Existence and global attractivity of positive periodic solutions of ...

Mathematical Biosciences 160 (1999) 47±61

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Existence and global attractivity of positive periodic solutions of periodic n-species Lotka± Volterra competition systems with several deviating arguments 1 Meng Fan *, Ke Wang, Daqing Jiang Department of Mathematics, Northeast Normal University, Changchun 130024, People's Republic of China Received 29 September 1998; received in revised form 15 February 1999; accepted 3 March 1999

Abstract In this paper, we study the existence and global attractivity of positive periodic solutions of periodic n-species Lotka±Volterra competition systems. By using the method of coincidence degree and Lyapunov functional, a set of easily veri®able sucient conditions are derived for the existence of at least one strictly positive (componentwise) periodic solution of periodic n-species Lotka±Volterra competition systems with several deviating arguments and the existence of a unique globally asymptotically stable periodic solution with strictly positive components of periodic n-species Lotka±Volterra competition system with several delays. Some new results are obtained. As an application, we also examine some special cases of the system we considered, which have been studied extensively in the literature. Some known results are improved and generalized. Ó 1999 Elsevier Science Inc. All rights reserved. AMS: 34K15; 92D25; 34C25; 34D20 Keywords: Positive periodic solutions; Global attractivity; Lotka±Volterra competition system; Deviating arguments; Coincidence degree; Lyapunov functional

*

Corresponding author. E-mail: [email protected] Supported by National Natural Sciences Foundation of People's Republic of China No.: 19871012. 1

0025-5564/99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 5 - 5 5 6 4 ( 9 9 ) 0 0 0 2 2 - X

48

M. Fan et al. / Mathematical Biosciences 160 (1999) 47±61

1. Introduction In recent years, the application of theories of functional dierential equations in mathematical ecology has developed rapidly. Various mathematical models have been proposed in the study of population dynamics, ecology and epidemic. Some of them are described as nonautonomous delay dierential equations. Many people are doing research on the dynamics of population with delays, which is useful for the control of the population of mankind, animals and the environment. One of the famous models for dynamics of population is the Lotka±Volterra competition system. Owing to its theoretical and practical signi®cance, the Lotka±Volterra systems have been studied extensively [1± 3,5,6,8±22]. To consider periodic environmental factors, it is reasonable to study the Lotka±Volterra system with periodic coecients. A very basic and important ecological problem associated with the study of multispecies population interaction in a periodic environment is the existence and global attractivity of periodic solutions. Such questions arise also in many other situations. In this paper, we investigate the following periodic n-species Lotka±Volterra competition system with several deviating arguments " # n X aij tyj t ÿ sij t ; i 1; 2; . . . ; n; 1:1 y_i t yi t ri t ÿ j1

where aij 2 CR; 0; 1; ri ; sij 2 CR; R; are x-periodic functions with 1 ri : x aij :

Zx

1 x

ri t dt > 0; 0

Zx aij t dt P 0;

1 Ri : x

Zx jri tj dt; 0

i; j 1; 2; . . . ; n:

1:2

0

In mathematical ecology, Eq. (1.1) denotes a model of the dynamics of an nspecies system in which each individual competes with all others of the system for a common resource and the intra-species and inter-species competition involves deviating arguments sij . The assumption of periodicity of the parameters ri ; aij ; sij is a way of incorporating the periodicity of the environment (e.g. seasonal eects of weather condition, food supplies, temperature, mating habits etc). The growth functions ri are not necessarily positive, since the environment ¯uctuates randomly; in bad conditions, ri may be negative. The existence and attractivity of periodic solutions of some special cases of Eq. (1.1) has been studied extensively in the literature [1±3,5,8± 11,15,16,18,19,21,22]. Shibata and Saito [17] examined a two-species delay Lotka±Volterra competition system. It has been shown that the time delays in a


49

two-species Lotka±Volterra competition system can lead to chaotic behaviour. To our knowledge, few papers have been published on the existence and global attractivity of periodic solutions of Eq. (1.1). The main purpose of this paper is to derive a set of `easily veri®able' sucient conditions for the existence and global attractivity of positive periodic solution of Eq. (1.1). The present paper is organized as follows. In Section 2, we study the existence of periodic solutions with strictly positive components of Eq. (1.1) by using the continuation theorem of coincidence degree theory proposed by Gaines and Mawhin [7]. In Section 3, we investigate the global attractivity of strictly positive periodic solutions of n-species Lotka±Volterra competition system with several delays, which is a special case of Eq. (1.1), by using the method of Lyapunov functional. We de®ne a suitable Lyapunov functional and obtain a set of easily veri®able sucient conditions. In Section 3, we also examine the uniform persistence of Eq. (1.1) and analyze the eect of the time delays on the uniform persistence. In Section 4, as an application, we study some special cases of Eq. (1.1), which have been studied extensively in the literature. The examples show that our general easily veri®able sucient conditions dier from the known results and improve and generalize some known results. 2. Existence of positive periodic solutions In this section, we study the existence of strictly positive (componentwise) periodic solutions of Eq. (1.1). For the reader's convenience, we shall ®rst summarize below a few concepts and results from Ref. [7] that will be basic for this section. Let X ; Z be normed vector spaces, L : Dom L X ! Z be a linear mapping, and N : X ! Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim Ker L codim Im L < 1 and Im L is closed in Z. If L is a Fredholm mapping of index zero there exist continuous projectors P : X ! X and Q : Z ! Z such that Im P Ker L; Ker Q Im L ImI ÿ Q. It follows that L j Dom L \ Ker P : I ÿ P X ! Im L is invertible. We denote the inverse of that map by KP . If X is an open bounded subset of if QN X is bounded and X , the mapping N will be called L-compact on X KP I ÿ QN : X ! X is compact. Since Im Q is isomorphic to Ker L, there exist isomorphisms J : Im Q ! Ker L. In the proof of our existence theorem below, we will use the continuation theorem of Gaines and Mawhin ([7], p. 40). Lemma 2.1 (Continuation theorem). Let L be a Fredholm mapping of index zero Suppose and let N be L-compact on X. (a) For each k 2 0; 1, every solution x of Lx kNx is such that x 62 oX;

50


(b) QNx 6 0 for each x 2 oX \ Ker L and degfJQN ; X \ Ker L; 0g 6 0: Then the equation Lx Nx has at least one solution lying in Dom L \ X. T

Lemma 2.2. The domain Rn fy1 ; . . . ; yn j yi > 0; i 1; 2; . . . ; ng is positive invariant with respect to Eq. (1.1). Proof. Since yi t yi 0 exp

8 t 0, that is T y1 0; . . . ; yn 0 2 Rn . The proof is complete. Theorem 2.1. Suppose that aii > 0 and n X aij rj exp f rj Rj xg; i 1; 2; . . . ; n ri > a jj j1 j6i

hold. Then Eq. (1.1) has at least one positive periodic solution of period x say T y t y1 t; . . . ; yn t Pand there exists a positive constant B such that n 2 1=2 ky k 6 B, where ky k i1 maxt20;x jyi tj . Proof. By Lemma 4.1.1 in Ref. [10] and the assumptions in Theorem 2.1, it is not dicult to show that the system of algebraic equations n X 2:2 aij exp uj ri ; i 1; 2; . . . ; n j1 T

has a unique solution u1 ; u2 ; . . . ; un 2 Rn . Making the change of variable yi t exp fxi tg;

i 1; 2; . . . ; n

then Eq. (1.1) is reformulated as n X aij t exp xj t ÿ sij t ; x_ i t ri t ÿ

2:3

i 1; 2; . . . ; n:

2:4

j1

Let X Z fxt x1 t; x2 t; . . . ; xn tT 2 CR; Rn ;

xt x xtg; 2:5


kxk

n X i1

2 !1=2 max jxi tj

t20;x

for any x 2 X or Z:

51

2:6

Then X and Z are both Banach spaces when they are endowed with the norm k k. Let 0 1 n X t ÿ a t exp x t ÿ s t r 0 1 B 1 1j j 1j C x1 B C j1 B C B . C B C . CB . Nx N B C for any x 2 X ; . @ . A B . C B n X C xn @ A rn t ÿ anj t exp xj t ÿ snj t j1

Lx x_ ;

1 Px x

Obviously,

Zx xt dt;

1 Qz x

x 2 X;

0

Zx zt dt;

z 2 Z: 2:7

0

Ker L fx j x 2 X ; x h; h 2 Rn g; 8 9 Zx < = Im L z j z 2 Z; zt dt 0 : ;

2:8

0

and

dim Ker L n codim Im L: 2:9 Since Im L is closed in Z, L is a Fredholm mapping of index zero. It is easy to show that P and Q are continuous projectors such that

Im P Ker L; Ker Q Im L ImI ÿ Q: 2:10 Furthermore, the generalized inverse (to L) KP : Im L ! Ker P \ Dom L is given by Zt Zx Zt 1 KP z zs ds ÿ zs ds dt: 2:11 x 0

Thus QN : X ! Z 0

0

0

# 1 B1 r1 t ÿ a1j t exp xj t ÿ s1j t dt C Bx C 0 1 B C j1 x1 B 0 C B . C B C .. B . C!B C @ . A B C . B x" # C B Z C n xn X B1 C @x rn t ÿ anj t exp xj t ÿ snj t dt A Zx "

0

n X

j1

52


KP I ÿ QN : X ! X 0 Zt "

# 1 n X B r1 s ÿ a1j s exp xj s ÿ s1j s ds C B C 0 1 j1 B 0 C x1 B C B . C B C .. B . C!B C @ . A B C . B " # C t B C Z xn n X B C @ rn s ÿ anj s exp xj s ÿ snj s ds A j1

0

0 Zx Zt B x1 B B 0 0 B B ÿB B B B Zx Zt B @1 x

0

0

"

n X

#

1

ds dt C C C C C .. C C . C " # C n X C rn s ÿ anj s exp xj s ÿ snj s ds dt A r1 s ÿ

a1j s exp xj s ÿ s1j s

j1

j1

0

# 1 Zx " n X B xt ÿ 12 r1 s ÿ a1j s exp xj s ÿ s1j s ds C B C j1 B C 0 B C B C .. C: ÿB B C . B # C B C Zx " n X B t 1 C @ ÿ rn s ÿ anj s exp xj s ÿ snj s ds A x 2 0

j1

2:12 Clearly, QN and KP I ÿ QN are continuous. Using the Arzela±Ascoli theorem, is compact for any open bounded it is not dicult to show that KP I ÿ QN X with any set X X . Moreover, QN X is bounded.Thus, N is L-compact on X open bounded set X X . The isomorphism J of Im Q onto Ker L can be the identity mapping, since Im Q Ker L. Now we reach the position to search for an appropriate open, bounded subset X for the application of the continuation theorem. Corresponding to the operator equation Lx kNx; k 2 0; 1, we have " # n X aij t exp xj t ÿ sij t ; k 2 0; 1; x_ i t k ri t ÿ j1

i 1; 2; . . . ; n:

2:13

Assume that x xt 2 X is a solution of Eq. (2.13) for a certain k 2 0; 1: Integrating Eq. (2.13) over the interval 0; x, we obtain


Zx " ri t ÿ

n X

aij t exp xj t ÿ sij t

53

# dt 0;

i 1; 2; . . . ; n;

j1

0

that is, n Z X

x

j1

aij t exp xj t ÿ sij t dt ri x;

i 1; 2; . . . ; n:

2:14

0

It follows from Eqs. (2.13) and (2.14) that Zx Zx n X j_xi tj dt k ri t ÿ aij t exp xj t ÿ sij t dt j1 0

Zx

0

jri tj dt

rj = ajj exp rj Rj x > < ri ÿ j1 aij = j6i

> :

aii

> ;

: Bi ;

i 1; 2; . . . ; n: From Eqs. (2.15) and (2.23), we have Zx xi t P xi gi ÿ j_xi j dt > Bi ÿ ri Ri x;

2:23

i 1; 2; . . . ; n:

2:24

0

Eqs. (2.19) and (2.24) imply ) ( ( ) ri max jxi tj < max ln ri Ri x ; Bi ÿ ri Ri x : Hi : t20;x aii 2:25 ÿPn 2 1=2 Clearly, Hi are independent of k. Set H H C, where C is taken i1 i suciently large such that the unique solution of Eq. (2.2) satis®es ku1 ; u2 ; . . . ; un T k < C, then kxk < H . T Let X : fx x1 ; . . . ; xn 2 X jkxk < H g. It is clear that X veri®es the requirement (a) in Lemma 2.1. When x 2 oX \ Ker L oX \ Rn ; x is a constant vector in Rn with kxk H . Then


0

# r1 t ÿ a1j t exp xj dt C C B C j1 B 0 C B C . QNx B C .. B x" # C B Z C n X B1 C @x rn t ÿ anj t exp xj dt A B1 Bx

0

Zx "

55

1

n X

j1

0

1 ÿ exp x a r 1j j C B 1 B C j1 B C .. C 6 0: B B C . B n X C @ A rn ÿ anj exp xj n X

2:26

j1

Furthermore, in view of the assumption in Theorem 2.1, it is easy to prove that degfJQNx; X \ Ker L; 0g ( n

aij nn exp sign ÿ 1 det

(

n X j1

)) uj

6 0:

2:27

By now we have proved that X veri®es all the requirements in Lemma 2.1. Set y t Hence, Eq. (2.4) has at least one x-periodic solution x t in X. i exp xi t ; then by Eq. (2.3) we know that y t y1 t; . . . ; yn tT is a positive x-periodic solution of Eq. (1.1). The boundedness of x t implies the existence of positive constant B in Theorem 2.1. The proof of Theorem 2.1 is complete. Remark 2.1. Theorem 2.1 is also valid for both the advanced type and the mixed type Lotka±Volterra competition system. From Theorem 2.1, we can see that the deviating arguments sij ; i; j 1; 2; . . . ; n have no eect on the existence of positive periodic solution of Eq. (1.1). Remark 2.2. Li [15] discussed the existence of positive periodic solutions of periodic N -species competition system with time delays, which is a special case of Eq. (1.1). Theorem 2.1 improves and extends the results in Ref. [15]. 3. Global asymptotic stability of positive periodic solutions We shall assume, henceforth, that sij ; i; j 1; . . . ; n are non-negative constants. That is, in this section we will con®ne ourself to periodic n-species Lotka±Volterra competition system with several delays, which is a special case of Eq. (1.1).

56

M. Fan et al. / Mathematical Biosciences 160 (1999) 47±61 T

De®nition 3.1. Let y t y1 t; . . . ; yn t be a strictly positive (componentwise) periodic solution of Eq. (1.1) and we say y t is globally asymptotically T stable (or attractive) if any other solution yt y1 t; . . . ; yn t of Eq. (1.1) together with the initial condition yi s /i s P 0;

s 2 ÿsi ; 0;

/i 0 > 0;

i 1; . . . ; n

3:1

where si max1 6 j 6 n sji and /i s is continuous on ÿsi ; 0, has the property lim

t!1

n X i1

jyi t ÿ yi tj 0:

3:2

It is immediate that if y t is globally asymptotically stable then y t is in fact unique. Lemma 3.1 ([4]). Let f be a nonnegative function de®ned on 0; 1 such that f is integrable on 0; 1 and is uniformly continuous on 0; 1. Then limt!1 f t 0: Theorem 3.1. Assume that the conditions in Theorem 2.1 hold. Furthermore, suppose that sii 0; i 1; 2; . . . ; n and min ajj t >

t20;x

n X i1 i6j

max aij t ;

t20;x

j 1; 2; . . . ; n:

3:3

Then system (1.1) has a unique periodic solution with strictly positive components, T say y t y1 t; . . . ; yn t , which is globally asymptotically stable and there exists a positive constant B > 0 such that ky k 6 B. Proof. By Theorem 2.1, there exists a strictly positive (componentwise) periodic T solution y t y1 t; . . . ; yn t of Eq. (1.1) and there is a positive constant B > 0 such that ky k 6 B. To complete the proof of Theorem 3.1, we only need to show that y t is globally asymptotically stable. Consider a Lyapunov functional vt de®ned by 2 3 t Z n n X X6 7 6j ln yi t ÿ ln y tj aij u sij jyj u ÿ yj uj du7 vt i 4 5; i1

j1 tÿsij j6i

t P 0: 3:4 Obviously


v0 6

n X i1

j ln yi 0 ÿ lnyi 0j n X

n X

! max aij t sup j/j t ÿ

t20;x

j1 j6i

vt P

t2ÿsij ;0

j ln yi t ÿ ln yi tj;

i1

yj tjsij

< 1;

n B X

B ÿ aii tjyi t ÿ y tj i @

i1

3:6

n B X B ÿ ajj tjyj t ÿ y tj j @ n X j1

where

jyj t ÿ yj tj;

0

B min ajj t ÿ l min B 1 6 j 6 n@t20;x

n X i1 i6j

1

i1 i6j

C aij t sij jyj t ÿ yj tjC A

n X

j1

6 ÿl

C aij t sij jyj t ÿ yj tjC A

n X

B ÿ ajj tjyj t ÿ y tj j @

0 6

j1 j6i

n B X j1

1

n X

0

3:5

t P 0:

A direct calculation of the right derivative D v of vt leads to 0 D vt 6

57

1 C max aij t jyj t ÿ yj tjC A

t20;x

i1 i6j

t P 0;

3:7

1

C max aij t C A > 0: t20;x

Integrating from 0 to t on both sides of Eq. (3.7) produces n Z X jyj s ÿ yj sj ds 6 v0 < 1; vt l t

j1

t P 0;

3:8

0

then n Z X

t

i1

0

and hence

jyi s ÿ yi sj ds 6 v0=l < 1; Pn

i1

t P 0;

3:9

jyi t ÿ yi tj 2 L1 0; 1: By Eqs. (3.5)±(3.8), one obtains

58

M. Fan et al. / Mathematical Biosciences 160 (1999) 47±61 n X i1

j ln yi t ÿ ln yi tj 6 vt 6 v0 < 1;

t P 0:

3:10

t P 0:

3:11

Therefore j ln yi t ÿ ln yi tj 6 v0; i 1; 2; . . . ; n;

Furthermore min yi t expfÿv0g 6 yi t 6 max yi t exp fv0g 0; Kt > 0: Noting Eq. (2.22) and applying Theorem 2.1 to Eq. (4.1), we obtain the following theorem, which is an immediate corollary of Theorem 2.1. Theorem 4.1. System (4.1) has at least one strictly positive x-periodic solution. Zhang and Gopalsamy [23] study a special case of Eq. (4.1) in which the period of the environmental parameters is related to the time delay in the selfregulating negative feed back mechanism. It has been proved that if st mx; m is a positive integer, then there exists a positive x-periodic solution of Eq. (4.1). Obviously, Theorem 4.1 improves the results in Ref. [23]. It is interesting that for periodic logistic equation, the time delay has no eect on the existence of positive periodic solutions. Application 2. Consider a two-species competition system in a periodic environment of the form dy1 t y1 tr1 t ÿ a11 ty1 t ÿ s11 t ÿ a12 ty2 t ÿ s12 t; dt dy2 t y2 tr2 t ÿ a21 ty1 t ÿ s21 t ÿ a22 ty2 t ÿ s22 t; 4:2 dt Rx where ri ; aij ; sij ; i; j 1; 2 are continuous x-periodic functions with 0 ri t dt > 0; aij t > 0: Eq. (4.2) is dierent from the periodic two-species competition system considered in the literature, since ri t are not always positive. Applying Theorems 2.1 and 3.1 to Eq. (4.2), we have the following results. Theorem 4.2. Suppose that r1 a22 > r2 a12 exp f r2 R2 xg; r2 a11 > r1 a21 exp fr1 R1 xg 4:3 hold. Then Eq. (4.2) has at least one strictly positive (componentwise) x-periodic solution. Theorem 4.3. Suppose that s11 t s22 t 0;

s12 t; s21 t are nonÿnegative constants

4:5

60


and r2 R2 xg; r1 a22 > r2 a12 exp f min a11 t > max a21 t;

t20;x

t20;x

r2 a11 > r1 a21 exp fr1 R1 xg;

min a22 t > max a12 t

t20;x

t20;x

4:6

hold. Then Eq. (4.2) has a unique x-periodic solution with strictly positive components, which is globally asymptotically stable. Gopalsamy [8] also examined a two-species periodic Lotka±Volterra system dy1 t y1 tr1 t ÿ a11 ty1 t ÿ a12 ty2 t ÿ x; dt dy2 t 4:7 y2 tr2 t ÿ a21 ty1 t ÿ x ÿ a22 ty2 t; dt where ri ; aij 2 CR; 0; 1; i; j 1; 2 are x-periodic functions. It has been proved that if r1l al22 > r2u au12 ; al11 > au21 ;

r2l al11 > r1u au21 ;

al22 > au12 :

4:8

Then Eq. (4.7) has a unique positive periodic solution which is globally asymptotically stable, where ril min ri t; t20;x

alij

min aij t; t20;x

riu max ri t; t20;x

auij

max aij t; t20;x

i; j 1; 2:

4:9

Obviously, our results for two-species competition system are dierent from the results obtained in Ref. [8]. For n-species competition system, our results are fresh and general. Acknowledgements The authors are very grateful to the referees for their careful reading of the manuscript and a number of excellent suggestions, which improve the representation of the present paper.

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61

[4] I. Barbalat, Systemes d'equations dierentielle d'oscillations nonlineaires, Rev. Roum. Math. Pures Appl. 4 (1959) 267. [5] J.M. Cushing, Two species competition in a periodic environment, J. Math. Biol. 10 (1980) 385. [6] H.I. Freedman, P. Waltman, Persistence in a model of three competitive populations, Math. Biosci. 73 (1985) 89. [7] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Dierential Equations, Springer, Berlin, 1977. [8] K. Gopalsamy, Globally asymptotic stability in a periodic Lotka±Volterra system, J. Austral. Math. Soc. Ser. B 24 (1982) 160. [9] K. Gopalsamy, Globally asymptotic stability in a periodic Lotka±Volterra system, J. Austral. Math. Soc. Ser. B 29 (1985) 66. [10] K. Gopalsamy, Stability and Oscillation in Delay Dierential Equations of Population Dynamics, Mathematicx and its Applications 74, Kluwer Academic, Dordrecht, 1992. [11] I. Gyori, G. Ladas, Oscillation Theory of Delay Dierential Equations, Oxford Science, Oxford, 1991. [12] J.K. Hale, A.S. Somolinos, Competition for ¯uctuating nutrient, J. Math. Biol. 18 (1983) 225. [13] P. Korman, Some new results on the periodic competition model, J. Math. Anal. Appl. 171 (1992) 131. [14] Y. Kuang, Delay Dierential Equations with Application in Population Dynamics, Academic Press, Boston, 1993. [15] Y.K. Li, Periodic solutions of N-species competition system with time delays, J. Biomath. 12 (1) (1997) 1. [16] R.M. May, W.J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975) 243. [17] A. Shibata, N. Saito, Time delays and chaos in two competition system, Math. Biosci. 51 (1980) 199. [18] H.L. Smith, Periodic solutions of periodic competitive and cooperative systems, SIAM J. Math. Anal. 17 (1986) 1289. [19] P. de Mottoni, A. Schiano, Competition systems with periodic coecients: a geometric approach, J. Math. Biol. 11 (1981) 319. [20] H.L. Smith, Periodic competitive dierential equations and the discrete dynamics of a competitive map, J. Dierent. Eq. 64 (1986) 165. [21] A. Trieo, C. Alrarez, A dierent consideration about the globally asymptotically stable solution of the periodic n-competing species problem, J. Math. Anal. Appl. 159 (1991) 44. [22] F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems, Results Math. 21 (1992) 224. [23] B.G. Zhang, K. Gopalsamy, Global attractivity and oscillations in a periodic delay-logistic equation, J. Math. Anal. Appl. 15 (1990) 274.