J. Math. Anal. Appl. 330 2007 24 33 www.elsevier.com/locate/jmaa Persistence global stability in discrete models of Lotka Volterra type Yoshiaki Muroya 1 Department of Mathematical Sciences, Waseda University, 3-4-1 Ohkubo Shinjuku-ku, Tokyo 169-8555, Japan Received 24 December 2002 Available online 22 August 2006 Submitted by H.L. Smith Abstract In this paper, we establish new sufficient conditions for global asymptotic stability of the positive equilibrium in the following discrete models of Lotka Volterra type: } N i p + 1 = N i p exp c i a i N i p a ij N j p k ij, p 0, 1 i n, N i p = N ip 0, p 0, N i0 > 0, 1 i n, where each N ip for p 0, each c i, a i a ij are finite ai > 0, a i + a ii > 0, 1 i n, k ij 0, 1 i, j n. Applying the former results [Y. Muroya, Persistence global stability for discrete models of nonautonomous Lotka Volterra type, J. Math. Anal. Appl. 273 2002 492 511] on sufficient conditions for the persistence of nonautonomous discrete Lotka Volterra systems, we first obtain conditions for the persistence of the above autonomous system, extending a similar technique to use a nonnegative Lyapunovlike function offered by Y. Saito, T. Hara W. Ma [Y. Saito, T. Hara, W. Ma, Necessary sufficient conditions for permanence global stability of a Lotka Volterra system with two delays, J. Math. Anal. Appl. 236 1999 534 556] for n = 2totheabovesystemforn 2, we establish new conditions for global asymptotic stability of the positive equilibrium. In some special cases that k ij = k jj,1 i, j n, n n a ji a jk = 0, i k, these conditions become a i > aji 2,1 i n, improve the well- E-mail address: ymuroya@waseda.jp. 1 Research was supported by Waseda University Grant for Special Research Projects 2005A-077 2006B-146, Scientific Research c, No. 16540207 of Japan Society for the Promotion of Science. 0022-247X/$ see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.07.070
Y. Muroya / J. Math. Anal. Appl. 330 2007 24 33 25 known stability conditions a i > n a ji,1 i n, obtained by K. Gopalsamy [K. Gopalsamy, Global asymptotic stability in Volterra s population systems, J. Math. Biol. 19 1984 157 168]. 2006 Elsevier Inc. All rights reserved. Keywords: Persistence; Global asymptotic stability; Discrete model of autonomous Lotka Volterra type 1. Introduction Consider the persistence global asymptotic stability of the following discrete models of Lotka Volterra type: } N i p + 1 = N i p exp c i a i N i p a ij N j p k ij, p 0, 1.1 N i p = N ip 0, p 0, N i0 > 0, 1 i n, where each N ip for p 0, each c i, a i a ij are finite ai > 0, a i + a ii > 0, 1 i n, 1.2 k ij 0, 1 i, j n. Recently, making the best use of the symmetry of the system an extended La Salle s invariance principle, Saito, Hara Ma [9] has shown necessary sufficient conditions for permanence global stability of a symmetrical Lotka Volterra type predator prey system with two delays. This improves the well-known sufficient condition on the global asymptotic stability of the positive equilibrium in the system obtained by Gopalsamy [4]. Saito [8] also established the necessary sufficient condition for global stability of a Lotka Volterra cooperative or competition system with delays for two species. On the other h, Xu Chen [10] has offer new techniques to obtain sufficient conditions of the persistence global stability for a time-dependent pure-delay-type Lotka Volterra predator prey model for three species. On the other h, Muroya [5,6] established conditions for the persistence global stability of delay differential system discrete system for n species, respectively, which are some extensions of the averaged condition offered by Ahmad Lazer [1,2]. In this paper, applying Lemma 2.2 Theorem 1.2 in Muroya [6] on sufficient conditions for the persistence of nonautonomous discrete Lotka Volterra systems to the discrete system 1.1 1.2, we first obtain conditions for the persistence of the above autonomous system, extending a similar technique to use a nonnegative Lyapunov-like function offered by Saito, Hara Ma [9] for n = 2 to the above system for n 2, we establish new conditions for global asymptotic stability of the positive equilibrium. This is a discrete version of Muroya [7]. In some special cases, these conditions improve the well-known stability result obtained by Gopalsamy [4]. Put a + ij = maxa ij, 0, a ij = mina ij, 0, 1.3 A 0 = diaga 1,a 2,...,a n, B = [ a ] ij, B + = [ a + ] ij D + = diag a + 11,a+ 22 nn,...,a+ are n n matrices, c =[c i ] is an n-dimensional vector, 1.4
26 Y. Muroya / J. Math. Anal. Appl. 330 2007 24 33 assume that A0 + D + + B is an M-matrix, A 0 + D + + B 1 c > 0 c >B + D + A 0 + D + + B 1 1.5 c, where a real n n matrix A =[a ij ] with a ij 0 for all i j is called an M-matrix if A is nonsingular A 1 0 see, for example, Berman Plemmons [3]. Applying Lemma 2.2 Theorem 2.2 in Muroya [6] on the sufficient conditions of the persistence of nonautonomous discrete Lotka Volterra systems to the system 1.1 1.2, we first obtain the following theorem. Theorem 1.1. See Muroya [6]. For the system 1.1 1.2, if the condition 1.5 is satisfied, then all solutions N i p, 1 i n, of the system are positive the system is persistent, that is, 0 < lim inf p 0 N ip lim sup N i p < +, 1 i n. 1.6 p 0 In particular, all solutions N i p, 1 i n, of the system are bounded above, that is, lim sup N i p N i, 1 i n, 1.7 p where N i, 1 i n, are defined by i 1 c i = c i aij N ci /a i, c i 1, j, Ñ i = c i /a i, N i = e c i 1 /a i, c i > 1. 1.8 By Theorem 1.1 extending a similar technique to use a nonnegative Lyapunov-like function offered by Saito, Hara Ma [9] for n = 2 to the above system for n 2, we get the following results. Theorem 1.2. For the system 1.1 1.2, in addition to 1.5 1.7, assume c i < 1, 1 i n, 1.9 suppose that there exists a positive equilibrium N = N1,N 2,...,N n a i > a ji a jk, 1 i n. 1.10 Then, the positive equilibrium N = N1,N 2,...,N n of 1.1 is globally asymptotically stable for any k ij 0, 1 i, j n. In particular, if k ij = k jj, 1 i, j n, a i > a ji a jk, 1 i n, 1.11 then the positive equilibrium N = N1,N 2,...,N n of 1.1 is globally asymptotically stable for any k ii 0, 1 i n. Moreover, if a ji a jk = 0, i k, 1.12
Y. Muroya / J. Math. Anal. Appl. 330 2007 24 33 27 then the last inequalities of 1.11 becomes a i > aji 2, 1 i n. 1.13 Thus, in the cases of 1.11 1.12, the condition 1.13 is weaker than the following sufficient condition on the global asymptotic stability of the positive equilibrium of the system a i > a ji, 1 i n, 1.14 which was obtained by Gopalsamy [4], this extends some of results in Saito, Hara Ma [9] for n = 2ton 2. The organization of this paper is as follows. In Section 2, applying the results in Muroya [6], we offer conditions for the persistence of system 1.1 1.2, using a nonnegative Lyapunovlike sequence, we establish conditions for the global asymptotic stability of positive equilibrium N = N1,N 2,...,N n of the system 1.1 1.2. 2. Proof of theorems In this section, we prove Theorems 1.1 1.2. Muroya [6] consider the following discrete system of nonautonomous Lotka Volterra type: } m N i p + 1 = N i p exp c i p a i pn i p aij l pn j p k l, l=0 2.1 p = 0, 1, 2,..., N i p = N ip 0, p 0, N i0 > 0, 1 i n, where each c i p, a i p aij l p are bounded for p 0 inf a ip > 0, aii 0 p 0, 1 i n, p 0 aij l p 0, 1 i j n, 0 l m, k 0 = 0, integers k l 0, 1 l m. For a given sequence gp} p=0,weset g M = sup gp p = 0, 1, 2,... }, g L = inf gp p = 0, 1, 2,... }, 2.3 for integers 0 p 1 <p 2,weset p 2 1 1 A[g,p 1,p 2 ]= gp. 2.4 p 2 p 1 p=p 1 The lower upper averages of gp, denoted by m[g] M[g], respectively, are defined by m[g]= lim inf A[g,p 1,p 2 ] p2 p 1 q } q 2.2 M[g]= lim q sup A[g,p 1,p 2 ] p 2 p 1 q }. 2.5
28 Y. Muroya / J. Math. Anal. Appl. 330 2007 24 33 Put aij l L = al ij L + al+ ij L, aij l M = al ij M + al+ m b ij L = aij l L, b ij M = l=0 l=0 ij M, al ij L 0 al+ ij L, al ij M 0 al+ ij M, b ij L = m l=0 a l ij L, m m aij l M b+ ij M = aij l+ M, 1 i, j n. 2.6 l=0 Let A L = diaga 1L,a 2L,...,a nl, BL = [ bij L], B + M = [ b ij + M], D L + = diag b + 11L,b+ 22L,...,b+ nnl D M + = diag b + 11M,b+ 22M,...,b+ nnm are n n matrices, c = [ m[c i ] ] c = [ M[c i ] ] are n-dimensional vectors. 2.7 Assume that AL + D L + + B L 1 c > 0 c > B + M D M + AL + D L + + B L 1 c, 2.8 put i 1 c im = c im bij N L j, Ñ i = c im /a il, cim /a N i = il, c im 1, 2.9 exp c im 1/a il, c im > 1. Muroya [6] obtained the following two results see Muroya [6, Lemma 2.2 Theorem 1.2]. Lemma 2.1. Assume that for Eq. 2.7 c M = c 1M,c 2M,...,c nm T, AL + BL 1cM > 0. 2.10 Then, any solution of the system 2.1 2.2 is bounded above, it holds that lim N ip N i, 1 i n, 2.11 p where N i, 1 i n, are defined by 2.9. Note that 2.8 implies 2.10. Lemma 2.2. For the system 2.1 2.2, if the condition 2.8 is satisfied, then all solutions N i p, 1 i n, of the system are bounded above. Moreover, if there exists a nonempty subset Q 1, 2,...,n} such that c il b + N ij M j > 0, for any i Q, 2.12 j/ Q
Y. Muroya / J. Math. Anal. Appl. 330 2007 24 33 29 then the system 2.1 2.2 is persistent for solutions, that is, 0 < lim inf p 0 N ip lim sup N i p < +, 1 i n. 2.13 p 0 Note that for the system 1.1 1.2, 1.5 corresponds to 2.8 in system 2.1 2.2 implies c > 0 for the set Q =1, 2,...,n}, it holds that c i j/ Q a + ij N j > 0, for any i Q, 2.14 which implies 2.12. Proof of Theorem 1.1. Put i 1 i 1 + j, i > j, l ij = j 1 j 1 + 2j i, i j, l=1 kij, l = l k l = ij, 0, otherwise. āij l = aij, l = l ij, 0, otherwise, Then, we have n 2 a ij N j t k ij = āij l N j t k l. Thus, the system 1.1 1.2 is a special autonomous case of system 2.1 2.2. We can apply the results in Lemmas 2.1 2.2 to Eqs. 1.1 1.2 obtain the conclusion of the theorem. This completes the proof. Proof of Theorem 1.2. Since by Theorem 1.1, the condition 1.9 implies that N i = Ñ i < 1/a i, 1 i n, we have that there is a positive integer p 0 such that for p p 0, N i p < N i,1 i n. Consider a nonnegative Lyapunov-like sequence vp} p=0 such that for p 0, Then, vp = Ni p 2a i 1 ln N i p/ni } Ni + N i a ji a jk p 1 q=p k ji Ni q Ni 2. vp + 1 vp Ni = 2a i p + 1 N i p N ln N } ip + 1 N i p Ni + a ji a jk p Ni 2 Ni p k ji Ni 2 }. 2.15
30 Y. Muroya / J. Math. Anal. Appl. 330 2007 24 33 Since N i p + 1 N i p = N i p exp ln N ip + 1 N i p = N i p ln N ip + 1 N i p } 1 + exp θ ln Nip+1 N i p 2! where for p p 0 1 i n, N i p exp θ ln N ip + 1 max N i p, N i p + 1 < 1, N i p a i one can verify that Ni 2a i p + 1 N i p N ln N } ip + 1 N i p 2a i Ni p Ni N i p + 1 ln + N i p by 2.1, we have that ln N ip + 1 = a i Ni p N i N i p ln N ip + 1 2 }, 0 <θ<1, N i p ln N ip + 1 2, 2.16 N i p a ij Nj p k ij Nj. We have that x 1 ln x 0, for any x>0. By Theorem 1.1, each N i p, 1 i n, are bounded above below by positive constants for p 0. Therefore, it follows from 1.6 that for any p k = maxk ij k ij 0, 1 i, j n}, 0 vp < +. Let p i = a i Ni p N q ij = a ij Nj p k ij Nj. Then, ln N ip+1 N i p ln N ip + 1 N i p = p i + n q ij, = 2p i p i = p i + q ij 2 q ij + j 1 qij 2 + 2 q ij q ik pi 2, for r ji = N j p k ij Nj p, we have that j 1 j 1 2 q ij q ik = 2 a ij a ik r ji r ki j=2 = j=2 j 1 j=2 a ij j=2 j 1 a ij a ik rji 2 + r2 ki j=2 n 1 a ik rji 2 + a ik a ij j=k+1 r 2 ki
= Y. Muroya / J. Math. Anal. Appl. 330 2007 24 33 31 i=2 a ji i 1 n 1 a jk rji 2 + a ji a jk a ji Therefore, j 1 qij 2 + 2 q ij q ik j=2 k i a jk rij 2. 2a i N i p + 1 N i p Ni ln N } ip + 1 N i p a ji a jk rij 2 = Thus, by 2.15, we obtain a ji a jk p 2 i Ni a ji a jk p k ji N 2 vp + 1 vp δ i r 2 ij k=i+1 ai 2 Ni p Ni 2. } Ni ai 2 a ji a jk p N Ni p Ni 2, where by 1.10, } δ = min ai 2 a ji a jk > 0. 1 i n Then, vp + 1 + δ p=0 p q=0 Ni p Ni Ni q Ni 2 v0, for any p 0, 2 v0 δ < +, from which we conclude that n N i p N i 2 = 0. This result implies that the positive equilibrium N = N 1,N 2,...,N n of 1.1 is globally asymptotically stable for any k ij 0, 1 i, j n. 2 i r 2 ki
32 Y. Muroya / J. Math. Anal. Appl. 330 2007 24 33 In particular, if 1.11 holds, then for r j = r jj = N j p k jj Nj,1 j n, we have that j 1 qij 2 + 2 } j 1 q ij q ik = aij 2 r2 j + 2 a ij r j a ik r k j=2 = j=2 a ij r j a ik r k = a ij a ik r j r k rj a ij a ik r 2 j r k a ij a + r2 k ik 2 r 2 j = a ij a ik 2 + r 2 a ij a k ik 2 = a ij a ik r j 2 = a ji a jk r 2 i. Thus, by 2.15, we obtain } vp + 1 vp ai 2 Ni a ji a jk p N i δ 1 Ni p Ni 2, where by 1.11, } δ 1 = min ai 2 a ji a jk > 0. 1 i n Then, vp + 1 + δ 1 p=0 p q=0 Ni q Ni 2 v0, for any p 0, Ni p Ni 2 v0 < +, δ 1 from which we conclude that n N i p N i 2 = 0. This result implies that the positive equilibrium N = N 1,N 2,...,N n of 1.1 is globally asymptotically stable for any k ii 0, 1 i n. Moreover, if 1.12 holds, then it is evident that the last inequalities of 1.11 becomes 1.13. 2
Y. Muroya / J. Math. Anal. Appl. 330 2007 24 33 33 Acknowledgment The author thanks the anonymous referee for his or her valuable comments to make this paper in the present form. References [1] S. Ahmad, A.C. Lazer, Necessary sufficient average growth in a Lotka Volterra system, Nonlinear Anal. 34 1998 191 228. [2] S. Ahmad, A.C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka Volterra system, Nonlinear Anal. 40 2000 37 49. [3] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. [4] K. Gopalsamy, Global asymptotic stability in Volterra s population systems, J. Math. Biol. 19 1984 157 168. [5] Y. Muroya, Persistence global stability for nonautonomous Lotka Volterra delay differential systems, Comm. Appl. Nonlinear Anal. 8 2002 31 45. [6] Y. Muroya, Persistence global stability for discrete models of nonautonomous Lotka Volterra type, J. Math. Anal. Appl. 273 2002 492 511. [7] Y. Muroya, Persistence global stability in Lotka Volterra delay differential systems, Appl. Math. Lett. 17 2004 795 800. [8] Y. Saito, The necessary sufficient condition for global stability of Lotka Volterra cooperative or competition system with delays, J. Math. Anal. Appl. 268 2002 109 124. [9] Y. Saito, T. Hara, W. Ma, Necessary sufficient conditions for permanence global stability of a Lotka Volterra system with two delays, J. Math. Anal. Appl. 236 1999 534 556. [10] R. Xu, L. Chen, Persistence global stability for a delayed nonautonomous predator prey system without dominating instantaneous negative feedback, J. Math. Anal. Appl. 262 2001 50 61.