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Average conditions for permanence and extinction in nonautonomous Gilpin–Ayala competition model
Nonlinear Analysis: Real World Applications 7 (2006) 895 – 915 www.elsevier.com/locate/na
Average conditions for permanence and extinction in nonautonomous Gilpin–Ayala competition model Fengde Chen College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China Received 16 February 2005; accepted 25 April 2005
Abstract In this paper, we consider a general nonautonomous n-species Gilpin–Ayala competitive system, which is more general and more realistic than classical Lotka–Volterra competition model. By means of Ahmad and Lazer’s definitions of lower and upper averages of a function, we first give the average conditions for the permanence and global attractivity of the system. Next, for each r n the average conditions on the coefficients are provided to guarantee that r of the species in the system are permanent while the remaining n − r species are driven to extinction. Examples show the feasibility of the main results. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: primary 34C05; 34C25; secondary 92D25; 34D20; 34D40 Keywords: Gilpin–Ayala competitive system; Lower average; Upper average; Permanence; Extinction; Global attractivity
1. Introduction Traditional Lotka–Volterra competition system can be expressed as follows: ⎡ ⎤ n aij (t)xj (t)⎦ , i = 1, 2, . . . , n. x˙i (t) = xi (t) ⎣bi (t) − j =1
E-mail addresses: [email protected], [email protected]. 1468-1218/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2005.04.007
(1.1)
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The model has been studied extensively (see [1–4,22–30,34,33,20,19,6,11,14,10,15] and the references cited therein). Many excellent results concerned with the permanence, extinction and global attractivity of periodic solution or almost periodic solution of system (1.1) are obtained. Recently, many scholars focus on the general nonautonomous case of system (1.1). By introducing a notation of the upper and lower averages of a function, Ahmad and Lazer [3] obtained sufficient condition which guarantee the permanence and global attractivity of system (1.1). Zhao et al. [34] obtained some excellent results which generalized the main results of [3]. Corresponding to the permanence of system (1.1), several scholars also investigated the uniform persistence and extinction for partial species, see for example Ahmad [4], Montes de Oca and Zeeman [24,23], Teng [26] and Zhao [33]. On the other hand, in 1973, Ayala et al. [5] conducted experiments on fruit fly dynamics to test the validity of 10 models of competitions. One of the models accounting best for the experimental results is given by x2 (t) x1 (t) 1 x˙1 (t) = r1 x1 (t) 1 − , − 12 K1 K2 x1 (t) x2 (t) 2 . (1.2) x˙2 (t) = r2 x2 (t) 1 − − 21 K2 K1 In order to fit data in their experiments and to yield significantly more accurate results, Gilpin and Ayala [17] claimed that a slightly more complicated model was needed and proposed the following competition model: ⎞ ⎛ i n xj (t) (t) x i ⎠ , i = 1, 2, . . . , n, x˙i (t) = ri xi (t) ⎝1 − (1.3) − bij Ki Kj j =1,j =i
where xi is the population density of the ith species, ri is the intrinsic exponential growth rate of the ith species, Ki is the environment carrying capacity of species i in the absence of competition, i provides a nonlinear measure of intra-specific interference, and bij provides a measure of interspecific interference. Goh and Agnew [18] and Liao and Li [21] obtained sufficient conditions which guarantee the global asymptotic stability of system (1.3). Fan and Wang [16] further considered the nonautonomous case of system (1.3), and obtained a set of sufficient conditions which guarantee the existence of the positive periodic solution of the system. Recently, we [12] further considered a multispecies nonlinear predator–prey competition system and investigated the stability property of the system. For more works on nonlinear ecosystems, one could refer to [5,16–18,21,12,7–9,13,8,32,31] and the references cited therein. However, to the best of the author’s knowledge, to this day, still no scholars investigate the permanence and extinction of the following general nonautonomous n-species Gilpin–Ayala competitive system ⎤ ⎡ n x˙i (t) = xi (t) ⎣bi (t) − (1.4) aij (t)(xj (t))ij ⎦ , i = 1, 2, . . . , n, j =1
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where bi (t), 1i n and aij (t), i, j = 1, 2, . . . , n are continuous for c t < + ∞, ij are positive constants. The aim of this paper is to investigate the qualitative properties of general nonautonomous system (1.4). First, we introduce the following notations and definitions. Given a function g(t) defined on [c, +∞), we set gM = sup{g(t)|c t < + ∞},
gL = inf{g(t)|c t < + ∞}.
From now on, we assume that the coefficients of system (1.4) satisfy bi (t) > 0, biM < + ∞, biL > 0, aij (t) > 0, aij M < + ∞, aij L > 0,
(1.5)
i.e., the coefficients of system (1.4) are bounded above and below by strictly positive constants. According to Ahmad and Lazer [3], we define the lower and upper averages of a function g which is continuous and bounded above and below on [c, +∞); if c t1 < t2 we set t2 1 g(s) ds. A[g, t1 , t2 ] = t2 − t1 t1 Definition 1.1. The lower and upper averages of g, denoted by m[g] and M[g], respectively, are defined by m[g] = lim inf{A[g, t1 , t2 ]|t1 − t2 s} s→+∞
and M[g] = lim sup{A[g, t1 , t2 ]|t1 − t2 s}. s→+∞
Since the set {A[g, t1 , t2 ]|t1 − t2 s} gets smaller as s increases, the limits exist; and since gL A[g, t1 , t2 ] gM , it follows that gL m[g]M[g] gM .
(1.6)
Definition 1.2. System (1.4) is called permanent, if for any positive solution X(t) = (x1 (t), x2 (t), . . . , xn (t))T of (1.4), there exist positive constants i , ki and T such that for t T i xi (t)ki ,
(i = 1, 2, . . . , n).
Definition 1.3. System (1.4) is said to be globally attractive if any two positive solution Y (t) = (y1 (t), . . . , yn (t))T and X(t) = (x1 (t), . . . , xn (t))T of system (1.4) satisfy lim |xi (t) − yi (t)| = 0,
t→+∞
i = 1, 2, . . . , n.
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Lemma 1.1. Both the positive and nonnegative cones of R n are invariant with respect to (1.4). It follows from Lemma 1.1 that any solution of (1.4) with positive (nonnegative) initial value remains positive (nonnegative). For the logistic equation x(t) ˙ = x(t)(bi (t) − aii (t)(x(t))ii ).
(1.7)
From Lemma 2.1 of [31], we have: Lemma 1.2. Suppose that bi (t) and aii (t) satisfy (1.5), then any positive solutions of Eq. (1.7) are defined on [t0 , +∞) for some t0 ∈ [c, +∞), bounded above and below by positive constants and globally attractive. Let us denote by Xi0 (t) any positive solutions of Eq. (1.7), then from Lemma 1.1, Lemma 1.2 and the definition of lower and upper averages of a function, M[Xi0 ] and m[Xi0 ] exist. Lemma 1.3. Let X(t) = (x1 (t), . . . , xn (t)) be a solution of system (1.4) with xi (t0 ) > 0 for some t0 ∈ [c, +∞) and i = 1, 2, . . . , n. Let Xi (t) be the solution of (1.7) such that Xi (t0 )xi (t0 ), then for i = 1, 2, . . . , n, Xi (t)xi (t),
t t0 .
The proof of Lemma 1.3 is similar to that given by Tineo and Alvarez [30, Proposition 2.1], and the detail are omitted here. The paper is arranged as follows: We state the main results Theorems 2.1–2.4 in Section 2, and prove Theorem 2.1 in Section 3, Theorems 2.2–2.4 in Section 4. Two examples in Section 5 show the feasibility of the main results of this paper.
2. Main results Now we state the main results of this paper. Theorem 2.1. Under assumption (1.5), suppose that the average conditions hold: ⎡ ⎤ n M ⎣bi (t) − aij (t)(Xj 0 (t))ij ⎦ > 0, i = 1, 2, . . . , n,
(2.1)
j =1,j =i
where Xi0 (t) is any positive solution of (1.7), then for solution of system (1.4) the following are true: (I) Let X(t) = (x1 (t), . . . , xn (t))T be any solution of system (1.4) with xi (t0 ) > 0, i = 1, 2, . . . , n, for some t0 c, then there exist positive constants i , ki and T
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such that for t T i xi (t)ki ,
(i = 1, 2, . . . , n),
(2.2)
i.e., system (1.4) is permanent; (II) Let X(t) = (x1 (t), . . . , xn (t))T and Y (t) = (y1 (t), . . . , yn (t))T be any two positive solutions of system (1.4). In addition to (2.1), assume further that: (A) ii maxj {j i }, i, j = 1, 2, . . . , n; (B) There exist positive constants (0 < < mini {i }), i , i = 1, 2, . . . , n, and such that for all t c and i = 1, 2, . . . , n, i ii aii (t) >
n
j j i aj i (t) + ,
(2.3)
j =1,j =i
then lim |xi (t) − yi (t)| = 0,
t→+∞
i = 1, 2, . . . , n,
i.e., system (1.4) is globally attractive. Remark 1. Under the assumption ij ≡ 1 in system (1.4), i.e., we consider Lotka–Volterra system (1.1), then conditions (A) and (B) in (II) of Theorem 2.1 could be replaced by: (C) There exist positive constants i , i = 1, 2, . . . , n, and such that i aii (t) >
n
j aj i (t) + .
j =1,j =i
Obviously, Theorem 2.1 generalizes the main results of Theorem 2.1 of [34]. Concerned with the extinction of partial species in system (1.4), we have: Theorem 2.2. Let r be a given integer and 1r < n. If for any k > r there is a ik < k such that for any j k ik j = kj , and the inequality M[bk (t)] < inf m[bik (t)]
(2.4)
akj (t) t t0 aik j (t)
(2.5)
holds. Let X(t)=(x1 (t), x2 (t), . . . , xn (t)) be any solution of system (1.4) with xi (t0 ) > 0, i= 1, 2, . . . , n, for some t0 ∈ [c, +∞), then for all i > r, xi (t) → 0 exponentially as t → +∞. Concerned with the permanence of partial species xi (t) (i = 1, 2, . . . , r) of system (1.4), we can prove the following result.
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Theorem 2.3. Suppose that all the conditions of Theorem 2.2 hold, if for each i r the inequality ⎡ ⎤ r M ⎣bi (t) − (2.6) aij (t)(Xj 0 (t)) ij ⎦ > 0, i = 1, 2, . . . , r, j =1,j =i
where Xi0 (t) is any positive solution of (1.7), then the species xi (i r) are uniformly persistent, i.e., there are positive constants i , ki and T such that for t T i xi (t)ki
for all i r.
(2.7)
For each r n, let H r denote the r-dimensional coordinate subspace on which xr+1 , . . . , xn vanish. We use the variable u to denote the restriction of system (1.4) to H r , ⎤ ⎡ r u˙ i (t) = ui (t) ⎣bi (t) − (2.8) aij (t)(uj (t))ij ⎦ , i = 1, 2, . . . , r. j =1
Concerned with the global attractivity of partial species xi (t) (i = 1, 2, . . . , r) of system (1.4), we can prove the following result. Theorem 2.4. Suppose that all the conditions of Theorems 2.2 and 2.3 hold. Assume further that: (A ) ii maxj {j i }, i, j = 1, 2, . . . , r; (B ) There exist positive constants (0 < < min1 i r {i }), i , i =1, 2, . . . , r and such that for all t c and i = 1, 2, . . . , n, ii
i aii (t) >
r
j j i aj i (t) + ,
(2.9)
j =1,j =i
then for any positive solution X(t)=(x1 (t), x2 (t), . . . , xn (t)) and any positive solution U (t) = (u1 (t), u2 (t), . . . , ur (t)) of subsystem (2.8), one has lim (xj (t) − uj (t)) = 0,
t→+∞
lim xj (t) = 0,
t→+∞
j = 1, 2, . . . , r,
j = r + 1, . . . , n.
Remark 2. Theorems 2.2–2.4 generalize the main results of Zhao [33, Theorems 1.1 and 1.2]. We mention here that for general nonautonomous Lotka–Volterra system (1.1), Teng [26] also obtained some similar results as that of Zhao [33]. It is in this sense, our results can also be seen as the generalization of Theorems 1–3 of [26]. Remark 3. If system (1.4) is a periodic system, i.e., bi (t), aij (t), i, j = 1, 2, . . . , n, are the continuous T-periodic function, then Xj 0 (t) in conditions (2.1) and (2.6) can be replaced by the unique positive T-periodic solution Xj∗0 (t) of (1.7).
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3. Proof of Theorem 2.1 The idea for our proof comes from Zhao et al. [34,31] and Chen [12,13]. Proof of Theorem 2.1. (I) Let X(t) = (x1 (t), . . . , xn (t))T , t0 and Xi (t), i = 1, 2, . . . , n, be as in Lemma 1.3. From Lemma 1.1 one could deduce that if xi (t0 ) > 0, i = 1, 2, . . . , n, then X(t) is a positive solution of system (1.4). From Lemma 1.2 if Xi (t0 ) > 0, then Xi (t) is positive solution of (1.7) for t t0 . From Lemma 1.3 we know that xi (t0 ) Xi (t0 ) implies xi (t)Xi (t),
t t0 .
By Lemma 1.2, for any positive solution Xi0 (t) of (1.7) and a sufficiently small > 0, there exists a Ti1 > t0 such that for t Ti1 xi (t)Xi (t) < Xi0 (t) + ,
i = 1, 2, . . . , n.
(3.1)
Let ki = sup{Xi0 (t) + |t t0 }. Then ki does not depend on any solution of system (1.4). Thus for i = 1, 2, . . . , n, xi (t)ki
for all t Ti1 .
(3.2)
Note the fact M[Xi0 (t) + c] = M[Xi0 (t)] + c, where c is a constant. From the boundedness of Xi0 (t), i = 1, 2, . . . , n, and condition (2.1) we can choose above small enough such that ⎡ ⎤ n M ⎣bi (t) − (3.3) aij (t)(Xj 0 (t) + )ij − aii (t)ii ⎦ > j =1,j =i
for all i = 1, 2, . . . , n, i.e., ⎧ ⎡ n ⎨ 1 t2 ⎣ lim sup bi (t) − aij (t)(Xj 0 (t) + )ij t→+∞ ⎩ t2 − t1 t1 j =1,j =i ⎫ ⎤ ⎬ − aii (t)ii ⎦ dt|t2 − t1 s > , ⎭ which deduces that ⎧ ⎡ n ⎨ t2 ⎣bi (t) − lim sup aij (t)(Xj 0 (t) + )ij t→+∞ ⎩ t1 j =1,j =i ⎫ ⎤ ⎬ − aii (t)ii ⎦ dt|t2 − t1 s = +∞. ⎭ Clearly, does not depend on any solution of system (1.4).
(3.4)
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Now we claim that there exist a i satisfying 0 < i < + ∞ and a Ti1 Ti1 such that for all t Ti1 .
xi (t)i
First, if there exists a Ti1 Ti1 such that for all t Ti1 ,
xi (t)
then the claim follows immediately. Therefore, we can suppose that there is a sequence q i → +∞ as q → +∞ such that for q = 1, 2, . . . q
xi (i ) < .
(3.5)
We can divide (3.5) into two cases: Case 1: There exists a Ti0 Ti1 such that xi (t) <
for all t Ti0 .
Case 2: u = xi (t), t ∈ [Ti1 , +∞) oscillates around u = infinitely, i.e., there exist two k < t k such that x (t k ) = x (t k ) = and x (t) < for any t ∈ (t k , t k ). sequence ti1 i i1 i i2 i i2 i1 i2 Assume that Case 1 occurs, then from system (1.4) and (3.1) it follows that ⎡ ⎤ t n ⎣bi (s) − xi (t) = xi (Ti0 ) exp aij (s)(xj (s))ij ⎦ ds Ti0
> xi (Ti0 ) exp
t
Ti0
j =1
⎡ ⎣bi (s) −
⎤
n
aij (s)(Xj 0 (s) + )ij − aii (s)ii ⎦ ds.
j =1,j =i
So by (3.4) there exists a sequence {ti } such that xi (ti ) → +∞ as ti → +∞, this contradicts with (3.2). Case 1 does not hold. k − t k . Integrating (1.4) on [t k , t k ], it follows Suppose that Case 2 holds, and let sik = ti2 i1 i1 i2 k k = xi (ti2 ) = xi (ti1 ) exp
> exp
k ti2
k ti1
⎡ ⎣bi (s) −
k ti2
k ti1
⎡ ⎣bi (s) −
n
n j =1
⎤ aij (s)(xj (s))ij ⎦ ds ⎤
aij (s)(Xj 0 (s) + )ij − aii (s)ii ⎦ ds.
j =1,j =i
If sik → +∞ as k → +∞, then by (3.4) as in Case 1 above the inequality would lead k to a contradiction. So the sequence {sik }+∞ k=1 is positive and bounded. If there exists a ti0 ∈ k , t k ) such that (ti1 i2 k xi (ti0 ) < exp(−Ai Bi ),
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where
903
⎫ ⎧ n ⎬ ⎨ Ai = sup bi (s) − aij (s)(Xj 0 (s) + )ij − aii (s)ii s ∈ [c, +∞) ; ⎭ ⎩ j =1,j =i Bi = sup{sik |k = 1, 2, . . .}.
k , t k ] leads to Obviously, 0 < Ai , Bi < + ∞. Integrating both sides of (1.4) on [ti1 i0
exp(−Ai Bi ) k k > xi (ti0 ) = xi (ti1 ) exp
> exp
k ti0
k ti1
k ti0
k ti1
⎡
⎡ ⎣bi (s) −
n
⎣bi (s) −
n
⎤ aij (s)(xj (s))ij ⎦ ds
j =1
⎤
aij (s)(Xj 0 (s) + )ij − aii (s)ii ⎦ ds
j =1,j =i
> exp(−Ai Bi ). k , tk ) This contradiction means that for t ∈ (ti1 i2
xi (t) exp(−Ai Bi ). From the above discussion we know that k , t k ), exp(−Ai Bi ), t ∈ (ti1 +∞ i2 k k xi (t) , t∈ / k=1 (ti1 , ti2 ). Let i = exp(−Ai Bi ), then i does not depend on any solution of (1.4), and so for all t Ti1 xi (t)i .
(3.6)
Now take T = maxi {Ti1 }, from (3.2) and (3.6), for t T , one has i xi (t)ki ,
(i = 1, 2, . . . , n),
(3.7)
i.e., system (1.4) is permanent; (II) Let X(t) = (x1 (t), . . . , xn (t)) be any solutions of (1.4) with xi (t0 ) > 0, 1 i n, for some t0 c, then from the first part of Theorem 2.1 there exist positive constants i , ki , i = 1, 2, . . . , n, and enough large T > 0 such that for all t T i xi (t)ki ,
(i = 1, 2, . . . , n).
Let be a positive constant with min1 i n {i } and yi (t) = xi (t)/. Then system (1.4) is transformed into ⎤ ⎡ n y˙i (t) = yi (t) ⎣bi (t) − (3.8) aij (t)ij (yj (t))ij ⎦ . j =1
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Obviously, y(t) = (x1 (t)/, . . . , xn (t)/) is the positive solution of system (3.8). Also, system (1.4) is equivalent to system (3.8), which means that (3.8) is permanent under the conditions of Theorem 2.1, and to prove system (1.4) is globally attractive is equivalent to prove system (3.8) is globally attractive. Let y(t) = (y1 (t), . . . , yn (t)), v(t) = (v1 (t), . . . , vn (t)) be any two solutions of system (3.8) satisfying the initial conditions yi (t0 ) > 0 and vi (t0 ) > 0 (i = 1, 2, . . . , n). Set V (t) =
n
i | ln yi (t) − ln vi (t)|.
i=1
Calculating the upper right derivative of V (t) along the solutions of system (1.4), we have D + V (t) =
n
⎡ i sgn(xi (t) − yi (t)) ⎣−
⎤ aij (t)ij ((yj (t))ij − (vj (t))ij )⎦
j =1
i=1
−
n
n
i ii aii (t)|(yi (t))ii − (vi (t))ii |
i=1
+
n
n
j j i aj i (t)|(yi (t))j i − (vi (t))j i |.
i=1 j =1,j =i
From yi (t) = xi (t)/, i = 1, 2, . . . , n, we know yi (t)1
for t T .
Observe that y = |a x − bx | is an increasing function for a 1, a b and x > 0. For ii maxj {j i } and t T , we get |(yi (t))j i − (vi (t))j i ||(yi (t))ii − (vi (t))ii |,
i, j = 1, 2, . . . , n.
(3.9)
Therefore, for t T , it follows from (2.3) that D + V (t) −
n
⎛ ⎝i ii aii (t) −
i=1 n
−
n
⎞ j j i aj i (t)⎠ |(yi (t))ii − (vi (t))ii |
j =1,j =i
|(yi (t))ii − (vi (t))ii | < 0.
(3.10)
i=1
By using (3.10), the rest of the proof is similar to the analysis of [31, p. 575] and we omit the detail here.
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4. Proof of Theorems 2.2–2.4 The idea for our proof comes from [2,24,26,33]. Applying Lemma 1.2 and the differential inequality theorem, we can easily prove the following result. Lemma 4.1. There exists a constant B > 0 such that lim sup xi (t) B, t→+∞
i = 1, 2, . . . , n,
for any positive solution X(t) = (x1 (t), x2 (t), . . . , xn (t)) of system (1.4). Proof of Theorem 2.2. Let X(t) = (x1 (t), . . . , xn (t)) be a solution of system (1.4) with xi (t0 ) > 0, i = 1, 2, . . . , n for some t0 ∈ [c, +∞). We prove this theorem by induction. We first prove xn (t) → 0 as t → +∞. Let i = in be given by (2.4) and (2.5). We note that n
x˙i (t) aij (t)(xj (t))ij , =bi (t)− xi (t) j =1
n
x˙n (t) anj (t)(xj (t))nj . =bn (t)− xn (t)
(4.1)
j =1
By (2.4), for j n, we have ij = nj .
(4.2)
By (2.5) we can choose , > 0 such that for j n, anj (t) M[bn (t)] t t0 . < < inf
aij (t) m[bi (t)]
(4.3)
Let Vn (t) = xi− (t)xn (t). From (4.1) and (4.2) it follows that ⎡ V˙n (t) = Vn (t) ⎣ bn (t) − bi (t) + ⎡ = Vn (t) ⎣ bn (t) − bi (t) +
n j =1 n j =1
⎤ (aij (t) − anj (t))(xj (t))ij ⎦ ⎤ anj (t) −
aij (t) (xj (t))ij ⎦ .
aij (t)
By (4.3), we can choose > 0 such that for t t0 and j n, anj (t) − < − < 0.
aij (t) This implies that V˙n (t)Vn (t)[ bn (t) − bi (t)].
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Integrating this inequality from t0 to t ( t0 ), it follows t [ bn (s) − bi (s)] ds. Vn (t)Vn (t0 ) exp
(4.4)
t0
By Lemma 4.1 we know that there exist > 0 such that xi (t)
for all i = 1, 2, . . . , n and
t t0 .
(4.5)
Therefore, (4.4) implies that 1 t [ bn (s) − bi (s)] ds, xn (t) < C exp
t0
(4.6)
where C = / (xi (t0 ))−/ xn (t0 ) > 0. From (4.3) it follows
M[bn (t)] − m[bi (t)] < 0. Already, Zhao and Jiang [33, p. 670] showed that the above inequality implies that t [ bn (s) − bi (s)] ds = −∞. lim t→+∞
(4.7)
t0
(4.7) together with (4.6) shows that xn (t) → 0 exponentially as t → +∞. For any integer k > r, assume that we have obtained xi (t) → 0 as t → +∞ for all i > k. Now we prove that xk (t) → 0 as t → 0. Let i = ik be given by (2.4) and (2.5). By (2.4), for j k, we have ij = kj .
(4.8)
By (2.5) we can choose , > 0 such that for j k, akj (t) M[bk (t)] . < < inf t t 0 m[bi (t)] aij (t)
(4.9)
We note that n
x˙i (t) = bi (t) − aij (t)(xj (t))ij , xi (t) j =1
n
x˙k (t) = bk (t) − akj (t)(xj (t))kj . xk (t) j =1
(4.10) Let
Vk (t) = xi− (t)xk (t).
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From (4.9) and (4.10) it follows that ⎡ V˙k (t) = Vk (t) ⎣ bk (t) − bi (t) +
n
⎤ (aij (t) − akj (t))(xj (t))ij ⎦
j =1
⎡ = Vk (t) ⎣ bk (t) − bi (t) +
k
aij (t)
j =1 n
+
907
akj (t) − (xj (t))ij aij (t) ⎤
(aij (t) − akj (t))(xj (t))ij ⎦ .
j =k+1
By (4.9), we can choose > 0 such that for t t0 and j k, akj (t) − < − < 0. aij (t) This implies that ⎡
n
V˙k (t)Vk (t) ⎣ bk (t) − bi (t) +
⎤ (aij (t) − akj (t))(xj (t))ij ⎦ .
(4.11)
j =k+1
Since xi (t) → 0 exponentially as t → +∞ for all i > k, by the boundedness of aij (t) (i, j = 1, 2, . . . , n) on [c, +∞), we obtain lim
t→+∞
n
(aij (t) − akj (t))(xj (t))ij = 0.
(4.12)
j =k+1
Hence, for any small > 0, there exists a T ∗ such that n
(aij (t) − akj (t))(xj (t))ij <
for all t T ∗ .
(4.13)
j =k+1
By using (4.13), integrating (4.11) from T ∗ to t ( T ∗ ), it follows t ∗ [ bk (s) − bi (s) + ] ds. Vk (t)Vk (T ) exp T∗
By the same procedure as above for xn (t), we finally get that for t T ∗ 1 t xk (t) < C exp [ bk (s) − bi (s) + ] ds, t0 where C = / (xi (T ∗ ))−/ xk (T ∗ ) > 0.
(4.14)
(4.15)
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From (4.9) it follows M[bk (t)] − m[bi (t)] < 0. This implies that for enough small positive constant , one has M[bk (t)] − m[bi (t)] + < 0. Therefore, M[ bk (t) − bi + ] M[bk (t)] + M[−bi (t)] + = M[bk (t)] − m[bi (t)] + < 0. Then similarly to the analysis of Zhao and Jiang [33, p. 670], the above inequality implies that t lim (4.16) [ bk (s) − bi (s) + ] ds = −∞. t→+∞ T ∗
(4.16) together with (4.15) shows that xk (t) → 0 exponentially as t → +∞. Finally, by the induction we obtain that xi (t) → 0 exponentially as t → +∞ for all i > r. The proof is complete. Proof of Theorem 2.3. From system (1.4) we have x˙i (t)xi (t)[bi (t) − aii (t)(xi (t))ii ]. By the differential inequality theorem and Lemma 1.2, for any > 0 there exists a Ti1 > t0 such that for t > Ti1 and i = 1, 2, . . . , r, xi (t) < Xi0 (t) + , where Xi0 (t) is any strictly positive solution of (1.7). Let ki = sup{Xi0 (t) + |t t0 }. Then ki does not depend on any solution of (1.7). Thus for i = 1, 2, . . . , r, xi (t)ki
for all t Ti1 .
(4.17)
Note the fact M[Xi0 (t) + c] = M[Xi0 (t)] + c, where c is a constant. From the boundedness of Xi0 (t), i = 1, 2, . . . , r, and condition (2.6) we can choose 1 small enough such that ⎡ ⎤ r n M ⎣bi (t) − aij (t)(Xj 0 (t) + 1 )ij − aij (t)1ij − aii (t)1ii ⎦ j =1,j =i
> 1
j =r+1
(4.18)
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for all i = 1, 2, . . . , r, i.e., ⎧ ⎡ r ⎨ 1 t2 ⎣bi (t) − aij (t)(Xj 0 (t) + 1 )ij lim sup t→+∞ ⎩ t2 − t1 t1 j =1,j =i ⎫ ⎤ n ⎬ − aij (t)1ij − aii (t)1ii ⎦ dt|t2 − t1 s > 1 , ⎭ j =r+1
which deduces that ⎧ ⎡ r ⎨ t2 ⎣bi (t) − lim sup t→+∞ ⎩ t1
aij (t)(Xj 0 (t) + 1 )ij
j =1,j =i
−
n
aij (t)1ij
⎤
− aii (t)1ii ⎦ dt|t2
j =r+1
− t1 s
⎫ ⎬ ⎭
= +∞.
(4.19)
Clearly, 1 does not depend on any solution of system (1.4). By assumptions (2.4) and (2.5), from Theorem 2.2, for above 1 > 0 there exists T2 > t0 such that for t > T2 0 < xi (t) < 1 ,
i = r + 1, . . . , n.
(4.20)
Now we claim that there exists i > 0 and T > max{Ti1 , Ti2 , . . . , Tir , T2 } such that for all t >T xi (t) > i ,
i = 1, 2, . . . , r.
By using (4.19) and (4.20), the proof of above claim follows that of the proof of the first part of Theorem 2.1 with slight modification and we omit the detail here. The proof is complete. Before we prove Theorem 2.4, we need the following lemma. Lemma 4.2. Suppose that condition (2.6) in Theorem 2.3 holds, then subsystem (2.8) is permanent. Lemma 4.2 can be proved by using the same method given in the proof of Theorem 2.1 and we omit the detail here. Proof of Theorem 2.4. Let X(t) = (x1 (t), x2 (t), . . . , xn (t)) be any positive solution of system (1.4) and u(t) = (u1 (t), . . . , ur (t)) be any positive solution of subsystem (2.8). By Theorem 2.2, we have xi (t) → 0 as t → +∞ for all i > r. By Theorem 2.3 and Lemma 4.2, there are positive constants i , ki and enough large T > 0 such that i xi (t), ui (t)ki ,
(i = 1, 2, . . . , r) for all t > T .
(4.21)
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Let be a positive constant with min1 i r {i } and yi (t) = xi (t)/, vi (t) = ui (t)/ Then system (1.4) is transformed into ⎤ ⎡ n (4.22) aij (t)ij (yj (t))ij ⎦ . y˙i (t) = yi (t) ⎣bi (t) − j =1
And system (2.8) is transformed into ⎤ ⎡ r aij (t)ij (vj (t))ij ⎦ . v˙i (t) = vi (t) ⎣bi (t) −
(4.23)
j =1
Obviously, y(t) = (x1 (t)/, . . . , xn (t)/) is the positive solution of system (4.22) and v(t) = (u1 (t)/, . . . , ur (t)/) is the positive solution of system (4.23). Also, system (1.4) is equivalent to system (4.22) and system (2.8) is equivalent to system (4.23), which means that to end the proof of Theorem 2.4, it is enough to show that lim (yj (t) − vj (t)) = 0,
t→+∞
j = 1, 2, . . . , r.
From (4.21) we have i yi (t),
vi (t)
ki ,
(i = 1, 2, . . . , r) for all t > T .
Consider the Lyapunov function as follows: Vr (t) =
r
i | ln yi (t) − ln vi (t)|.
i=1
Calculating the upper right derivative of Vr (t), we have D + Vr (t) r − i ii aii (t)|(yi (t))ii − (vi (t))ii | i=1
+
r
r
j j i aj i (t)|(yi (t))j i − (vi (t))j i | + g(t)
i=1 j =1,j =i
for all t t0 , where g(t) =
r i=1
i
n
ij aij (t)(yj (t))ij .
j =r+1
From yi (t) = xi (t)/, i = 1, 2, . . . , n, we know that yi (t)1
for t T .
(4.24)
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Since y =|a x −bx | is an increasing function for a 1, a b and x > 0. For ii maxj {j i } and t T , we get |(yi (t))j i − (vi (t))j i ||(yi (t))ii − (vi (t))ii |,
i, j = 1, 2, . . . , r.
(4.25)
Therefore, by using the mean value theorem, for t T , it follows from (2.9), (4.21) and (4.25) that D + Vr (t) −
r
⎛ ⎝i ii aii (t) −
−
⎞ j j i aj i (t)⎠ |(yi (t))ii − (vi (t))ii | + g(t)
j =1,j =i
i=1 r
r
|(yi (t))ii − (vi (t))ii | + g(t)
i=1
−
r
ii ( i (t))ii −1 |yi (t) − vi (t)| + g(t)
i=1
i ii −1 ki ii −1 |yi (t) − vi (t)| + g(t) − , ii min i=1 r i ii −1 ki ii −1 , | ln yi (t) − ln vi (t)| + g(t) − ii min ki
r
i=1
− Vr (t) + g(t), ii −1 , where i (t) lies between yi (t) and vi (t) and = min1 i r {−1 i ii min{(i /) −1 ii }/ki }. Applying the differential inequality theorem and the variation of con(ki /) stants formula of solutions of first-order linear differential equation, we have t Vr (t)e−(t−T ) g(s)e(s−T ) ds + Vr (T ) . (4.26) T
Since g(t) → 0 as t → +∞, it is not hard to prove Vr (t) → 0 as t → +∞. This implies that for all i r yi (t) − vi (t) → 0
as t → +∞,
which is equivalent to xi (t) − ui (t) → 0 The proof is complete.
as t → +∞.
5. Examples Following two examples show the feasibility of our results.
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Example 1. Consider the two-species competitive system x˙1 (t) = x1 (t)[4 − (2 + cos t)x1 (t) − (x2 (t))2 ], x˙2 (t) = x2 (t)[3 − (x1 (t))2/3 − (2 + sin t)x2 (t)].
(5.1)
In this case, b1 (t) = 4, b2 (t) = 3, a11 (t) = 2 + cos t, a12 (t) = 1, a21 (t) = 1, a22 (t) = 2 + sin t, 11 = 1, 12 = 2, 21 = 23 , 22 = 1. The corresponding Logistic equations of (5.1) is as follows: x˙1 (t) = x1 (t)[4 − (2 + cos t)x1 (t)],
(5.2)
x˙2 (t) = x2 (t)[3 − (2 + sin t)x2 (t)].
(5.3)
Already, Zhao et al. [34, pp. 274–275] had showed that system (5.2) has a unique positive periodic solution x1∗ (t) = 34(17 + 2 sin t + 8 cos t)−1
(5.4)
and system (5.3) has a unique positive periodic solution x2∗ (t) = 30(20 + 9 sin t − 3 cos t)−1 .
(5.5)
By using computation tool Maple 6, one could easily obtain the following estimate: M[b1 (t) − a12 (t)(x2∗ (t))2 ] = 4 − M[(30(20 + 9 sin t − 3 cos t)−1 )2 ] ≈ 4 − 3.297933395 = 0.702066605 > 0,
(5.6)
M[b2 (t) − a21 (t)(x1∗ (t))2/3 ] = 3 − M[(34(17 + 2 sin t + 8 cos t)−1 )2/3 ] ≈ 3 − 1.709498686 = 1.290501314 > 0.
(5.7)
and
The above two inequalities show that conditions (2.1) of Theorem 2.1 hold, thus, system (5.1) is permanent. Example 2. Consider the following three-species competitive system: x˙1 (t) = x1 (t)[4 − (2 + cos t)x1 (t) − (x2 (t))2 − x3 (t)], x˙2 (t) = x2 (t)[3 − (x1 (t))2/3 − (2 + sin t)x2 (t) − (x3 (t))2 ], x˙3 (t) = x3 (t)[2 − 2(x1 (t))2/3 − 6x2 (t) − (3 + sin t)(x3 (t))2 ].
(5.8)
In this case, b1 (t)=4, b2 (t)=3, b3 (t)=2, a11 (t)=2+cos t, a12 (t)=1, a13 (t)=1, a21 (t)= 1, a22 (t) = 2 + sin t, a23 (t) = 1, a31 (t) = 2, a32 (t) = 6, a33 (t) = 3 + sin t, 11 = 1, 12 = 2, 13 = 1, 21 = 31 = 23 , 22 = 32 = 1, 23 = 33 = 2. One could easily see that for
F. Chen / Nonlinear Analysis: Real World Applications 7 (2006) 895 – 915
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3.5 3 2.5 2 1.5 1 0.5 0 2
4
6
8
10 12 14 16 18 20 t
Fig. 1. Dynamics of system (5.8) with initial values x1 (0) = 2, x2 (0) = 0.7, x3 (0) = 0.5 and t ∈ [0, 20].
species x2 , x3 , condition (2.4) of Theorem 2.2 holds; Also, we have a31 (t) M[b3 ] 2 = < 2 = inf t 0 , m[b2 ] 3 a (t) 21 M[b3 ] 2 a32 (t) = < 2 inf t 0 , m[b2 ] 3 a22 (t) M[b3 ] 2 a33 (t) t 0 . = < 2 inf a23 (t) m[b2 ] 3 Therefore (2.5) holds for r = 2, which means that species x3 (t) → 0 exponentially as t → +∞. Now let us consider the following subsystem of system (5.8): x˙1 (t) = x1 (t)[4 − (2 + cos t)x1 (t) − (x2 (t))2 ], x˙2 (t) = x2 (t)[3 − (x1 (t))2/3 − (2 + sin t)x2 (t)].
(5.9)
This is the system we had investigated in Example 1. Also, from (5.6) and (5.7) we see that conditions (2.6) of Theorem 2.3 hold, thus species x1 , x2 is permanent. Fig. 1 shows the dynamics behavior of system (5.8). Acknowledgements The author is grateful to an anonymous referee for his excellent suggestions, which greatly improve the presentation of the paper. Also, this work was supported by the National Natural Science Foundation of China (Tian Yuan Foundation) (10426010), the Foundation of Science and Technology of Fujian Province for Young Scholars (2004J0002) and the Foundation of Fujian Education Bureau (JA04156). References [1] S. Ahmad, On the nonautonomous Volterra–Lotka competition equations, Proc. Amer. Math. Soc. 117 (1993) 199–204. [2] S. Ahmad, Extinction of species in nonautonomous Lotka–Volterra systems, Proc. Amer. Math. Soc. 127 (1999) 2905–2910.
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[3] S. Ahmad, A.C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka–Volterra system, Nonlinear Anal. 40 (2000) 37–49. [4] S. Ahmad, F. Montes de Oca, Extinction in nonautonomous T-periodic competitive Lotka–Volterra system, Appl. Math. Comput. 90 (1998) 155–166. [5] F.J. Ayala, M.E. Gilpin, J.G. Eherenfeld, Competition between species: theoretical models and experimental tests, Theor. Popul. Biol. 4 (1973) 331–356. [6] Y. Chen, New results on positive periodic solutions of a periodic integro-differential competition system, Appl. Math. Comput. 153 (2) (2004) 557–565. [7] F.D. Chen, Persistence and periodic orbits for two-species non-autonomous diffusion Lotka–Volterra models, Appl. Math. J. Chinese Univ. Ser. B. 19 (4) (2004) 359–366. [8] F.D. Chen, On a nonlinear non-autonomous predator–prey model with diffusion and distributed delay, J. Comput. Appl. Math. 80 (1) (2005) 33–49. [9] F.D. Chen, Periodicity in a nonlinear predator–prey system with state dependent delays, Acta Mathematicae Applicatae Sinica, English Series 21 (1) (2005) 49–60. [10] F.D. Chen, Positive periodic solutions of neutral Lotka–Volterra system with feedback control, Appl. Math. Comput. 162 (3) (2005) 1279–1302. [11] F.D. Chen, The permanence and global attractivity of Lotka–Volterra competition system with feedback controls, Nonlinear Anal.: Real World Appl., in press. [12] F.D. Chen, On a periodic multi-species ecological model, Appl. Math. Comput., in press. [13] F.D. Chen, S.J. Lin, Periodicity in a logistic type system with several delays, Comput. Math. Appl. 48 (1–2) (2004) 35–44. [14] F.D. Chen, F.X. Lin, X.X. Chen, Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control, Appl. Math. Comput. 158 (1) (2004) 45–68. [15] F.D. Chen, D.X. Sun, F.X. Lin, Periodicity in a food-limited population model with toxicants and state dependent delays, J. Math. Anal. Appl. 288 (1) (2003) 132–142. [16] M. Fan, K. Wang, Global periodic solutions of a generalized n-species Gilpin–Ayala competition model, Comput. Math. Appl. 40 (2000) 1141–1151. [17] M.E. Gilpin, F.J. Ayala, Global models of growth and competition, Proc. Nat. Acad. Sci. USA 70 (1973) 3590–3593. [18] B.S. Goh, T.T. Agnew, Stability in Gilpin and Ayala’s model of competition, J. Math. Biol. 4 (1977) 275–279. [19] H.F. Huo, W.T. Li, Periodic solutions of a periodic Lotka–Volterra system with delay, Appl. Math. Comput. 156 (3) (2004) 787–803. [20] Y.K. Li, Y. Kuang, Periodic solutions of periodic delay Lotka–Volterra equations and systems, J. Math. Anal. Appl. 255 (1) (2001) 260–280. [21] X.X. Liao, J. Li, Stability in Gilpin–Ayala competition models with diffusion, Nonlinear Anal. TMA 28 (1997) 1751–1758. [22] B. Lisena, Global stability in periodic competitive systems, Nonlinear Anal.: Real World Appl. 5 (4) (2004) 613–627. [23] F. Montes de Oca, M.L. Zeeman, Balancing survival and extinction in nonautonomous competitive Lotka–Volterra systems, J. Math. Anal. Appl. 192 (1995) 360–370. [24] F. Montes de Oca, M.L. Zeeman, Extinction in nonautonomous competitive Lotka–Volterra systems, Proc. Amer. Math. Soc. 124 (1996) 3677–3687. [25] Y. Muroya, Boundedness and partial survival of species in nonautonomous Lotka–Volterra systems, Nonlinear Anal.: Real World Appl. 6 (2) (2005) 263–272. [26] Z. Teng, On the nonautonomous Lotka–Volterra N-species competing systems, Appl. Math. Comput. 114 (2000) 175–185. [27] Z. Teng, The almost periodic Komogorov competitive systems, Nonlinear Anal. 42 (2000) 1221–1230. [28] A. Tineo, On the asymptotic behavior of some population models, J. Math. Anal. Appl. 167 (1992) 516–529. [29] A. Tineo, An iterative scheme for the N-competing species problem, J. Differential Equations 116 (1995) 1–15. [30] A. Tineo, C. Alvarz, A different consideration about the globally asymptotically stable solution of the periodic n-competing species problem, J. Math. Anal. Appl. 159 (1991) 44–55.
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[31] J.D. Zhao, W.C. Chen, The qualitative analysis of N-species nonlinear prey-competition systems, Appl. Math. Comput. 149 (2004) 567–576. [32] J.D. Zhao, J.F. Jiang, Permanence in nonautonomous Lotka–Volterra system with predator–prey, Appl. Math. Comput. 152 (2004) 99–109. [33] J.D. Zhao, J.F. Jiang, Average conditions for permanence and extinction in nonautonomous Lotka–Volterra system, J. Math. Anal. Appl. 299 (2004) 663–675. [34] J.D. Zhao, J.F. Jiang, A.C. Lazer, The permanence and global attractivity in a nonautonomous Lotka–Volterra system, Nonlinear Anal.: Real World Appl. 5 (4) (2004) 265–276.
Average conditions for permanence and extinction in nonautonomous Gilpin–Ayala competition model Fengde Chen College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China Received 16 February 2005; accepted 25 April 2005
Abstract In this paper, we consider a general nonautonomous n-species Gilpin–Ayala competitive system, which is more general and more realistic than classical Lotka–Volterra competition model. By means of Ahmad and Lazer’s definitions of lower and upper averages of a function, we first give the average conditions for the permanence and global attractivity of the system. Next, for each r n the average conditions on the coefficients are provided to guarantee that r of the species in the system are permanent while the remaining n − r species are driven to extinction. Examples show the feasibility of the main results. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: primary 34C05; 34C25; secondary 92D25; 34D20; 34D40 Keywords: Gilpin–Ayala competitive system; Lower average; Upper average; Permanence; Extinction; Global attractivity
1. Introduction Traditional Lotka–Volterra competition system can be expressed as follows: ⎡ ⎤ n aij (t)xj (t)⎦ , i = 1, 2, . . . , n. x˙i (t) = xi (t) ⎣bi (t) − j =1
E-mail addresses: [email protected], [email protected]. 1468-1218/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2005.04.007
(1.1)
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The model has been studied extensively (see [1–4,22–30,34,33,20,19,6,11,14,10,15] and the references cited therein). Many excellent results concerned with the permanence, extinction and global attractivity of periodic solution or almost periodic solution of system (1.1) are obtained. Recently, many scholars focus on the general nonautonomous case of system (1.1). By introducing a notation of the upper and lower averages of a function, Ahmad and Lazer [3] obtained sufficient condition which guarantee the permanence and global attractivity of system (1.1). Zhao et al. [34] obtained some excellent results which generalized the main results of [3]. Corresponding to the permanence of system (1.1), several scholars also investigated the uniform persistence and extinction for partial species, see for example Ahmad [4], Montes de Oca and Zeeman [24,23], Teng [26] and Zhao [33]. On the other hand, in 1973, Ayala et al. [5] conducted experiments on fruit fly dynamics to test the validity of 10 models of competitions. One of the models accounting best for the experimental results is given by x2 (t) x1 (t) 1 x˙1 (t) = r1 x1 (t) 1 − , − 12 K1 K2 x1 (t) x2 (t) 2 . (1.2) x˙2 (t) = r2 x2 (t) 1 − − 21 K2 K1 In order to fit data in their experiments and to yield significantly more accurate results, Gilpin and Ayala [17] claimed that a slightly more complicated model was needed and proposed the following competition model: ⎞ ⎛ i n xj (t) (t) x i ⎠ , i = 1, 2, . . . , n, x˙i (t) = ri xi (t) ⎝1 − (1.3) − bij Ki Kj j =1,j =i
where xi is the population density of the ith species, ri is the intrinsic exponential growth rate of the ith species, Ki is the environment carrying capacity of species i in the absence of competition, i provides a nonlinear measure of intra-specific interference, and bij provides a measure of interspecific interference. Goh and Agnew [18] and Liao and Li [21] obtained sufficient conditions which guarantee the global asymptotic stability of system (1.3). Fan and Wang [16] further considered the nonautonomous case of system (1.3), and obtained a set of sufficient conditions which guarantee the existence of the positive periodic solution of the system. Recently, we [12] further considered a multispecies nonlinear predator–prey competition system and investigated the stability property of the system. For more works on nonlinear ecosystems, one could refer to [5,16–18,21,12,7–9,13,8,32,31] and the references cited therein. However, to the best of the author’s knowledge, to this day, still no scholars investigate the permanence and extinction of the following general nonautonomous n-species Gilpin–Ayala competitive system ⎤ ⎡ n x˙i (t) = xi (t) ⎣bi (t) − (1.4) aij (t)(xj (t))ij ⎦ , i = 1, 2, . . . , n, j =1
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where bi (t), 1i n and aij (t), i, j = 1, 2, . . . , n are continuous for c t < + ∞, ij are positive constants. The aim of this paper is to investigate the qualitative properties of general nonautonomous system (1.4). First, we introduce the following notations and definitions. Given a function g(t) defined on [c, +∞), we set gM = sup{g(t)|c t < + ∞},
gL = inf{g(t)|c t < + ∞}.
From now on, we assume that the coefficients of system (1.4) satisfy bi (t) > 0, biM < + ∞, biL > 0, aij (t) > 0, aij M < + ∞, aij L > 0,
(1.5)
i.e., the coefficients of system (1.4) are bounded above and below by strictly positive constants. According to Ahmad and Lazer [3], we define the lower and upper averages of a function g which is continuous and bounded above and below on [c, +∞); if c t1 < t2 we set t2 1 g(s) ds. A[g, t1 , t2 ] = t2 − t1 t1 Definition 1.1. The lower and upper averages of g, denoted by m[g] and M[g], respectively, are defined by m[g] = lim inf{A[g, t1 , t2 ]|t1 − t2 s} s→+∞
and M[g] = lim sup{A[g, t1 , t2 ]|t1 − t2 s}. s→+∞
Since the set {A[g, t1 , t2 ]|t1 − t2 s} gets smaller as s increases, the limits exist; and since gL A[g, t1 , t2 ] gM , it follows that gL m[g]M[g] gM .
(1.6)
Definition 1.2. System (1.4) is called permanent, if for any positive solution X(t) = (x1 (t), x2 (t), . . . , xn (t))T of (1.4), there exist positive constants i , ki and T such that for t T i xi (t)ki ,
(i = 1, 2, . . . , n).
Definition 1.3. System (1.4) is said to be globally attractive if any two positive solution Y (t) = (y1 (t), . . . , yn (t))T and X(t) = (x1 (t), . . . , xn (t))T of system (1.4) satisfy lim |xi (t) − yi (t)| = 0,
t→+∞
i = 1, 2, . . . , n.
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Lemma 1.1. Both the positive and nonnegative cones of R n are invariant with respect to (1.4). It follows from Lemma 1.1 that any solution of (1.4) with positive (nonnegative) initial value remains positive (nonnegative). For the logistic equation x(t) ˙ = x(t)(bi (t) − aii (t)(x(t))ii ).
(1.7)
From Lemma 2.1 of [31], we have: Lemma 1.2. Suppose that bi (t) and aii (t) satisfy (1.5), then any positive solutions of Eq. (1.7) are defined on [t0 , +∞) for some t0 ∈ [c, +∞), bounded above and below by positive constants and globally attractive. Let us denote by Xi0 (t) any positive solutions of Eq. (1.7), then from Lemma 1.1, Lemma 1.2 and the definition of lower and upper averages of a function, M[Xi0 ] and m[Xi0 ] exist. Lemma 1.3. Let X(t) = (x1 (t), . . . , xn (t)) be a solution of system (1.4) with xi (t0 ) > 0 for some t0 ∈ [c, +∞) and i = 1, 2, . . . , n. Let Xi (t) be the solution of (1.7) such that Xi (t0 )xi (t0 ), then for i = 1, 2, . . . , n, Xi (t)xi (t),
t t0 .
The proof of Lemma 1.3 is similar to that given by Tineo and Alvarez [30, Proposition 2.1], and the detail are omitted here. The paper is arranged as follows: We state the main results Theorems 2.1–2.4 in Section 2, and prove Theorem 2.1 in Section 3, Theorems 2.2–2.4 in Section 4. Two examples in Section 5 show the feasibility of the main results of this paper.
2. Main results Now we state the main results of this paper. Theorem 2.1. Under assumption (1.5), suppose that the average conditions hold: ⎡ ⎤ n M ⎣bi (t) − aij (t)(Xj 0 (t))ij ⎦ > 0, i = 1, 2, . . . , n,
(2.1)
j =1,j =i
where Xi0 (t) is any positive solution of (1.7), then for solution of system (1.4) the following are true: (I) Let X(t) = (x1 (t), . . . , xn (t))T be any solution of system (1.4) with xi (t0 ) > 0, i = 1, 2, . . . , n, for some t0 c, then there exist positive constants i , ki and T
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such that for t T i xi (t)ki ,
(i = 1, 2, . . . , n),
(2.2)
i.e., system (1.4) is permanent; (II) Let X(t) = (x1 (t), . . . , xn (t))T and Y (t) = (y1 (t), . . . , yn (t))T be any two positive solutions of system (1.4). In addition to (2.1), assume further that: (A) ii maxj {j i }, i, j = 1, 2, . . . , n; (B) There exist positive constants (0 < < mini {i }), i , i = 1, 2, . . . , n, and such that for all t c and i = 1, 2, . . . , n, i ii aii (t) >
n
j j i aj i (t) + ,
(2.3)
j =1,j =i
then lim |xi (t) − yi (t)| = 0,
t→+∞
i = 1, 2, . . . , n,
i.e., system (1.4) is globally attractive. Remark 1. Under the assumption ij ≡ 1 in system (1.4), i.e., we consider Lotka–Volterra system (1.1), then conditions (A) and (B) in (II) of Theorem 2.1 could be replaced by: (C) There exist positive constants i , i = 1, 2, . . . , n, and such that i aii (t) >
n
j aj i (t) + .
j =1,j =i
Obviously, Theorem 2.1 generalizes the main results of Theorem 2.1 of [34]. Concerned with the extinction of partial species in system (1.4), we have: Theorem 2.2. Let r be a given integer and 1r < n. If for any k > r there is a ik < k such that for any j k ik j = kj , and the inequality M[bk (t)] < inf m[bik (t)]
(2.4)
akj (t) t t0 aik j (t)
(2.5)
holds. Let X(t)=(x1 (t), x2 (t), . . . , xn (t)) be any solution of system (1.4) with xi (t0 ) > 0, i= 1, 2, . . . , n, for some t0 ∈ [c, +∞), then for all i > r, xi (t) → 0 exponentially as t → +∞. Concerned with the permanence of partial species xi (t) (i = 1, 2, . . . , r) of system (1.4), we can prove the following result.
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F. Chen / Nonlinear Analysis: Real World Applications 7 (2006) 895 – 915
Theorem 2.3. Suppose that all the conditions of Theorem 2.2 hold, if for each i r the inequality ⎡ ⎤ r M ⎣bi (t) − (2.6) aij (t)(Xj 0 (t)) ij ⎦ > 0, i = 1, 2, . . . , r, j =1,j =i
where Xi0 (t) is any positive solution of (1.7), then the species xi (i r) are uniformly persistent, i.e., there are positive constants i , ki and T such that for t T i xi (t)ki
for all i r.
(2.7)
For each r n, let H r denote the r-dimensional coordinate subspace on which xr+1 , . . . , xn vanish. We use the variable u to denote the restriction of system (1.4) to H r , ⎤ ⎡ r u˙ i (t) = ui (t) ⎣bi (t) − (2.8) aij (t)(uj (t))ij ⎦ , i = 1, 2, . . . , r. j =1
Concerned with the global attractivity of partial species xi (t) (i = 1, 2, . . . , r) of system (1.4), we can prove the following result. Theorem 2.4. Suppose that all the conditions of Theorems 2.2 and 2.3 hold. Assume further that: (A ) ii maxj {j i }, i, j = 1, 2, . . . , r; (B ) There exist positive constants (0 < < min1 i r {i }), i , i =1, 2, . . . , r and such that for all t c and i = 1, 2, . . . , n, ii
i aii (t) >
r
j j i aj i (t) + ,
(2.9)
j =1,j =i
then for any positive solution X(t)=(x1 (t), x2 (t), . . . , xn (t)) and any positive solution U (t) = (u1 (t), u2 (t), . . . , ur (t)) of subsystem (2.8), one has lim (xj (t) − uj (t)) = 0,
t→+∞
lim xj (t) = 0,
t→+∞
j = 1, 2, . . . , r,
j = r + 1, . . . , n.
Remark 2. Theorems 2.2–2.4 generalize the main results of Zhao [33, Theorems 1.1 and 1.2]. We mention here that for general nonautonomous Lotka–Volterra system (1.1), Teng [26] also obtained some similar results as that of Zhao [33]. It is in this sense, our results can also be seen as the generalization of Theorems 1–3 of [26]. Remark 3. If system (1.4) is a periodic system, i.e., bi (t), aij (t), i, j = 1, 2, . . . , n, are the continuous T-periodic function, then Xj 0 (t) in conditions (2.1) and (2.6) can be replaced by the unique positive T-periodic solution Xj∗0 (t) of (1.7).
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3. Proof of Theorem 2.1 The idea for our proof comes from Zhao et al. [34,31] and Chen [12,13]. Proof of Theorem 2.1. (I) Let X(t) = (x1 (t), . . . , xn (t))T , t0 and Xi (t), i = 1, 2, . . . , n, be as in Lemma 1.3. From Lemma 1.1 one could deduce that if xi (t0 ) > 0, i = 1, 2, . . . , n, then X(t) is a positive solution of system (1.4). From Lemma 1.2 if Xi (t0 ) > 0, then Xi (t) is positive solution of (1.7) for t t0 . From Lemma 1.3 we know that xi (t0 ) Xi (t0 ) implies xi (t)Xi (t),
t t0 .
By Lemma 1.2, for any positive solution Xi0 (t) of (1.7) and a sufficiently small > 0, there exists a Ti1 > t0 such that for t Ti1 xi (t)Xi (t) < Xi0 (t) + ,
i = 1, 2, . . . , n.
(3.1)
Let ki = sup{Xi0 (t) + |t t0 }. Then ki does not depend on any solution of system (1.4). Thus for i = 1, 2, . . . , n, xi (t)ki
for all t Ti1 .
(3.2)
Note the fact M[Xi0 (t) + c] = M[Xi0 (t)] + c, where c is a constant. From the boundedness of Xi0 (t), i = 1, 2, . . . , n, and condition (2.1) we can choose above small enough such that ⎡ ⎤ n M ⎣bi (t) − (3.3) aij (t)(Xj 0 (t) + )ij − aii (t)ii ⎦ > j =1,j =i
for all i = 1, 2, . . . , n, i.e., ⎧ ⎡ n ⎨ 1 t2 ⎣ lim sup bi (t) − aij (t)(Xj 0 (t) + )ij t→+∞ ⎩ t2 − t1 t1 j =1,j =i ⎫ ⎤ ⎬ − aii (t)ii ⎦ dt|t2 − t1 s > , ⎭ which deduces that ⎧ ⎡ n ⎨ t2 ⎣bi (t) − lim sup aij (t)(Xj 0 (t) + )ij t→+∞ ⎩ t1 j =1,j =i ⎫ ⎤ ⎬ − aii (t)ii ⎦ dt|t2 − t1 s = +∞. ⎭ Clearly, does not depend on any solution of system (1.4).
(3.4)
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F. Chen / Nonlinear Analysis: Real World Applications 7 (2006) 895 – 915
Now we claim that there exist a i satisfying 0 < i < + ∞ and a Ti1 Ti1 such that for all t Ti1 .
xi (t)i
First, if there exists a Ti1 Ti1 such that for all t Ti1 ,
xi (t)
then the claim follows immediately. Therefore, we can suppose that there is a sequence q i → +∞ as q → +∞ such that for q = 1, 2, . . . q
xi (i ) < .
(3.5)
We can divide (3.5) into two cases: Case 1: There exists a Ti0 Ti1 such that xi (t) <
for all t Ti0 .
Case 2: u = xi (t), t ∈ [Ti1 , +∞) oscillates around u = infinitely, i.e., there exist two k < t k such that x (t k ) = x (t k ) = and x (t) < for any t ∈ (t k , t k ). sequence ti1 i i1 i i2 i i2 i1 i2 Assume that Case 1 occurs, then from system (1.4) and (3.1) it follows that ⎡ ⎤ t n ⎣bi (s) − xi (t) = xi (Ti0 ) exp aij (s)(xj (s))ij ⎦ ds Ti0
> xi (Ti0 ) exp
t
Ti0
j =1
⎡ ⎣bi (s) −
⎤
n
aij (s)(Xj 0 (s) + )ij − aii (s)ii ⎦ ds.
j =1,j =i
So by (3.4) there exists a sequence {ti } such that xi (ti ) → +∞ as ti → +∞, this contradicts with (3.2). Case 1 does not hold. k − t k . Integrating (1.4) on [t k , t k ], it follows Suppose that Case 2 holds, and let sik = ti2 i1 i1 i2 k k = xi (ti2 ) = xi (ti1 ) exp
> exp
k ti2
k ti1
⎡ ⎣bi (s) −
k ti2
k ti1
⎡ ⎣bi (s) −
n
n j =1
⎤ aij (s)(xj (s))ij ⎦ ds ⎤
aij (s)(Xj 0 (s) + )ij − aii (s)ii ⎦ ds.
j =1,j =i
If sik → +∞ as k → +∞, then by (3.4) as in Case 1 above the inequality would lead k to a contradiction. So the sequence {sik }+∞ k=1 is positive and bounded. If there exists a ti0 ∈ k , t k ) such that (ti1 i2 k xi (ti0 ) < exp(−Ai Bi ),
F. Chen / Nonlinear Analysis: Real World Applications 7 (2006) 895 – 915
where
903
⎫ ⎧ n ⎬ ⎨ Ai = sup bi (s) − aij (s)(Xj 0 (s) + )ij − aii (s)ii s ∈ [c, +∞) ; ⎭ ⎩ j =1,j =i Bi = sup{sik |k = 1, 2, . . .}.
k , t k ] leads to Obviously, 0 < Ai , Bi < + ∞. Integrating both sides of (1.4) on [ti1 i0
exp(−Ai Bi ) k k > xi (ti0 ) = xi (ti1 ) exp
> exp
k ti0
k ti1
k ti0
k ti1
⎡
⎡ ⎣bi (s) −
n
⎣bi (s) −
n
⎤ aij (s)(xj (s))ij ⎦ ds
j =1
⎤
aij (s)(Xj 0 (s) + )ij − aii (s)ii ⎦ ds
j =1,j =i
> exp(−Ai Bi ). k , tk ) This contradiction means that for t ∈ (ti1 i2
xi (t) exp(−Ai Bi ). From the above discussion we know that k , t k ), exp(−Ai Bi ), t ∈ (ti1 +∞ i2 k k xi (t) , t∈ / k=1 (ti1 , ti2 ). Let i = exp(−Ai Bi ), then i does not depend on any solution of (1.4), and so for all t Ti1 xi (t)i .
(3.6)
Now take T = maxi {Ti1 }, from (3.2) and (3.6), for t T , one has i xi (t)ki ,
(i = 1, 2, . . . , n),
(3.7)
i.e., system (1.4) is permanent; (II) Let X(t) = (x1 (t), . . . , xn (t)) be any solutions of (1.4) with xi (t0 ) > 0, 1 i n, for some t0 c, then from the first part of Theorem 2.1 there exist positive constants i , ki , i = 1, 2, . . . , n, and enough large T > 0 such that for all t T i xi (t)ki ,
(i = 1, 2, . . . , n).
Let be a positive constant with min1 i n {i } and yi (t) = xi (t)/. Then system (1.4) is transformed into ⎤ ⎡ n y˙i (t) = yi (t) ⎣bi (t) − (3.8) aij (t)ij (yj (t))ij ⎦ . j =1
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Obviously, y(t) = (x1 (t)/, . . . , xn (t)/) is the positive solution of system (3.8). Also, system (1.4) is equivalent to system (3.8), which means that (3.8) is permanent under the conditions of Theorem 2.1, and to prove system (1.4) is globally attractive is equivalent to prove system (3.8) is globally attractive. Let y(t) = (y1 (t), . . . , yn (t)), v(t) = (v1 (t), . . . , vn (t)) be any two solutions of system (3.8) satisfying the initial conditions yi (t0 ) > 0 and vi (t0 ) > 0 (i = 1, 2, . . . , n). Set V (t) =
n
i | ln yi (t) − ln vi (t)|.
i=1
Calculating the upper right derivative of V (t) along the solutions of system (1.4), we have D + V (t) =
n
⎡ i sgn(xi (t) − yi (t)) ⎣−
⎤ aij (t)ij ((yj (t))ij − (vj (t))ij )⎦
j =1
i=1
−
n
n
i ii aii (t)|(yi (t))ii − (vi (t))ii |
i=1
+
n
n
j j i aj i (t)|(yi (t))j i − (vi (t))j i |.
i=1 j =1,j =i
From yi (t) = xi (t)/, i = 1, 2, . . . , n, we know yi (t)1
for t T .
Observe that y = |a x − bx | is an increasing function for a 1, a b and x > 0. For ii maxj {j i } and t T , we get |(yi (t))j i − (vi (t))j i ||(yi (t))ii − (vi (t))ii |,
i, j = 1, 2, . . . , n.
(3.9)
Therefore, for t T , it follows from (2.3) that D + V (t) −
n
⎛ ⎝i ii aii (t) −
i=1 n
−
n
⎞ j j i aj i (t)⎠ |(yi (t))ii − (vi (t))ii |
j =1,j =i
|(yi (t))ii − (vi (t))ii | < 0.
(3.10)
i=1
By using (3.10), the rest of the proof is similar to the analysis of [31, p. 575] and we omit the detail here.
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4. Proof of Theorems 2.2–2.4 The idea for our proof comes from [2,24,26,33]. Applying Lemma 1.2 and the differential inequality theorem, we can easily prove the following result. Lemma 4.1. There exists a constant B > 0 such that lim sup xi (t) B, t→+∞
i = 1, 2, . . . , n,
for any positive solution X(t) = (x1 (t), x2 (t), . . . , xn (t)) of system (1.4). Proof of Theorem 2.2. Let X(t) = (x1 (t), . . . , xn (t)) be a solution of system (1.4) with xi (t0 ) > 0, i = 1, 2, . . . , n for some t0 ∈ [c, +∞). We prove this theorem by induction. We first prove xn (t) → 0 as t → +∞. Let i = in be given by (2.4) and (2.5). We note that n
x˙i (t) aij (t)(xj (t))ij , =bi (t)− xi (t) j =1
n
x˙n (t) anj (t)(xj (t))nj . =bn (t)− xn (t)
(4.1)
j =1
By (2.4), for j n, we have ij = nj .
(4.2)
By (2.5) we can choose , > 0 such that for j n, anj (t) M[bn (t)] t t0 . < < inf
aij (t) m[bi (t)]
(4.3)
Let Vn (t) = xi− (t)xn (t). From (4.1) and (4.2) it follows that ⎡ V˙n (t) = Vn (t) ⎣ bn (t) − bi (t) + ⎡ = Vn (t) ⎣ bn (t) − bi (t) +
n j =1 n j =1
⎤ (aij (t) − anj (t))(xj (t))ij ⎦ ⎤ anj (t) −
aij (t) (xj (t))ij ⎦ .
aij (t)
By (4.3), we can choose > 0 such that for t t0 and j n, anj (t) − < − < 0.
aij (t) This implies that V˙n (t)Vn (t)[ bn (t) − bi (t)].
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Integrating this inequality from t0 to t ( t0 ), it follows t [ bn (s) − bi (s)] ds. Vn (t)Vn (t0 ) exp
(4.4)
t0
By Lemma 4.1 we know that there exist > 0 such that xi (t)
for all i = 1, 2, . . . , n and
t t0 .
(4.5)
Therefore, (4.4) implies that 1 t [ bn (s) − bi (s)] ds, xn (t) < C exp
t0
(4.6)
where C = / (xi (t0 ))−/ xn (t0 ) > 0. From (4.3) it follows
M[bn (t)] − m[bi (t)] < 0. Already, Zhao and Jiang [33, p. 670] showed that the above inequality implies that t [ bn (s) − bi (s)] ds = −∞. lim t→+∞
(4.7)
t0
(4.7) together with (4.6) shows that xn (t) → 0 exponentially as t → +∞. For any integer k > r, assume that we have obtained xi (t) → 0 as t → +∞ for all i > k. Now we prove that xk (t) → 0 as t → 0. Let i = ik be given by (2.4) and (2.5). By (2.4), for j k, we have ij = kj .
(4.8)
By (2.5) we can choose , > 0 such that for j k, akj (t) M[bk (t)] . < < inf t t 0 m[bi (t)] aij (t)
(4.9)
We note that n
x˙i (t) = bi (t) − aij (t)(xj (t))ij , xi (t) j =1
n
x˙k (t) = bk (t) − akj (t)(xj (t))kj . xk (t) j =1
(4.10) Let
Vk (t) = xi− (t)xk (t).
F. Chen / Nonlinear Analysis: Real World Applications 7 (2006) 895 – 915
From (4.9) and (4.10) it follows that ⎡ V˙k (t) = Vk (t) ⎣ bk (t) − bi (t) +
n
⎤ (aij (t) − akj (t))(xj (t))ij ⎦
j =1
⎡ = Vk (t) ⎣ bk (t) − bi (t) +
k
aij (t)
j =1 n
+
907
akj (t) − (xj (t))ij aij (t) ⎤
(aij (t) − akj (t))(xj (t))ij ⎦ .
j =k+1
By (4.9), we can choose > 0 such that for t t0 and j k, akj (t) − < − < 0. aij (t) This implies that ⎡
n
V˙k (t)Vk (t) ⎣ bk (t) − bi (t) +
⎤ (aij (t) − akj (t))(xj (t))ij ⎦ .
(4.11)
j =k+1
Since xi (t) → 0 exponentially as t → +∞ for all i > k, by the boundedness of aij (t) (i, j = 1, 2, . . . , n) on [c, +∞), we obtain lim
t→+∞
n
(aij (t) − akj (t))(xj (t))ij = 0.
(4.12)
j =k+1
Hence, for any small > 0, there exists a T ∗ such that n
(aij (t) − akj (t))(xj (t))ij <
for all t T ∗ .
(4.13)
j =k+1
By using (4.13), integrating (4.11) from T ∗ to t ( T ∗ ), it follows t ∗ [ bk (s) − bi (s) + ] ds. Vk (t)Vk (T ) exp T∗
By the same procedure as above for xn (t), we finally get that for t T ∗ 1 t xk (t) < C exp [ bk (s) − bi (s) + ] ds, t0 where C = / (xi (T ∗ ))−/ xk (T ∗ ) > 0.
(4.14)
(4.15)
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F. Chen / Nonlinear Analysis: Real World Applications 7 (2006) 895 – 915
From (4.9) it follows M[bk (t)] − m[bi (t)] < 0. This implies that for enough small positive constant , one has M[bk (t)] − m[bi (t)] + < 0. Therefore, M[ bk (t) − bi + ] M[bk (t)] + M[−bi (t)] + = M[bk (t)] − m[bi (t)] + < 0. Then similarly to the analysis of Zhao and Jiang [33, p. 670], the above inequality implies that t lim (4.16) [ bk (s) − bi (s) + ] ds = −∞. t→+∞ T ∗
(4.16) together with (4.15) shows that xk (t) → 0 exponentially as t → +∞. Finally, by the induction we obtain that xi (t) → 0 exponentially as t → +∞ for all i > r. The proof is complete. Proof of Theorem 2.3. From system (1.4) we have x˙i (t)xi (t)[bi (t) − aii (t)(xi (t))ii ]. By the differential inequality theorem and Lemma 1.2, for any > 0 there exists a Ti1 > t0 such that for t > Ti1 and i = 1, 2, . . . , r, xi (t) < Xi0 (t) + , where Xi0 (t) is any strictly positive solution of (1.7). Let ki = sup{Xi0 (t) + |t t0 }. Then ki does not depend on any solution of (1.7). Thus for i = 1, 2, . . . , r, xi (t)ki
for all t Ti1 .
(4.17)
Note the fact M[Xi0 (t) + c] = M[Xi0 (t)] + c, where c is a constant. From the boundedness of Xi0 (t), i = 1, 2, . . . , r, and condition (2.6) we can choose 1 small enough such that ⎡ ⎤ r n M ⎣bi (t) − aij (t)(Xj 0 (t) + 1 )ij − aij (t)1ij − aii (t)1ii ⎦ j =1,j =i
> 1
j =r+1
(4.18)
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for all i = 1, 2, . . . , r, i.e., ⎧ ⎡ r ⎨ 1 t2 ⎣bi (t) − aij (t)(Xj 0 (t) + 1 )ij lim sup t→+∞ ⎩ t2 − t1 t1 j =1,j =i ⎫ ⎤ n ⎬ − aij (t)1ij − aii (t)1ii ⎦ dt|t2 − t1 s > 1 , ⎭ j =r+1
which deduces that ⎧ ⎡ r ⎨ t2 ⎣bi (t) − lim sup t→+∞ ⎩ t1
aij (t)(Xj 0 (t) + 1 )ij
j =1,j =i
−
n
aij (t)1ij
⎤
− aii (t)1ii ⎦ dt|t2
j =r+1
− t1 s
⎫ ⎬ ⎭
= +∞.
(4.19)
Clearly, 1 does not depend on any solution of system (1.4). By assumptions (2.4) and (2.5), from Theorem 2.2, for above 1 > 0 there exists T2 > t0 such that for t > T2 0 < xi (t) < 1 ,
i = r + 1, . . . , n.
(4.20)
Now we claim that there exists i > 0 and T > max{Ti1 , Ti2 , . . . , Tir , T2 } such that for all t >T xi (t) > i ,
i = 1, 2, . . . , r.
By using (4.19) and (4.20), the proof of above claim follows that of the proof of the first part of Theorem 2.1 with slight modification and we omit the detail here. The proof is complete. Before we prove Theorem 2.4, we need the following lemma. Lemma 4.2. Suppose that condition (2.6) in Theorem 2.3 holds, then subsystem (2.8) is permanent. Lemma 4.2 can be proved by using the same method given in the proof of Theorem 2.1 and we omit the detail here. Proof of Theorem 2.4. Let X(t) = (x1 (t), x2 (t), . . . , xn (t)) be any positive solution of system (1.4) and u(t) = (u1 (t), . . . , ur (t)) be any positive solution of subsystem (2.8). By Theorem 2.2, we have xi (t) → 0 as t → +∞ for all i > r. By Theorem 2.3 and Lemma 4.2, there are positive constants i , ki and enough large T > 0 such that i xi (t), ui (t)ki ,
(i = 1, 2, . . . , r) for all t > T .
(4.21)
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Let be a positive constant with min1 i r {i } and yi (t) = xi (t)/, vi (t) = ui (t)/ Then system (1.4) is transformed into ⎤ ⎡ n (4.22) aij (t)ij (yj (t))ij ⎦ . y˙i (t) = yi (t) ⎣bi (t) − j =1
And system (2.8) is transformed into ⎤ ⎡ r aij (t)ij (vj (t))ij ⎦ . v˙i (t) = vi (t) ⎣bi (t) −
(4.23)
j =1
Obviously, y(t) = (x1 (t)/, . . . , xn (t)/) is the positive solution of system (4.22) and v(t) = (u1 (t)/, . . . , ur (t)/) is the positive solution of system (4.23). Also, system (1.4) is equivalent to system (4.22) and system (2.8) is equivalent to system (4.23), which means that to end the proof of Theorem 2.4, it is enough to show that lim (yj (t) − vj (t)) = 0,
t→+∞
j = 1, 2, . . . , r.
From (4.21) we have i yi (t),
vi (t)
ki ,
(i = 1, 2, . . . , r) for all t > T .
Consider the Lyapunov function as follows: Vr (t) =
r
i | ln yi (t) − ln vi (t)|.
i=1
Calculating the upper right derivative of Vr (t), we have D + Vr (t) r − i ii aii (t)|(yi (t))ii − (vi (t))ii | i=1
+
r
r
j j i aj i (t)|(yi (t))j i − (vi (t))j i | + g(t)
i=1 j =1,j =i
for all t t0 , where g(t) =
r i=1
i
n
ij aij (t)(yj (t))ij .
j =r+1
From yi (t) = xi (t)/, i = 1, 2, . . . , n, we know that yi (t)1
for t T .
(4.24)
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Since y =|a x −bx | is an increasing function for a 1, a b and x > 0. For ii maxj {j i } and t T , we get |(yi (t))j i − (vi (t))j i ||(yi (t))ii − (vi (t))ii |,
i, j = 1, 2, . . . , r.
(4.25)
Therefore, by using the mean value theorem, for t T , it follows from (2.9), (4.21) and (4.25) that D + Vr (t) −
r
⎛ ⎝i ii aii (t) −
−
⎞ j j i aj i (t)⎠ |(yi (t))ii − (vi (t))ii | + g(t)
j =1,j =i
i=1 r
r
|(yi (t))ii − (vi (t))ii | + g(t)
i=1
−
r
ii ( i (t))ii −1 |yi (t) − vi (t)| + g(t)
i=1
i ii −1 ki ii −1 |yi (t) − vi (t)| + g(t) − , ii min i=1 r i ii −1 ki ii −1 , | ln yi (t) − ln vi (t)| + g(t) − ii min ki
r
i=1
− Vr (t) + g(t), ii −1 , where i (t) lies between yi (t) and vi (t) and = min1 i r {−1 i ii min{(i /) −1 ii }/ki }. Applying the differential inequality theorem and the variation of con(ki /) stants formula of solutions of first-order linear differential equation, we have t Vr (t)e−(t−T ) g(s)e(s−T ) ds + Vr (T ) . (4.26) T
Since g(t) → 0 as t → +∞, it is not hard to prove Vr (t) → 0 as t → +∞. This implies that for all i r yi (t) − vi (t) → 0
as t → +∞,
which is equivalent to xi (t) − ui (t) → 0 The proof is complete.
as t → +∞.
5. Examples Following two examples show the feasibility of our results.
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Example 1. Consider the two-species competitive system x˙1 (t) = x1 (t)[4 − (2 + cos t)x1 (t) − (x2 (t))2 ], x˙2 (t) = x2 (t)[3 − (x1 (t))2/3 − (2 + sin t)x2 (t)].
(5.1)
In this case, b1 (t) = 4, b2 (t) = 3, a11 (t) = 2 + cos t, a12 (t) = 1, a21 (t) = 1, a22 (t) = 2 + sin t, 11 = 1, 12 = 2, 21 = 23 , 22 = 1. The corresponding Logistic equations of (5.1) is as follows: x˙1 (t) = x1 (t)[4 − (2 + cos t)x1 (t)],
(5.2)
x˙2 (t) = x2 (t)[3 − (2 + sin t)x2 (t)].
(5.3)
Already, Zhao et al. [34, pp. 274–275] had showed that system (5.2) has a unique positive periodic solution x1∗ (t) = 34(17 + 2 sin t + 8 cos t)−1
(5.4)
and system (5.3) has a unique positive periodic solution x2∗ (t) = 30(20 + 9 sin t − 3 cos t)−1 .
(5.5)
By using computation tool Maple 6, one could easily obtain the following estimate: M[b1 (t) − a12 (t)(x2∗ (t))2 ] = 4 − M[(30(20 + 9 sin t − 3 cos t)−1 )2 ] ≈ 4 − 3.297933395 = 0.702066605 > 0,
(5.6)
M[b2 (t) − a21 (t)(x1∗ (t))2/3 ] = 3 − M[(34(17 + 2 sin t + 8 cos t)−1 )2/3 ] ≈ 3 − 1.709498686 = 1.290501314 > 0.
(5.7)
and
The above two inequalities show that conditions (2.1) of Theorem 2.1 hold, thus, system (5.1) is permanent. Example 2. Consider the following three-species competitive system: x˙1 (t) = x1 (t)[4 − (2 + cos t)x1 (t) − (x2 (t))2 − x3 (t)], x˙2 (t) = x2 (t)[3 − (x1 (t))2/3 − (2 + sin t)x2 (t) − (x3 (t))2 ], x˙3 (t) = x3 (t)[2 − 2(x1 (t))2/3 − 6x2 (t) − (3 + sin t)(x3 (t))2 ].
(5.8)
In this case, b1 (t)=4, b2 (t)=3, b3 (t)=2, a11 (t)=2+cos t, a12 (t)=1, a13 (t)=1, a21 (t)= 1, a22 (t) = 2 + sin t, a23 (t) = 1, a31 (t) = 2, a32 (t) = 6, a33 (t) = 3 + sin t, 11 = 1, 12 = 2, 13 = 1, 21 = 31 = 23 , 22 = 32 = 1, 23 = 33 = 2. One could easily see that for
F. Chen / Nonlinear Analysis: Real World Applications 7 (2006) 895 – 915
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3.5 3 2.5 2 1.5 1 0.5 0 2
4
6
8
10 12 14 16 18 20 t
Fig. 1. Dynamics of system (5.8) with initial values x1 (0) = 2, x2 (0) = 0.7, x3 (0) = 0.5 and t ∈ [0, 20].
species x2 , x3 , condition (2.4) of Theorem 2.2 holds; Also, we have a31 (t) M[b3 ] 2 = < 2 = inf t 0 , m[b2 ] 3 a (t) 21 M[b3 ] 2 a32 (t) = < 2 inf t 0 , m[b2 ] 3 a22 (t) M[b3 ] 2 a33 (t) t 0 . = < 2 inf a23 (t) m[b2 ] 3 Therefore (2.5) holds for r = 2, which means that species x3 (t) → 0 exponentially as t → +∞. Now let us consider the following subsystem of system (5.8): x˙1 (t) = x1 (t)[4 − (2 + cos t)x1 (t) − (x2 (t))2 ], x˙2 (t) = x2 (t)[3 − (x1 (t))2/3 − (2 + sin t)x2 (t)].
(5.9)
This is the system we had investigated in Example 1. Also, from (5.6) and (5.7) we see that conditions (2.6) of Theorem 2.3 hold, thus species x1 , x2 is permanent. Fig. 1 shows the dynamics behavior of system (5.8). Acknowledgements The author is grateful to an anonymous referee for his excellent suggestions, which greatly improve the presentation of the paper. Also, this work was supported by the National Natural Science Foundation of China (Tian Yuan Foundation) (10426010), the Foundation of Science and Technology of Fujian Province for Young Scholars (2004J0002) and the Foundation of Fujian Education Bureau (JA04156). References [1] S. Ahmad, On the nonautonomous Volterra–Lotka competition equations, Proc. Amer. Math. Soc. 117 (1993) 199–204. [2] S. Ahmad, Extinction of species in nonautonomous Lotka–Volterra systems, Proc. Amer. Math. Soc. 127 (1999) 2905–2910.
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