Applicable Analysis Vol. 83, No. 4, April 2004, pp. 411–431
Harmless and Profitless Delays in Discrete Competitive Lotka^Volterra Systems SHENGQIANG LIUa,b,*, LANSUN CHENa,y and RAVI P. AGARWALc,z a
Academy of Mathematics and System Sciences, Academia Sinica, Beijing, 100080, P.R. China; Department of Mathematics, University of Turku, 20014,Turku, Finland; c Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA b
Communicated by R.P. Gibert (Received 15 March 2003) Recently, time-delayed discrete population dynamical systems have received much attention. Many authors are interested in studying the effects of time-delays on the dynamical behaviors of discrete systems. Among them, Saito et al. [Y. Saito, W. Ma and T. Hara (2001). Necessary and sufficient condition for permanence of a Lotka–Volterra discrete system with delays. J. Math. Anal. Appl., 256, 162–174; Y. Saito, T. Hara and W. Ma (2002). Harmless delays for permanence and impersistence of Lotka–Volterra discrete predator–prey system. Nonlinear Analysis, 50, 703–715.], Tang and Xiao [S. Tang and Y. Xiao (2001). Permanence in Kolmogorov-Type systems of delay difference equations. J. Diff. Eqns. Appl., 7, 1–15.] have considered the two-species Lotka–Volterra discrete system with time-delays, and they conclude that time-delays therein are harmless for permanence. How will time-delays affect the dynamical behaviors of the general Lotka–Volterra discrete systems? In this article, we discuss a general n-species discrete competitive Lotka–Volterra system with delayed density dependence and delayed interspecific competition. We obtain some new results about the effect of time-delays on permanence, extinction and balancing survival. We conclude that under some conditions, the inclusion, exclusion and change of time-delays do not affect the conditions for the permanence, extinction and balancing survival of species. We also find that time-delays are harmless for both the permanence and balancing survival of species, in addition to being profitless to the extinction of species. In particular, when n ¼ 2, the extinction and permanence of this system are corresponded to some inequalities that only involve the coefficients therein. Importantly, permanence and extinction in this two-species system are determined only by three elements: growth rate, density dependence and interspecific competition rate. Keywords: Permanence; Extinction; Balancing survival; Discrete Lotka–Volterra systems; Time-delays AMS Subject Classifications: 34K15; 92D25; 34C25; 34D20
1 INTRODUCTION Difference equations are frequently used in modeling the interactions of populations with nonoverlapping generations. Among the very basic and important ecological *Corresponding author. Fax: + 358 2 333 6595; E-mail:
[email protected] y E-mail:
[email protected] z E-mail:
[email protected] ISSN 0003-6811 print: ISSN 1563-504X online ß 2004 Taylor & Francis Ltd DOI: 10.1080/00036810310001643202
412
S. LIU et al.
problems associated with the populations dynamical systems, there are some critical examples, namely, the long-time coexistence of some or all of the species involved, and the elimination future of some populations. Such problems in the nondelayed discrete systems have attracted much attention and many resolutions have been obtained [1,5,14,17,24,36]. However, recent studies of the dynamics of natural populations indicate that the density-dependent population regulation sometimes takes place over many generations (e.g. [3,6–10,12,13,15,16,18,19,26–29,34,35] and the references therein). Therefore, it is realistic to introduce time-delays into these nondelayed discrete models and consider the delayed discrete models. This gives rise to a new question: How will the introduction of time-delays into the discrete models affect their permanence, extinction and partial survival? Many different models are constructed and thus different answers are obtained [9,12,19,30,31,33]. Ginzburg and Taneyhill [12] developed a two-dimensional model of delayed difference equation which relates the average quality of individuals to patterns of abundance. The delayed density dependence is caused by transmission of quality between generations through maternal effects. They proved that the delayed model can produce patterns of population fluctuations. Crone [9] presented a nondelayed model and reveals that the inclusion of delays changes the shape of population cycles (flip vs Hopf bifurcations) and decreases the range of parameters which are used to predict stable equilibria. Keeling et al. [19] proposed that delayed density dependence should be one of the reasons for stabilizing the natural enemy–victim interactions and allowing the long-term coexistence of the two species. As for the Lotka–Volterra systems, Saito et al. [30,31] and Tang et al. [33] have studied the two-species Lotka–Volterra difference system with time-delays and prove that delays therein are harmless for the permanence of the system. Then how will time-delays affect the asymptotic behaviors for the general Lotka–Volterra systems? To answer the question, in this article, we construct and study the discrete n-species Lotka–Volterra systems with delayed density dependence and delayed interspecific interaction. Our model takes the following form: ( xi ðm þ 1Þ ¼ xi ðmÞ exp bi
p n X X
) aijðkÞ xj ðm
ijðkÞ Þ
,
i ¼ 1, 2, . . . , n,
ð1Þ
j¼1 k¼1
where xi (m) represents the density of population i at the mth generation; ijðkÞ is the nonnegative integer delay to the competition rate between the species i and j; bi represents the growth rate of species i and aijðkÞ measures the influence of the ðm ijðkÞP Þth generation of population j on population i. It is assumed that aijðkÞ 0, ðkÞ bi > 0, pk¼1 aðkÞ ii > 0 and aii > 0 ði, j ¼ 1, . . . , n, k ¼ 1, . . . , pÞ. Using to denote ðkÞ maxfij : i, j ¼ 1, . . . , n, k ¼ 1, . . . , pg, we also assume system (1) satisfies the following initial conditions: xi ðvÞ 0,
xi ð0Þ > 0,
v ¼ 1, . . . , , i ¼ 1, 2, . . . , n:
We define the following terms: aij ¼
p X k¼1
aðkÞ ij ,
A ¼ ðaij Þnn ,
B ¼ ðb1 , . . . , bn ÞT , i, j ¼ 1, . . . , n;
HARMLESS AND PROFITLESS DELAYS
413
Rþ n ¼ fðx1 , x2 , . . . , xn Þ: xi 0, i ¼ 1, . . . , ng þ with Int Rþ n being the internal region of Rn : And X is used to represent the solution to T (1) with X ¼ ðx1 , . . . , xn Þ , i, j ¼ 1, 2, . . . , n: In this article, species i is called permanent if there exist positive constants Ci > ci such that the ith component of any solution to (1) will ultimately enter and stay in the set ðci , Ci Þ, ði ¼ 1, . . . , nÞ. System (1) is called a permanent one (uniformly persistent) if all the species in (1) are permanent. Species i is defined to go extinct if the ith component of any solution to (1) ultimately converges to zero. System (1) is called an r-balancing survival one if r species are permanent while the remaining n r species are extinct. From the above definitions, we can regard that the permanence of system (1) is one extreme case of the r-balancing survival of system (1) with r ¼ n, that the extinction of all but one species in (1) is the other extreme case of it with r ¼ 1: Recently, Tang and Xiao [33], and Saito et al. [30] have studied the special case of system (1) with n ¼ 2 and p ¼ 1. Tang and Xiao [33] showed that the conditions for equilibrium, i.e., ð1Þ b1 að1Þ 21 < b2 a11 ,
ð1Þ b1 að1Þ 22 > b2 a12
ensure the permanence of this two-species system. Later, Saito et al. [30] further proved that the above conditions are also the necessary ones for permanence of the system. Hence time-delays therein are harmless for the permanence. In another paper, Saito et al. [31] considered the discrete Lotka–Volterra system of prey–predator type, and proved the permanence by some inequalities that only involve its coefficients; specifically, time-delays which are also harmless for the permanence of the two-species Lotka–Volterra system of prey–predator type. In this article, we establish the corresponding sufficient conditions for the permanence, extinction and balancing survival of system (1). We find that all of these sufficient conditions are independent of the time-delays, which means that the inclusion (exclusion) of time-delays into (out of) system (1) has no effect on the coexistence and extinction of species, and that time-delays are not only harmless for the permanence of some or all species of the system but also profitless to the extinction of species. Further, we also study the following special case of system (1) with n ¼ 2: ( xi ðm þ 1Þ ¼ xi ðmÞ exp bi
p 2 X X
) aðkÞ ij xj ðm
ijðkÞ Þ
,
i ¼ 1, 2:
ð2Þ
j¼1 k¼1
We find that the permanence and extinction of system (2) are determined only by some inequalities that merely involve its coefficients. That is, permanence and extinction in system (2) are connected and only connected with the three elements: growth rate bi, density dependence aðkÞ ii and the interspecific competition coefficients aðkÞ ði, j ¼ 1, 2, j ¼ 6 i, k ¼ 1, . . . , pÞ: ij Our proof of the permanence results also points out the way to obtain permanence region of system (1), which is formulated in terms of parameters of the system. In practice, one can choose the parameters according to the formulae so that the density of the n-species will eventually lie in desired region.
414
S. LIU et al.
The above definitions of permanence, extinction and balancing survival for difference systems are equivalent to those usually for differential systems (see, e.g., [22,23,25,32]). Permanence has emerged as one of the most pertinent stability concepts for ecological models. The two main approaches to permanence that have been developed are boundary flow analysis and Lyapunov function. These techniques have been exploited with some success for continuous models [11,20–23,32,37], but much less for discrete models [24,30,31,33,36]. In this article, we will extend the methods employed in [11,20–23,32,37] and construct the vector Lyapunov function to establish the boundary of the permanence region for system (1). Further, motivated by the arguments in [2,21,25], we construct the balancing survival extinction (BSE)-typed vector Lyapunov function to obtain the extinction and balancing survival; these BSE-typed Lyapunov functions are made up of two parts: one part to deduce the extinction, the other to obtain the survival permanence. In comparison with the continuous models (since mean value theorem does not hold in the discrete models) the proof of our main result is more difficult and is highly technical. This article is organized as follows. In the next section we present and prove our results for the permanence, extinction and balancing survival of system (1). In Section 3, we derive the equivalent conditions for the permanence and extinction of system (2). Discussion follows in the last section.
2
LONG-TIME BEHAVIORS FOR THE n-SPECIES SYSTEM
In this section, we present and prove our main results for the long-time behaviors of system (1). We have two theorems and two corollaries. The purpose of Theorem 1 is to give the sufficient conditions for the permanence of (1) as well as to obtain the eventual upper bound and lower bound for positive solution of (1), and hence to give a suitable permanence region explicitly. Corollary 1 obtains the special sufficient conditions for the permanence of system (1). Theorem 2 gives the sufficient conditions for the balancing survival of some species in (1). Corollary 2 establishes the conditions for the extinction of every species but one in (1). It seems that conclusions for system (1) are surprisingly equal to those for its corresponding continuous systems (e.g. [21–23,32]). To prove the main results, four lemmas are needed. Lemma 2.1 ensures that all the solutions to system (1) are positive and ultimately bounded. Lemma 2.2 shows the permanence of a delayed single-species discrete system. Lemma 2.3 implies that the sum of the densities of all species in system (1) eventually has its positive lower bound. Lemma 2.4 displays some relationship of the coefficients of system (1) under certain conditions. Below, we present our main results in this section. THEOREM 1
Assume
The Matrix A is nonsingular and the vector equation AX ¼ B has a positive solution X0 ¼ ðx10 , . . . , xn0 ÞT ; ðH2 Þ Let ðij Þnn denote the inverse matrix of A with ii > 0 and ij 0 for all i, j ¼ 1, . . . , n and j 6¼ i: Then system (1) is permanent.
ðH1 Þ
By Theorem 1 and the matrix theory, we directly derive the following corollary.
HARMLESS AND PROFITLESS DELAYS
COROLLARY 1
415
Assume
ðH3 Þ bi >
n X
aij
j6¼i
bj , ajj
i ¼ 1, . . . , n:
Then system (1) is permanent. Let 2 q < n, Aq ¼ ðaij Þqq , Bq ¼ ðb1 , . . . , bq ÞT , Xq ¼ ðx1 , . . . , xq ÞT : We have THEOREM 2
Assume
The Matrix Aq is nonsingular and the vector equation AqXq ¼ Bq admits a positive solution Xq0 ¼ ðx1q , . . . , xqq ÞT : ðH5 Þ Let ðij Þnn be the inverse matrix of Aq with ii > 0 and ij 0 for all i, j ¼ 1, . . . , n and j 6¼ i: ðH6 Þ For all q r n, there exists ir < r such that br air j bir ar j < 0 holds for all j ¼ 1, 2, . . . , r: Then system (1) is q-balancing survival, i.e., species 1, . . . , q are permanent while species q þ 1, . . . , n will go extinct.
ðH4 Þ
Let r ¼ n 1. Then Theorem 2 directly yields the following corollary. COROLLARY 2 ðH7 Þ
Assume
For each 2 r n, there exists a positive integer ir with ir < r and br air j bir arj < 0 holds for all j ¼ 1, 2, . . . , r: Then all species in (1) except species 1 are going extinct while species 1 is permanent.
Remark 1 Since conditions in Theorems 1 and 2 and Corollaries 1 and 2 are all independent of the delays ijðkÞ , once conditions for this propositions are satisfied, the inclusion, exclusion or the change of the time-delays will not affect the conclusions. Remark 2 Kuang [20], Tang and Kuang [32], Liu and Chen [21] obtained the conditions that ensure the permanence for the n-species Lotka–Volterra differential equations with delays. Their conclusions suggest that time-delays are harmless for the permanence of the continuous Lotka–Volterra system. Our results in Theorem 1 and Corollary 1 seem close to theirs. Remark 3 Theorem 1 and Corollary 2 can be regarded as the two extreme cases of Theorem 2 with r ¼ n and r ¼ 1 respectively. Noting that Lemma 2.3 yields that r 6¼ 0 for Theorem 2, then Theorem 2 unifies Theorem 1 and Corollary 2. Let ( C¼
bi
max expfbi ðiið1Þ 1in að1Þ ii
) þ 1Þg :
Then using the arguments similar to those in Lemma 2.1 in [30], we can prove the following lemma.
416
S. LIU et al.
LEMMA 2.1 (i)
Assume system (1) satisfies the positive initial conditions, then
Each solution of (1) is strictly positive and lim sup xi ðmÞ C,
m!1
(ii)
i ¼ 1, . . . , n;
xi ðmÞ < C for all i ¼ 1, . . . , n and m 0 if 0 < xi ðvÞ < C for all v ¼ , . . . , 0, i ¼ 1, . . . , n:
By Lemma 2.1, for each of the positive initial conditions to (1), there exists a positive integer m0 such that xi ðmÞ C for all m > m0 , i ¼ 1, . . . , n, where m0 depends on the initial conditions of (1). Consider the following discrete single-species system with delay. xðm þ 1Þ ¼ xðmÞ expfb axðm 0 Þg,
ð3Þ
where b > 0, a > 0, 0 is a nonnegative integer, and xðvÞ 0 with v ¼ 0 , 0 þ 1, . . . , 1, xð0Þ > 0: LEMMA 2.2
System (3) is permanent.
Proof Using Lemma 2.1, there exist a positive integer m00 and a constant C0 with C 0 ¼ ðb=aÞ expfbð0 þ 1Þg > 0 such that xðmÞ < C 0 for all m > m00 : Hence from (3), we have xðmÞ ¼ xðm þ 1Þ expfb þ axðm 0 Þg < xðm þ 1Þ expfb þ aC 0 g,
m > m00 þ 0 :
This yields xðm 0 Þ < xðm 0 þ 1Þ expfb þ aC 0 g < < xðmÞ expf0 ðb þ aC 0 Þg, m > m00 þ 0 : Therefore, we have xðm þ 1Þ > xðmÞ exp b a expf0 ðb þ aC 0 ÞgxðmÞ : Now from a well-known result in the theory of Logistic difference equation, we obtain the inequality limm!1 inf xðmÞ > 0, which proves Lemma 2.2. P LEMMA 2.3 For system (1), ni¼1 xi ðmÞ has an ultimately positive lower bound. Proof
Using Lemma 2.1, we get ( xi ðmÞ ¼ xi ðm þ 1Þ exp bi þ
p n X X
) aðkÞ ij xj ðm
ijðkÞ Þ
j¼1 k¼1
( xi ðm þ 1Þ exp bi þ
p n X X j¼1 k¼1
) aðkÞ ij C
,
m > m0 þ , i ¼ 1, . . . , n:
HARMLESS AND PROFITLESS DELAYS
417
This gives ( xi ðm
ijðkÞ Þ
xi ðmÞ exp
" ijðkÞ
bi þ
p n X X
#) aijðkÞ C
:
ð4Þ
j¼1 k¼1
Let ( C1 ¼ max exp
" ijðkÞ
bi þ
p n X X
#! aðkÞ ij C
) : 1 i n, m > m0 þ ,
j¼1 k¼1
n o C2 ¼ max aðkÞ ij C1 : 1 i, j n, 1 k p , Then (4) and (1) yield (
p n X X
xi ðm þ 1Þ xi ðmÞ exp bi
) aðkÞ ij C1 xj ðmÞ
j¼1 k¼1
(
xi ðmÞ exp min fbi g pC2 1in
n X
) xj ðmÞ ,
j¼1
from which it follows that n X i¼1
Let yðmÞ ¼
Pn
i¼1
xi ðm þ 1Þ
n X
( xi ðmÞ exp min fbi g pC2 1in
i¼1
n X
) xj ðmÞ :
ð5Þ
j¼1
xi ðmÞ: Then (5) becomes yðm þ 1Þ yðmÞ exp
min fbi g pC2 yðmÞ :
1in
Now using arguments similar to the Logistic differenceP equation, there exist a positive constant and a positive integer mn > m0 such that ni¼1 xi ðmÞ > for all m > mn : This completes the proof of Lemma 2.3. Remark 4 Lemma 2.3 implies that the sum of the whole populations for a discrete n-species competitive community has eventually a positive lower bound. Thus it is impossible that every species of system (1) goes extinct. Since assumptions ðH2 Þ and ðH5 Þ imply that A1 and A1 q are the M-matrices [4], following Berman and Plemmons [4] and Lemma 8.5.2 of Kuang [20], we have LEMMA 2.4 (i)
Assume ðH2 Þ holds true. Then there exist n positive constants di ði ¼ 1, 2, . . . , nÞ such that di ii >
n X j6¼i
dj jjk j,
i ¼ 1, 2, . . . , n:
418
(ii)
S. LIU et al.
Assume ðH5 Þ holds. Then there exist q positive constants ei ði ¼ 1, . . . , qÞ such that
ei ii >
q X
ej jjk j,
i ¼ 1, 2, . . . , q:
j6¼i
Proof of Theorem 1
Construct the n-dimensional vector Lyapunov function Vðm þ 1Þ ¼ ðV1 ðm þ 1Þ, . . . , Vn ðm þ 1ÞÞ
with
Vl ðm þ 1Þ ¼
8 > <
n Y
p X
n X
9 > =
m X
xi ðm þ 1Þdl li exp dl li aðkÞ ij xj ðsÞ , > > ; : i, j¼1 i¼1 k¼1 s¼mþ1 ðkÞ
l ¼ 1, . . . , n,
ij
ð6Þ where di is defined in Lemma 2.4, ij defined in ðH2 Þ: Then we have ( " #) p n n X X X Vl ðm þ 1Þ ðkÞ ðkÞ ¼ exp dl li bi aij xj ðm ij Þ Vl ðmÞ ð1Þ i¼1 j¼1 k¼1 ( exp
n X
dl li
i, j¼1
( " ¼ exp dl ( " ¼ exp dl
n X
p X
) aijðkÞ ½xj ðmÞ
xj ðm
ijðkÞ Þ
k¼1
li bi
n X
i¼1
i, j¼1
n X
n X
li bi
i¼1
li
p X
#) aðkÞ ij xj ðmÞ
k¼1
#) li aij xj ðmÞ
:
i, j¼1
From ðH1 Þ, we have X0 ¼ A1 B: Noting ðij Þnn ¼ A1 , it follows that xl0 > 0 and n X i¼1
( li aij ¼
1,
j ¼ l,
0,
j 6¼ l:
Pn
i¼1
bi li ¼
Thus Vl ðm þ 1Þ ¼ expfdl ðxl0 xl ðmÞÞg: Vl ðmÞ ð1Þ
ð7Þ
HARMLESS AND PROFITLESS DELAYS
419
Let (
(
M1 ¼ max expfbi g, exp
p n X X
)
)
aðkÞ ij C : 1 i n, m m0 þ ,
j¼1 k¼1
(
(
M2 ¼ max M1 , C, exp
n X
dl jli j
i, j¼1
0 ¼
1 min fxi , 1g, 2 1in 0
2 ¼
1 ðM2 Þ2 : 2
p X
) ijðkÞ aðkÞ ij C
) :1ln ,
k¼1
P n o d 1 ¼ min ðM2 Þ2 C i6¼l l li d0l ll ð1 þ M1 Þdl ll , 1ln
From Lemma 2.1, 0 < xi ðmÞ < C, ði ¼ 1, . . . , nÞ for all m m0 , thus M1 , M2 , 1 , 2 are bounded and positive constants. The definition of M1 implies that xi ðmÞ ðM1 Þ1 < xi ðm þ 1Þ < xi ðmÞ M1 ,
m m0 þ , i ¼ 1, . . . , n:
Define WðmÞ ¼ ðW1 ðmÞ, . . . , Wn ðmÞÞ where Wl ðmÞ ¼
n Y
xi ðmÞdl li ,
l ¼ 1, 2, . . . , n:
i¼1
Then Lemma 2.1 with the definition of M2 yields that M21 Wl ðmÞ Vl ðmÞ Wl ðmÞ M2 , m > m0 þ , l ¼ 1, . . . , n:
ð8Þ
Now, we define the following sets S ¼ fX 2 Rnþ : xl ðmÞ C, l ¼ 1, . . . , n, m > m0 þ g, S1 ¼ fX 2 Rnþ : Wl ðmÞ 1 , xl ðmÞ C, l ¼ 1, . . . , n, m > m0 þ g, S2 ¼ fX 2 Rnþ : Wl ðmÞ 2 , xl ðmÞ C, l ¼ 1, . . . , n, m > m0 þ g, S3 ¼ fX 2 Rnþ : Vl ðmÞ 1 M2 , xl ðmÞ C, l ¼ 1, . . . , n, m > m0 þ g: By the definitions of 1 , 2 and (8), we get S3 S1 S2 S: Now we shall prove S2 Int Rnþ : Since 8XðmÞ 2 S2 ðm > m0 þ Þ, we have n Y i¼1
Wi ðmÞ ð2 Þn ,
i:e:,
n Y i¼1
xi ðmÞ
Pn l¼1
dl li
ð2 Þn :
420
S. LIU et al.
Using Lemmas 2.1 and 2.4, we have
xi ðmÞ
Pn
d l¼1 l li
n
ð2 Þ
n Y
! Pn
l¼1
dl lj
ð2 Þn C n
xj ðmÞ
Pn l¼1
dl lj
> 0,
i ¼ 1, . . . , n:
j6¼i
Thus Pn 1 Pn dl li l¼1 n d n l lj l¼1 xi ðmÞ #i ¼ ð2 Þ C
for all m > m0 þ :
Let ¼ fðx1 , x2 , . . . , xn Þ: #i xi < C, i ¼ 1, 2, . . . , ng: Then we have XðmÞ 2 , which shows that S2 IntRnþ . Next we shall prove that the orbit X(m) enters and remains in the region S2 for a sufficiently large m. For this, we need to prove the following three claims. CLAIM 1
Assume Xðm00 Þ 2 S1 for some m00 > m0 þ . Then XðmÞ 2 S2 for all m m00 :
Suppose it is false. Then there must exist some m2 > m1 m00 such that Xðm1 Þ 2 S1 , Xðm2 Þ 2 SnS2 and XðmÞ 2 S2 nS1 for all m1 < m < m2 (the set (m1, m2) may be empty). We have two cases to consider. Consider the first case m2 ¼ m1 þ 1: Since Xðm2 Þ 2 SnS2 , then there exists l 2 f1, . . . , ng such that Wl ðm2 Þ < 2 : By the definitions of Wl ðmÞ, M1 , M2 , 1 , 2 and noticing li 0 for all i 6¼ l, we get ½xl ðm2 Þ
dl ll
< 2
n Y
l li xd ðmÞ i
< 1
M22
i6¼l
n Y
C
dl li
i6¼l
0 1 þ M1
dl ll ,
which implies xl ðm2 Þ < 0 =ð1 þ M1 Þ: Then by (1) and the definitions of M1 , M2 , we have ( xl ðm1 Þ ¼ xl ðm2 Þ exp bl þ
p n X X j¼1 k¼1
) aðkÞ lj xj ðm2
ljðkÞ Þ
0 M1 < 0 : 1 þ M1
Using (7), we get ðVl ðm2 Þ=Vl ðm1 ÞÞ > expfdl ðxl0 0 Þg > 1, i.e., Vl ðm2 Þ > Vl ðm1 Þ: Noticing Wl ðm1 Þ 1 , we find Vl ðm2 Þ Wl ðm2 Þ M2 < 2 M2 < 1 ðM2 Þ1 Wl ðm1 Þ ðM2 Þ1 Vl ðm1 Þ: But this is a contradiction, and hence Claim 1 holds in the case m2 ¼ m1 þ 1: Now consider the second case m2 > m1 þ 1. By the assumption of this claim, Wi ðm1 Þ 1 and there exists some integer l 2 f1, . . . , ng such that Wl ðm2 Þ < 2 : Let m3 ¼ maxfm: m m1 , Wl ðmÞ 1 g: This implies m1 m3 < m2 , Wl ðm3 Þ 1 while
HARMLESS AND PROFITLESS DELAYS
421
Wl ðmÞ < 1 for all m3 < m m2 : Using the arguments analogous to those for the case m2 ¼ m1 þ 1, we have xl ðmÞ <
0 , 1 þ M1
m3 < m m2 :
This yields xl ðm3 Þ xl ðm3 þ 1Þ M1 < 0 : And so by (7), we get Vl ðm3 Þ < Vl ðm3 þ 1Þ < Vl ðm3 þ 2Þ < < Vl ðm2 1Þ < Vl ðm2 Þ: But from (8), we have Vl ðm2 Þ Wl ðm2 Þ M2 < 2 M2 < 1 M21 W1 ðm3 Þ M21 Vl ðm3 Þ, a contradiction, which proves Claim 1. CLAIM 2 There exists some positive constant 3 ¼ 3 ðXð0ÞÞ such that xl ðmÞ 3 for all m 0 and l ¼ 1, 2, . . . , n: Let M00 ¼ supfxi ðmÞj 1 i n, m g, # ( " ) p n X X ðkÞ ðkÞ 0 aij xj ðm ij Þ : m , 1 i, j n , M1 ¼ sup exp bi i¼1 k¼1 9 3 > p = n m 1 X X X 6 7 ðkÞ 0 M2 ¼ sup exp4 dl jli j aij xj ðsÞ5: m , 1 i, j n , > > ; : i, j¼1 k¼1 s¼m ðkÞ 8 > <
2
ij
( 0 < 3 ¼ min
1ln
ðM00 Þ
P
d i6¼l l li
d0l ll
ðM20 Þ2
ð1 þ
M10 Þdl ll ,
) n Y dl li ½xi ð0Þ , i¼1
0 < 4 3 ðM20 Þ2 : Clearly, M00 , M10 , M20 , 3 , 4 are positive constants. Define S10 ¼ fX 2 Rnþ : Wl ðmÞ 3 , xl ðmÞ M00 , l ¼ 1, . . . , n, m 0g, S20 ¼ fX 2 Rnþ : Wl ðmÞ 4 , xl ðmÞ M00 , l ¼ 1, . . . , n, m 0g Using the arguments similar to those in Claim 1, we have S10 S20 Int Rnþ : Since Xð0Þ 2 S10 , arguments similar to those in Claim 1, give XðmÞ 2 S20 , ðm 0Þ: This completes the proof of Claim 2. Note that 3 depends on the initial values of system (1). CLAIM 3
There exists some m > m0 þ such that XðmÞ 2 S3 .
Suppose it is false, namely, XðmÞ 2 SnS3 for all m > m0 þ . Define VðmÞ ¼ min1ln fVl ðmÞg: Then it follows from the definition of S3 that VðNÞ < 1 M2 holds
422
S. LIU et al.
for any N > m0 þ . Let 1 p1 , p2 n and VðNÞ ¼ Vp1 ðNÞ, VðN þ 1Þ ¼ Vp2 ðNÞ: Using (8), we get Wp2 ðN þ 1Þ Vp2 ðN þ 1Þ M2 < 1 ðM2 Þ2 : Hence it follows from the definition of Wp2 ðN þ 1Þ that
1=dp p p P 2 2 2 d 2 i6¼p2 p2 p2 i < xp2 ðN þ 1Þ < 1 ðM2 Þ C
0 , 1 þ M1
m > m0 þ :
Now by (1), xp2 ðNÞ < xp2 ðN þ 1Þ M1 < 0 , thus VðN þ 1Þ Vp2 ðN þ 1Þ Vp2 ðN þ 1Þ Vp2 ðNÞ Vp2 ðN þ 1Þ ¼ ¼ VðNÞ Vp1 ðNÞ Vp2 ðNÞ Vp1 ðNÞ Vp2 ðNÞ p2 ¼ expfdp2 ðx0 xp2 ðNÞÞg > expfdp2 0 g exp min di 0 > 1: 1in
Therefore, ðVðm þ 1Þ=VðmÞÞ expfmin1in di 0 g for all m > m0 þ , which directly implies VðmÞ ! 1 as m ! 1: But by Claim 2 and the definition of V(m), V(m) should be ultimately bounded, which is a contradiction. This completes the proof of Claim 3. Claims 1 and 3 yield Theorem 1, and this completes the proof of Theorem 1. Remark 5 Arguments in Theorem 1 suggest that any solution of system (1) will ultimately enter and remain in the positive region : Since can be formulated in terms of the parameters of system (1), we can obtain explicit estimates for the eventual upper bound and eventual lower bound of the species x1 , . . . , xn : Proof of Theorem 2 Note that ðH5 Þ and Lemma 2.4 imply that the matrix ðij Þqq admits positive constants e1 , e2 , . . . , eq such that ei ii >
X
ej jji j,
i ¼ 1, . . . , q:
j6¼i, 1 jq
We construct the BSE-type vector function UðmÞ ¼ ðU1 ðmÞ, . . . , Un ðmÞÞ with 8 8 9 > < P = q q P p m > Q P > er ri > > ½x ðmÞ exp e x ðsÞ , i r ri j > > : i, j¼1 k¼1 ; ðkÞ > i¼1 s¼mþ1 > ij < 8 Ur ðmÞ ¼ 0 19 > > > > bir p r m 1 m 1 = < > X ðvÞ X XB X ðvÞ > xr ðmÞ C > > , a b x ðsÞ a b x ðsÞ exp @ A > r j i j r rj ir j > b > > > ; : j¼1 v¼1 s¼m ðvÞ : xirr ðmÞ s¼m ðvÞ ir j
where ir , ri are defined in ðH6 Þ:
rj
r ¼ q,
r > q,
HARMLESS AND PROFITLESS DELAYS
CLAIM 1
423
xn ðmÞ ! 0 exponentially as m ! 1:
Denoting k ¼ in, then by ðH6 Þ, k < n, we get ( ) p n X X Un ðm þ 1Þ ðvÞ ðvÞ ðvÞ ðvÞ ¼ exp bk anj xj ðm nj Þ þ bn akj xj ðm kj Þ Un ðmÞ ð1Þ j¼1 v¼1 9 8 > > p p n X m n X m = < X X X X ðvÞ ðvÞ exp anj bk xj ðsÞ þ akj bn xj ðsÞ > > ; : j¼1 v¼1 s¼mðvÞ þ1 j¼1 v¼1 s¼m ðvÞ þ1 nj
exp
8 > p n X m 1 : j¼1
( ¼ exp
n X
kj
aðvÞ nj bk xj ðsÞ
j¼1
v¼1 s¼m ðvÞ nj
bk
j¼1
p X v¼1
aðvÞ nj
bn
p X
9 > = ðvÞ akj bn xj ðsÞ > ; v¼1 s¼m ðvÞ
p n X m 1 X X
)
! aðvÞ kj
kj
xj ðmÞ
v¼1
(
) n X ¼ exp ðbk anj bn akj Þxj ðmÞ :
ð10Þ
j¼1
Let ¼ min1 jn fbk anj bn akj g. Then it follows from ðH6 Þ that is a positive constant. So ( ) n X Un ðm þ 1Þ exp xj ðmÞ expf g < 1, Un ðmÞ ð1Þ j¼1
m > mn ,
where mn is defined in Lemma 2.3. Thus it follows that Un ðm þ jÞ < Un ðm þ j 1Þ expfg < < Un ðmÞ expfj g,
m > mn ,
i.e., limm!1 Un ðm þ 1Þ ! 0 exponentially. Now, by the definition of Un(m) and the bound of xi ðmÞ, ði ¼ 1, 2, . . . , nÞ, we find xn ðmÞ ! 0 exponentially as m ! 1: CLAIM 2 Assume there exists some integer rðq < r nÞ such that limm!1 xl ðmÞ ¼ 0 exponentially for all r l n. Then limm!1 xr1 ðmÞ ¼ 0 exponentially. Let ir1 ¼ 0. Then we have
Ur1 ðmÞ ¼
0 xbr1 ðmÞ br1 x0 ðmÞ
8 0 19 > > p = n X m 1 m 1 < X X ðkÞ B X C ðkÞ ar1j b0 xj ðsÞ a0j br1 xj ðsÞA : exp @ > > ; : j¼1 k¼1 s¼m ðkÞ s¼m ðkÞ r1j
0j
ð11Þ
424
S. LIU et al.
Using the arguments similar to those in the above Claim 1, we obtain ( ) r1 X Ur1 ðm þ 1Þ ¼ exp ðb0 ar1j br1 a0j Þxj ðmÞ Ur1 ðmÞ ð1Þ j¼1 ( exp
" p n X X j¼r
ðkÞ b0 ar1 xj ðm
k¼1
ðkÞ r1j Þ
br1
p X
#) aðkÞ 0j xj ðm
0jðkÞ Þ
:
k¼1
ð12Þ Let r ¼ min1 jr1 fb0 ar1j br1 a0j g. Then it follows from ðH6 Þ that r > 0. Since limm!1 xj ðmÞ ¼ 0ðr j nÞ exponentially, there exists a sufficiently large mr > mn such that ! p p n X X X ðkÞ ðkÞ ðkÞ ðkÞ b0 ar1 xj ðm r1j Þ br1 a0j xj ðm 0j Þ < r =4 j¼r k¼1 k¼1 P P and nj¼r xj ðmÞ < =2Pfor all m > mr . But Lemma 2.3 implies that nj¼1 xj ðmÞ > for r1 all m > mn , and so j¼1 xj ðmÞ > =2, m > mr : Using (12), we get Ur1 ðm þ 1Þ expfr =2 þ r =4g ¼ expfr =4g < 1, U ðmÞ r1
m > mr :
ð1Þ
Hence Ur1 ðmÞ ! 0 exponentially as m ! 1. By (11) and the boundedness of xi(m), we obtain xr1 ðmÞ ! 0 exponentially as m ! 1: This proves Claim 2. Claims 1 and 2 imply that limm!1 xr ðmÞ ¼ 0 exponentially for all r ¼ q, q þ 1, . . . , n. Now, we show x1 ðmÞ, . . . , xr ðmÞ are permanent. From (12) it follows that ( ) p n X X Ul ðm þ 1Þ q ðkÞ ðkÞ ¼ exp el ðxl xl ðmÞÞ el lj alj xj ðm lj Þ , Ul ðmÞ ð1Þ j¼qþ1 k¼1
l ¼ 1, . . . , q:
From the exponential extinction of species q þ 1, . . . , n, there exists N > m0 þ such that p n X 1 X ðkÞ ðkÞ el lj alj xj ðm lj Þ < min xql min el , 3 1lq j¼qþ1 k¼1 1lq
m N:
Hence q
2xl Ul ðm þ 1Þ exp e ðmÞ , x l l Ul ðmÞ ð1Þ 3
l ¼ q þ 1, . . . , n, m > N:
HARMLESS AND PROFITLESS DELAYS
425
Now using arguments similar to those in Claims 1–3 of Theorem 1, it follows that species 1, 2, . . . , q are permanent. This proves system (1) is a q-balancing survival.
3
LONG TIME BEHAVIORS FOR THE TWO-SPECIES SYSTEM
The purpose of this section is to establish and prove the necessary–sufficient conditions for the permanence and extinction of system (2). We have three theorems. Theorems 3 and 4 give the necessary–sufficient conditions for the extinction of species and Theorem 5 gives the necessary–sufficient conditions for the permanence of system (2). Liu and Chen [22] have considered the counterpart of system (2) for the following continuous system: 8 ! p 2 X X > > ðkÞ ðkÞ > x_ 1 ðtÞ ¼ x1 ðtÞ b1 a1j xj ðt 1j Þ , > > > < j¼1 k¼1 ! > p 2 > > > x_ ðtÞ ¼ x ðtÞ b X X aðkÞ x ðt ðkÞ Þ , > > 2 2 2j j 2j : 2
ð13Þ
j¼1 k¼1
ðkÞ where bi , aðkÞ ij are the same as those in (2) and ij are the nonnegative constants. In [22], the following three propositions are proved:
PROPOSITION 1 [22] (H7)
Species 2 in (13) extincts if and only if
b1 a21 b2 a11 > 0 and b1 a22 b2 a12 0 or b1 a21 b2 a11 0 and b1 a22 b2 a12 > 0,
holds true. PROPOSITION 2 [22] ðH8 Þ
Species 1 in (13) extincts if and only if
b1 a21 b2 a11 < 0 and b1 a22 b2 a12 0 or b1 a21 b2 a11 0 and b1 a22 b2 a12 < 0
holds true. PROPOSITION 3 [22] ðH9 Þ
System (13) is permanent if and only if
b1 a21 b2 a11 < 0 and
b1 a22 b2 a12 > 0
holds true. Propositions 1–3 imply that the permanence, and the extinction of system (13) are determined only by the three factors: growth rate, intraspecific and interspecific competition. Roughly speaking, the two competing species can coexist if the intraspecific competition of both species is ‘‘stronger’’ than the interspecific competition. When the intraspecific competition of species A is ‘‘stronger’’ than its interspecific competition while the intraspecific competition of species B is ‘‘weaker’’ than its interspecific competition, then species B will extinct while species A will remain permanent. For the discrete system (2), when p ¼ 1, Tang and Xiao [33] proved that ðH9 Þ ensures the permanence of system (2). Saito [30] further proved that ðH9 Þ (p ¼ 1) is also the necessary condition for the coexistence of (2). In what follows, we will extend the results
426
S. LIU et al.
of Saito. We will also study the extinction of species in system (2). Our results are summarized in the following four theorems: THEOREM 3
Species 2 in system (2) extincts if and only if ðH7 Þ holds true.
THEOREM 4
Species 1 in system (2) extincts if and only if ðH8 Þ holds true.
THEOREM 5
System (2) is permanent if and only if ðH9 Þ holds.
Remark 6 Applying Theorem 5 to system (2), we directly deduce the main results established in [30,33]. Moreover, Theorems 3–5 imply that the permanence, extinction and balancing survival in system (2) are caused only by the relationship between bi and aðkÞ ij ði, j ¼ 1, 2, k ¼ 1, 2, . . . , pÞ, i.e., the growth rate, the intraspecific and the interspecific competition rates. Remark 7 Results in Theorems 3–5 for the discrete system (2) are similar to those for the continuous system (13). The coefficients of the system (2) can be classified in the following five cases. Case Case Case Case Case
(I) ðH7 Þ; (II) ðH8 Þ; (III) ðH9 Þ; (IV) b1 a21 > b2 a11 and b1 a22 < b2 a12 ; (V) b1 a21 ¼ b2 a11 and b1 a22 ¼ b2 a12 .
LEMMA 4.1 Assume ðH7 Þ holds. Then species 2 in system (2) goes extinction while species 1 keeps permanent. Proof Construct the two-dimensional BSE-type vector function UðmÞ ¼ ðU1 ðmÞ, U2 ðmÞÞ with
U2 ðmÞ ¼
xb21 ðmÞ xb12 ðmÞ
9 8 > > p p 2 X m 1 2 m 1 = < X X ðvÞ X X X ðvÞ a2j b1 xj ðsÞ þ a1j b2 xj ðsÞ exp > > ; : j¼1 v¼1 s¼m ðvÞ j¼1 v¼1 s¼m ðvÞ 2j
1j
and U1 ðmÞ ¼ x1 ðmÞ. Then we have U2 ðm þ 1Þ ¼ expfðb1 a21 b2 a11 Þx1 ðmÞ ðb1 a22 b2 a12 Þx2 ðmÞg: U2 ðmÞ ð2Þ
ð14Þ
We need to consider the following two cases: CLAIM (i)
b1 a21 b2 a11 and b1 a22 > b2 a12 :
Using (14), we get U2 ðm þ 1Þ < expfðb1 a22 b2 a12 Þx2 ðmÞg < 1: U2 ðmÞ ð2Þ
ð15Þ
HARMLESS AND PROFITLESS DELAYS
427
Hence the decreasing sequence fU2 ðmÞg1 m¼1 must converge to a nonnegative constant 1 : Now, we show 1 ¼ 0: If this is false, i.e., 1 > 0, we have lim
m!1
U2 ðm þ 1Þ ¼ 1, U2 ðmÞ
lim expfðb1 a22 b2 a12 Þx2 ðmÞg ¼ 1,
m!1
and hence, limm!1 x2 ðmÞ ¼ 0: The definition of U2 ðmÞ and the factor that x1 ðmÞ, x2 ðmÞ are ultimately bounded leads to limm!1 x1 ðmÞ ¼ 0; thus limm!1 ðx1 ðmÞ þ x2 ðmÞÞ ¼ 0, which contradicts Lemma 2.3. Then we have 1 ¼ 0, i.e., limm!1 x2 ðmÞ ¼ 0: By Lemma 2.3, x1 ðmÞ keeps permanent, which completes the proof of Claim (i). CLAIM (ii)
b1 a21 > b2 a11 and b1 a22 b2 a12 .
We have U2 ðm þ 1Þ < expfðb1 a21 b2 a11 Þx1 ðmÞg < 1: U2 ðmÞ ð2Þ Thus the decreasing sequence fU2 ðmÞg1 m¼1 converges to a nonnegative constant 2 : We will show that 2 ¼ 0: If it is false, i.e., 2 > 0, then limm!1 ðU2 ðm þ 1Þ=U2 ðmÞ ¼ 1Þ, thus it follows that limm!1 expfðb1 a21 b2 a11 Þx1 ðmÞg ¼ 1, and hence, limm!1 x1 ðmÞ ¼ 0: Now from the definition of U2 ðmÞ and the boundedness of x1 ðmÞ and x2 ðmÞ, we have limm!1 x2 ðmÞ ¼ 0, which contradicts Lemma 2.3. Hence 2 ¼ 0. The definition of U2 ðmÞ directly yields that limm!1 x2 ðmÞ ¼ 0: Proving Claim (ii), this completes the proof of Lemma 3.1. Proof of Theorem 3 Note that Lemma 4.1 yields the sufficiency of Theorem 3, so we just need to prove its necessity. Suppose it is false, i.e., ðH7 Þ does not hold true. It follows from Lemma 4.2 and Corollary 1 that system (2) does not belong to Cases (II) and (III). Then system (2) should belong to the remaining two cases: Cases (IV) and (V). Assume system (2) belongs to Case (V), i.e., b1 a21 ¼ b2 a11 and b1 a22 ¼ b2 a12 holds true, then any point in the set fðx1 , x2 Þ 2 Int R2þ : a11 x1 ðmÞ þ a12 x2 ðmÞ ¼ b1 , m g will remain fixed. But this contradicts the extinction of species 2, therefore system (2) does not belong to Case (V). Thus system (2) must belong to the remaining Case (IV). Let
Q2 ðmÞ ¼ ½x1 ðmÞa22 ½x2 ðmÞa12
9 8 > > p 2 X m 1 = < X X exp dC1i xj ðsÞ , > > ; : i, j¼1 k¼1 s¼m ðkÞ ij
where d ¼ jðaij Þj22 , C11 ¼ a22 =d, C12 ¼ a12 =d, C21 ¼ a21 =d, C22 ¼ a11 =d. we have Q2 ðm þ 1Þ ¼ expfðb2 a12 b1 a22 Þ þ ða12 a21 a11 a22 Þx1 ðmÞg: Q2 ðmÞ ð2Þ
Then
ð16Þ
428
S. LIU et al.
Let xi ðvÞ < C for all v ¼ , . . . , 0, i ¼ 1, 2: Then Lemma 2.1, implies xi ðmÞ < C, i ¼ 1, 2, m 1: Further, let x1 ð0Þ <
b2 a12 b1 a22 ½x2 ð0Þa12 =a21 C a12 =a21 exp 8p d maxfjC1j jgC : j¼1, 2 2ða12 a21 a11 a22 Þ
Using (16), it follows that Q2 ð1Þ ¼ expfðb2 a12 b1 a22 Þ þ ða12 a21 a11 a22 Þx1 ð0Þ < 1: Q2 ð0Þð2Þ Now, we claim Q2 ðm þ 1Þ 0 such that Q2 ðm þ 1Þ = 7 exp dC1i xj ðsÞ 5 > > ; :i, j¼1 k¼1 ðkÞ m0 8 > p 2 X 0: From the definition of Q2 ð0Þ, we have x1 ðmÞ <
b2 a12 b1 a22 2ða12 a21 a11 a22 Þ
ðm 0Þ:
Using (16), we find Q2 ðm þ 1Þ < expfðb2 a12 b1 a22 Þ=2g < 1, Q2 ðmÞ ð2Þ
ðm 0Þ:
This shows limm!1 Q2 ðmÞ ¼ 0, thus limm!1 x1 ðmÞ ¼ 0, a contradiction. Hence system (2) does not belong to Case (IV). Therefore system (2) must satisfy Case (I), this proves Theorem 3. The proof of Theorem 4 is similar. Remark 3 The strict inequalities in ðH6 Þ are necessary for Theorem 2 to ensure the exponential extinction of species q, q þ 1, . . . , n: But for the two-species system (2), the extinction instead of the exponentially extinction is enough to result in the extinction of some species. Then Theorems 3 and 4 hold true under comparatively weaker conditions. Proof of Theorem 5 Since Corollary 1 directly implies the sufficiency, we just need to prove the necessity. Suppose ðH9 Þ does not hold true. Then system (2) must satisfy one of the Cases (I), (II), (IV) or (V). Theorems 3 and 4 imply that system (2) does not belong to Cases (I) and (II). Further, by the arguments similar to that of Theorem 3, we can prove if (IV) holds, then some initial conditions can lead to the extinction of species 1, which is a contradiction. Hence, system (2) must satisfy Case (V). It follows from the permanence of system (2) that there exist positive constants 1 , 2 and sufficiently large integer m > 0 such that x1 ðmÞ > 1 and x2 ðmÞ > 2 for all m m : But by Case (V), all the points in the set fðx1 , x2 Þ 2 Int R2þ : a11 x1 ðmÞ þ a12 x2 ðmÞ ¼ b1 , m g will be fixed for any m, a contradiction, thus system (2) does not belong to Case (V). Hence it follows that system (2) can only satisfy Case (III). Proving the necessity.
4
DISCUSSION
Many authors have studied the effects of time-delays on dynamical behaviors of population difference systems. Crone [9] has shown that the inclusion of time-lags can dramatically change the dynamics and lead to chaos and cyclicity. Further, in [8] the authors have proved that inclusion of delays into the density dependence can destabilize the dynamics which may be stabilized by the nondelayed density dependence. Ginzburg and Taneybill [12] obtained that delays can produce patterns of population fluctuation. Keeling et al. [19] show that time-lags might be one of the causes to stabilize the natural enemy–victim interactions and allow the long-term coexistence of
430
S. LIU et al.
the two species. Recently, Tang and Xiao [33] and Saito et al. [30] studied the effects of time-delays on the two-species competitive systems and they proved that time-delays are harmless for the uniform persistence of system. Saito et al. [31] displayed similar conclusions for the two-species predator–prey systems. Different from the above studies, we consider the effects of time-delays on the permanence, extinction and balancing survival of the n-species systems. We obtain sufficient conditions for these dynamical behaviors. All of these conditions are independent of the time-delays. This implies that if the nondelayed n-species system is permanent (see the conditions for Theorem 1 and Corollary 1), then its corresponding delayed system is also permanent; if some of the species of the nondelayed n-species systems are extinct (survival persistent) (see conditions for Theorem 2 and Corollary 2), then its related delayed system is also extinct (survival persistent). On the other hand, under the corresponding conditions, if the delayed system is permanent or some of its species are extinct (balancing survival), then its related nondelayed system will also be permanent or some of its species will also be extinct (balancing survival). Therefore under proper conditions, time-delays are both harmless for the permanence and profitless to the extinction of species in system (1). Further, we show that the permanence and extinction of the two-species system (2) are shown by some coefficients inequalities (Theorems 3–5), i.e., the extinction of species 1 is equivalent to ðH7 Þ and the extinction of species 2 is shown by ðH8 Þ, the system (2) is permanent if and only if ðH9 Þ holds true. Therefore, the permanence and extinction of species of system (2) are determined and only determined by the relationship of following three components: the growth rate bi, the density dependence ðkÞ aðkÞ ii and the interspecific competition rate aij . Therefore, we arrive at the conclusion: Though time-delays may dramatically affect the dynamics of the population systems [8,9,12,19] and some times even lead to some complicated dynamical behaviors such as chaos [9], they should be excluded from the causes of permanence and extinction for Lotka–Volterra systems.
Acknowledgments The authors would like to extend their appreciation to Prof. Lin He and Dr.Yi–jie Liu for their help and comments on the first draft of this article. We also thank Dr. Sanyi Tang, Dr. Wanbiao Ma, Prof.Wendi Wang, Prof. M.J. Keeling, Prof. L.R. Ginzbur and Prof. John E. Franke for sending their reprints/preprints to us. This work was supported by the Chinese Postdoctoral Science Foundation, Academy of Finland and by the National Natural Science Foundation of China (No. 10171106). References [1] R.P. Agarwal (2000). Difference Equations and Inequalities, 2nd Edn. Marcel Dekker Inc., New York. [2] S. Ahmad and F.M. Oca (1988). Extinction in nonautonomous T–periodic competitive Lotka–Volterra systems. Appl. Math. Comput., 90, 155–166. [3] R. Arditi and L.R. Ginzbur (1989). Coupling in predator–prey dynamics: ratio–dependence. J. Theor. Biol., 139, 311–326. [4] A. Berman and R.J. Plemmons (1994). Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York. [5] D.M. Chan and J.E. Franke (2000). Extinction, weak extinction and persistence in a discrete, competitive Lotka–Volterra model. Inter. J. Appl. Math. Compu. Sci., 10, 7–36.
HARMLESS AND PROFITLESS DELAYS
431
[6] D. Chitty (1960). Population processes in the vole and their relevance to general theory. Can. J. Zool., 38, 99–113. [7] D. Chitty (1996). Do Lemmings Commit Suicide? Beautiful Hypotheses and Ugly Facts. Oxford University Press, New York. [8] E.E. Crone and D.R. Taylor (1996). Complex dynamics in experimental populations of an annual plant, cardamine pensylvanica. Ecology, 77, 289–299. [9] E.E. Crone (1997). Delayed density dependence and the stability of interacting populations and subpopulations. Theor. Popu. Biol., 51, 67–76. [10] E.E. Crone (1997). Parental environmental effects and cyclical dynamics in plant populations. Am. Nat., 150, 708–729. [11] H.I. Freedman and S. Ruan (1995). Uniform persistence in functional differential equations. J. Diff. Eqns., 115, 173–192, (1995). [12] L.R. Ginzburg and D.E. Taneyhill (1994). Population-cycles of forest lepidoptera – a maternal effect hypothesis. J. Anim. Ecol., 63, 79–92. [13] L.R. Ginzburg (1998). Assuming reproduction to be a function of consumption raises doubts about some popular predator–prey models. J. Anim. Ecol., 67, 325–327. [14] J. Hofbauer, V. Hutson and W. Jansen (1987). Coexistence for system governed by difference equations of Lotka–Volterra type. J. Math. Biol., 25, 553–570. [15] M. Holyoak (1984). Identifying delayed density–dependence in time–series data. Oiks, 70, 296–304. [16] B. Hornfeldt (1994). Delayed density–dependence as a determinant of vole cycles. Ecology, 75, 791–806. [17] V. Hutson and W. Moran (1982). Persistence of species obeying difference equations. J. Math. Biol., 15, 203–212. [18] P. Inchausti and L.R. Ginzbur (1998). Small mammals cycles in northern Europe: patterns and evidence for a maternal effect hypothesis. J. Anim. Ecol., 67, 180–194. [19] M.J. Keeling,, B.W. Howard and S.W. Pacala (2000). Reinterpreting space, time lags, and functional responses in Ecological Models. Science, 290, 1758–1761. [20] Y. Kuang (1993). Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York. [21] S. Liu and L. Chen (2002). Permanence, extinction and balancing survival in nonautonomous Lotka– Volterra system with delays. Appl. Math. Comput., 129, 481–499. [22] S. Liu and L. Chen (2003). Necessary–sufficient conditions for permanence and extinction in Lotka– Volterra system with distributed delays. Appl. Math. Letters, 16, 911–917. [23] Z. Lu and Y. Takeuchi (1994). Permanence and global attractivity for competitive Lotka–Volterra system with delay. Nonlinear Analysis, 22, 847–856. [24] Z. Lu and W. Wang (1999). Permanence and global attractivity for Lotka–Volterra difference systems. J. Math. Biol., 39, 269–282. [25] F.M. Oca and M.L. Zeeman (1995). Balancing survival and extinction in nonautonomous competitive Lotka–Volterra systems. J. Math. Anal. Appl., 192, 360–370. [26] T. Mousseau and H. Dingle (1991). Maternal effects in insect life histories. Annu. Rev. Entomol., 35, 511–534. [27] T. Mousseau and C.W. Fox (1998). Maternal Effects as Adaptations, pp. 42–53. Oxford University Press, Oxford. [28] D. Roach and R. Wulff (1987). Maternal effects in plants. Annu. Rev. Ecol. Syst., 18, 209–236. [29] M. Rossiter (1994). Maternal effects hypothesis of herbivore outbreaks. Bioscience, 44, 752–762. [30] Y. Saito, W. Ma and T. Hara (2001). Necessary and sufficient condition for permanence of a Lotka– Volterra discrete system with delays. J. Math. Anal. Appl., 256, 162–174. [31] Y. Saito, T. Hara and W. Ma (2002). Harmless delays for permanence and impersistence of Lotka– Volterra discrete predator–prey system. Nonlinear Analysis, 50, 703–715. [32] B. Tang and Y. Kuang (1996). Permanence in Kolmogorov type systems of nonautonomous functional differential equation. J. Math. Anal. Appl., 197, 427–447. [33] S. Tang and Y. Xiao (2001). Permanence in Kolmogorov-Type systems of delay difference equations. J. Diff. Eqns. Appl., 7, 1–15. [34] P. Turchin (1990). Rarity of density dependence or population regulation with lags. Nature, 334, 660–663. [35] P. Turchin and A. Taylor (1992). Complex dynamics in ecological time series. Ecology, 73, 289–305. [36] W. Wang (1992). Persistence in a discrete model with delays. J. Southwest China Normal Univ., 17, 13–18. [37] W. Wang and Z. Ma (1991). Harmless delays for uniform persistence. J. Math. Anal. Appl., 158, 256–268.
