for j g L. Let u s Ž z,). and u 0 s Ž z 0 , )., where ) denotes other components of u and u 0 . Then for fixed sufficiently small z ) 0, we find that f t Ž z, w 0 . ª x as t ª q`. Since f j Ž x . ) 0, f j Ž f t Ž z, w 0 .. ) 0 for large enough t. But as t ª q`, f t Ž z, w .4j ª x j s 0, which is a contradiction. As a result, x j ) 0. The proof is completed. Remark 3.1. If the condition f P Ž0, w 0 . ) 0 in Lemma 3.5 is replaced by f LŽ u 0 , 0. ) 0, there are similar conclusions. If f LŽ u 0 , 0. ) 0 and f P Ž0, w 0 . ) 0 hold symmetrically, we obtain our main result, Theorem 2.1.
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Before proving our main result, we again stress that x s Ž u, ¨ . with l nyl ny p u g Rq , ¨ g Rq and x s Ž z, w . with z g Rq , w g Rqp , if necessary. Proof of Theorem 2.1. Ž1. By Lemma 3.5, for sufficiently small z ) 0, f t Ž z, w 0 . ª x as t tends to infinity, and x 1 ) 0.
Ž 3.3.
Similarly, fix sufficiently small ¨ such that Ž0, w 0 . FK Ž u 0 , ¨ . FK Ž u 0 , 0.; then Ž0, w 0 . FK f t Ž u 0 , ¨ . FK Ž u 0 , 0. for all t G 0. Hence f t Ž u 0 , ¨ . ª ˜ x as t ª q`, Ž0, w 0 . FK ˜ x F Ž u 0 , 0., and
˜x 2 G Ž u 0 , ¨ . 4 J ) 0.
Ž 3.4.
Let z ) 0, ¨ ) 0 be small enough such that Ž z, w 0 . FK Ž u 0 , ¨ .. Then f t Ž z, w 0 . FK f t Ž u 0 , ¨ . for t ) 0. So x FK ˜ x, namely, Ž x 1, x 2 . FK Ž ˜ x 1, ˜ x 2 .. This means that
˜x 1 G x 1 ) 0,
x2 G ˜ x 2 ) 0.
Ž 3.5.
Combining Ž3.3., Ž3.4., and Ž3.5., we conclude that x ) 0 and ˜ x ) 0. By Proposition 3.4, for x s Ž u, ¨ . s Ž z, w . and t ) 0, f t Ž x . F ŽŽ f tL Ž u..I , Ž f tP Ž w .. J . holds. Then for t G 0,
f t Ž z, w 0 . F Ž Ž f tL Ž u . . I , Ž f tP Ž w 0 . . J . and
f t Ž u 0 , ¨ . F Ž Ž f tL Ž u 0 . . I , Ž f tP Ž w . . J . . By a limit process, this derives x 2 F Ž w 0 . J and ˜ x 1 F Ž u 0 .I . It follows from 1 1 0 2 2 0 Ž3.5. that x F ˜ x F Ž u .I and ˜ x F x F Ž w . J . Therefore, 0 - x, ˜ xF ŽŽ u 0 .I , Ž w 0 . J .. Ž2. It follows from Lemma 3.5 and Remark 3.1 that if x ) 0 and Ž0, w 0 . FK x FK x, then f t Ž x . ª x as t ª q`, and ˜ x FK x FK Ž u 0 , 0., ft Ž x . ª ˜ x as t ª q`. Ž3. Let x s Ž z, w . ) 0 with w GK w 0 . Then from Ž3. of Lemma 3.5, 2 v Ž x . ; x q K. Similarly, if x s Ž u, ¨ . ) 0 with u FK 1 u 0 , then v Ž x . ; ˜x y K. If x ) 0 and Ž0, w 0 . FK x FK Ž u 0 , 0., then u FK 1 u 0 by rewriting x to be x s Ž u, ¨ .. This yields
vŽ x. ; ˜ x y K.
Ž 3.6.
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On the other hand, we have
vŽ x. ; x q K
Ž 3.7.
if x is rewritten to be x s Ž z, w . with w GK 2 w . Combining Ž3.6. and Ž3.7., we get v Ž x . ; w x, ˜ xxK . In addition, for x ) 0, we choose w g w 0 y K 2 , u g u 0 q K 1 , such that 0
Ž 0, w . FK x FK Ž u, 0 . . By Theorem 3.1, Ž0, f tP Ž w .. FK f t Ž x . FK Ž f tL Ž u., 0. for t G 0. This yields Ž0, w 0 . FK v Ž x . FK Ž u 0 , 0.. Let x ) 0, y g v Ž x .. Since Ž0, w 0 . FK v Ž x . FK Ž u 0 , 0., we have yi G 0 wi ) 0 for i g P l I. We claim that yi ) 0 for all i g P ; I. If contrary, denote i 0 g P such that yi 0 s 0. Then f t kŽ x .4i 0 ª 0 for some t k ª q`. Since f P Ž0, w 0 . ) 0, by the continuity of f, f P Ž0, w 1 . ) 0 for w 1 FK 2 w 0 and w 1 ) 0. The monotonicity of f implies that f P Ž0, w . ) 0 for w GK 2 w 1 and w ) 0. Again from the continuity of f, there exists some ˆz ) 0 such that f P Ž ˆz, w . ) 0, that is, f i Ž ˆz, w . ) 0 for every i g P, w GK 2 w 1 and w ) 0. Without loss of generality, we may fix some sufficiently large k 0 such that f t k Ž x . s ˆ x s Žˆ z, w ˆ . with w ˆ GK 2 w 1 and w ˆ ) 0. It follows from 0 f i 0Ž ˆ x . ) 0 that f t Ž ˆ x .4i 0 G ˆ x i 0 ) 0 for t G 0. It implies that f tqt Ž x .4i G ˆ x i 0 ) 0 for t G 0, which contradicts f t kŽ x .4i 0 ª 0 for some k0 0 t k ª q`. This proves the claim. Similarly, we can prove y j ) 0 for all j g J. Therefore y ) 0 for every n y g v Ž x ., namely, v Ž x . ; R˙q . Later we will conclude that v Ž x . ; w x, ˜ x x K for x ) 0. In fact, for every 0. 0 Ž Ž . Ž . Ž x ) 0, 0, w FK v x FK u , 0 and v x . ) 0 Ž; y g v Ž x ., y ) 0. hold. We can fix a sufficiently large t 0 such that Ž0, w 0 . FK f t 0Ž x . FK Ž u 0 , 0. and f t 0Ž x . ) 0. Let ˆ x s f t 0Ž x .. Since ˆ x ) 0 and Ž0, w 0 . FK ˆ x FK Ž u 0 , 0., 0 there exist sufficiently small z, ¨ such that Ž z, w . FK ˆ x FK Ž u 0 , ¨ .. It 0 0 follows that f t Ž z, w . FK f t Ž ˆ x . FK f t Ž u , ¨ . by Theorem 3.1. This deduces x FK v Ž x . FK ˜ x. But v Ž ˆ x . s v Ž x .. Hence, v Ž x . ; w x, ˜ x x K for x ) 0. If x FK x, then v Ž x . FK x 4 and v Ž x . ; w x, ˜ x x K . So v Ž x . s x 4 . x GK ˜ x implies v Ž x . GK ˜ x 4 and v Ž x . ; w x, ˜ x x K . Hence v Ž x . s ˜ x 4 . The proof is completed.
4. LOTKA]VOLTERRA SYSTEMS Consider the Lotka]Volterra system
˙x s diag Ž x . Ž r q Mx . , where M is a type K matrix as in Ž1.1..
n x g Rq ,
r g Rn,
Ž S.
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One of the most interesting problems concerning the Lotka]Volterra system is whether the two groups I and J can coexist in a globally asymptotically stable steady state. This problem was studied in w1, 4x. For such system, Smith w1x gave the following result Žsee w1, Theorem 4.1x.. SMITH’S THEOREM. If Ž SI . has a positi¨ e steady state x 01 that is unstable ny k Ž to Rq that is, f J Ž x 01 , 0. ) 0. and Ž S J . has a positi¨ e steady state x 02 that is k Ž unstable to Rq that is, f I Ž0, x 02 . ) 0., then r 1 ) 0, r 2 ) 0, and the system Ž S . has a unique steady state x that is globally asymptotically stable with n respect to R˙q . In w4x, Takeuchi and Adachi proved that if M is stable, then Ž S . has a unique nonnegative steady state x with x q s 0 for q g Q and x r ) 0 for r g N y Q, which attracts all solutions with initial conditions in x g n Rq : x r ) 0, for r g N y Q4 . They also proved that x is globally asymptotin cally stable relative to x g Rq : x r ) 0, for r g N y Q4 if and only if n
rq q
Ý
mqj x j F 0
for all q g Q.
js1
Therefore, to give the conditions that guarantee that the Lotka] n Volterra system has a globally asymptotically stable steady state in R˙q , it is natural to assume that M is stable, that is, the principal minors of Mq alternate in sign as follows:
Ž y1 .
k
q a11 .. .
aq k1
... ...
a1qk .. . ) 0, q ak k
1 F k F n,
q < < where aq i j s a i j for i / j and a ii s a ii . Under this assumption, the existence of a positive steady state for the Lotka]Volterra system implies its n global stability in R˙q . Theorem 2.2 just gives the existence of such a steady state. To apply Theorem 2.1 to Theorem 2.2, we only want to prove that Theorem 2.2 satisfies the condition Ž0, w 0 . FK Ž u 0 , 0.. Indeed, we choose w in R˙qp such that wi s Ž w 0 . i for i g P l I, wj is ny p sufficiently large for j g J ; P, and z g Rq such that z i s 0 for i g l P ; I. Similarly, choose u in R˙q such that u i is sufficiently large ny l for i g I ; L, u j s Ž u 0 . j for j g L l J, and ¨ g Rq such that ¨ j s 0 for j g L ; J. Let x s Ž z, w . and y s Ž u, ¨ .; then x FK y. Thus, f t Ž x . FK f t Ž y . for t G 0, that is, Ž0, f tP Ž w .. FK Ž f tL Ž u., 0.. Since w 0 is globally asymptotically
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stable relative to R˙qp and u 0 is globally asymptotically stable relative to l R˙q , then
Ž 0, ftP Ž w . . ª Ž 0, w 0 .
as t ª q`,
Ž ftL Ž u . , 0 . ª Ž u 0 , 0.
as t ª q`.
and
Hence Ž0, w 0 . FK Ž u 0 , 0.. Smith’s result cannot be applied to the system Ž1.2. in the Introduction, but our Theorem 2.2 can. In fact, set L s 1, 2, 34 ; then the subsystem Ž SL . is u ˙ s diag Ž u . Ž r L q ML u . ,
3 u g Rq ,
Ž SL .
where ML s
ž
y18 12 y3
12 y36 y1
y1 y1 y12
/
y6 r L s 108 . 60
ž /
and
Ž SL . has a unique positive steady state u0 s
ž
1699 3387 4188 , , 979 979 979
/
and f LŽ u 0 , 0. s 10770r979 ) 0. Set P s 2, 3, 44 . The subsystem is as w ˙ s diag Ž w . Ž r P q MP w . ,
3 w g Rq ,
Ž SP .
where MP s
ž
y36 y1 y1
y1 y12 6
y6 18 y24
/
and
rP s
108 60 . y6
ž /
Ž SP . has a unique positive steady state w 0 s Ž546r211, 1434r211, 283r211. and f P Ž0, w 0 . s 2154r211 ) 0. By Theorem 2.2, ŽS. has a globally asymptotically stable positive steady state. By calculation it is Ž1, 3, 6, 1.. We conjecture that the conditions presented in Theorem 2.2 are sufficient and necessary ones for a type K Lotka]Volterra system to have a globally asymptotically stable positive steady state in the positive orthant; that is, if Ž S . has a globally asymptotically stable positive steady state, then there exist two index subsets L, P ; N that satisfy Ž1. and Ž2. of Theorem 2.2.
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REFERENCES 1. H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM. J. Appl. Math. 46 Ž1986., 856]873. 2. M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets, SIAM J. Math. Anal. 13 Ž1982., 167]179. 3. M. W. Hirsch, Systems of differential equations which are competitive or cooperative. II. Convergence almost everywhere, SIAM J. Math. Anal. 16 Ž1985., 423]439. 4. Y. Takeuchi and N. Adachi, The existence of globally stable equilibria of ecosystems of the generalized Volterra type, J. Math. Biol. 10 Ž1980., 401]415. 5. J. F. Selgrade, Asymptotic behavior of solutions to single loop positive feedback systems, J. Differential Equations 38 Ž1980., 80]103.