We consider nonautonomous N-dimensional generalized Lotka-Volterra competition systems. Under certain conditions we show that there exists a unique solution u*  whose components are bounded above and below by positive constants on R, and u*  attracts any solution. If such system is periodic, so is u*

TANIGUCHI Kunihiko

University of Toyama Repository

Toyama Math. J.Vol. 36(2013 · 2014), 1-14Global asymptotic stability for a nonautonomousLotka-Volterra competition systemKunihiko TaniguchiAbstract. We consider nonautonomous N -dimensional generalizedLotka-Volterra competition systems. Under certain conditions weshow that there exists a unique solution u∗ whose components arebounded above and below by positive constants on R, and u∗ attractsany solution. If such system is periodic, so is u∗.1. Introduction and statements of the main resultsIn this paper we consider the system of differential equationsu′i = uiai(t)− N∑j=1bij(t)fij(ui, uj) , i = 1, . . . , N, N ≥ 2, (GLV)where the functions ai(t), 1 ≤ i ≤ N , and bij(t), 1 ≤ i, j ≤ N , are assumedto be continuous and bounded on R. For a bounded function g(t) on R, weput gM := supt∈R g(t), gL := inft∈R g(t). We assume thatbij(t) ≥ 0, t ∈ R, 1 ≤ i, j ≤ N ; (1.1)aiL > 0, biiL > 0, 1 ≤ i ≤ N. (1.2)2000 Mathematics Subject Classification. Primary 34D05; Seondary 92D25.Key words and phrases. Lotka-Volterra systems, Nonautonomous system, competi-tion system, permanence.12 Kunihiko TaniguchiFurthermore let the function fij(x, y), 1 ≤ i, j ≤ N , be continuouslydifferentiable on [0, ∞)2, and we impose the following condition on fij :fij(x, y) > 0, (x, y) ∈ R2+, 1 ≤ i, j ≤ N ;D1fij(x, y) ≥ 0, (x, y) ∈ R2+, 1 ≤ i, j ≤ N ;D2fij(x, y) > 0, (x, y) ∈ R2+, 1 ≤ i, j ≤ N ;fij(0, 0) = 0, 1 ≤ i, j ≤ N ;limx→∞fii(x, x) =∞, 1 ≤ i ≤ N,(1.3)where R2+ = (0, ∞)2 and Di, i = 1, 2, denotes the differentiation withrespect to the i-th variable.Throughout the paper we make use of the well-known fact (see e.g. [3],[11]) that if u = (u1, . . . , uN ) is a local solution of system (GLV) withu(t0) ∈ RN+ , then u can be extended to the interval [t0, ∞) and u(t) ∈ RN+for t ∈ [t0, ∞). Therefore in the sequel we may assume that all solutionsof system (GLV) exist near +∞ and are positive there.System (GLV) is a generalization of the following nonautonomous N -dimensional Lotka-Volterra competition system that S. Ahmad and A. C.Lazer [2] considered:u′i = uiai(t)− N∑j=1bij(t)uj , i = 1, . . . , N, N ≥ 2. (LV)In system (LV) the negative influence of the j-th species to the i-th speciesis regarded essentially as linear function, because the corresponding term inthe classical system (LV) is written as uiuj . But, as is often the case withmathematical modeling in biology, such a hypothesis is too restrictive fromthe biological point of view. In fact, K. Gopalsamy [9] considered systemu′i = uiai + n∑m=1N∑j=1bmijumj , i = 1, . . . , N, (1.4)where ai, bmij , 1 ≤ i, j ≤ N , 1 ≤ m ≤ n, are real constants. If n =1, then (1.4) reduces to the form of system (GLV). Motivated by thesefacts, we propose less restrictive system (GLV) than system (LV), and makeGlobal asymptotic stability for a nonautonomous Lotka-Volterra competition system 3an attempt to analyze it in order to generalize previous results to system(GLV). In fourthcoming papers we will continue to consider system (GLV)further.When system (LV) is an autonomous case for N = 2, it is well knownthe following (see [10]): If positive constants ai, bij , i, j = 1, 2, satisfya1 >b12a2b22, a2 >b21a1b11,then there exists a unique equilibrium point (u∗1, u∗2) ∈ R2+ that attractsany solution curve (u1(t), u2(t)) of system (LV) with (u1(t0), u2(t0)) ∈ R2+,that is,u1(t)→ u∗1 and u2(t)→ u∗2 as t→∞.For system (LV) S. Ahmad and A. C. Lazer [2] have shown a general-ization of the above fact. They supposed conditions (1.1), (1.2) and theconditionaiL >∑j 6=ibijM(ajbjj)M, 1 ≤ i ≤ N. (1.5)Under these conditions they have shown the following: Let u = (u1, . . . , uN )be a solution of system (LV) with u(t0) ∈ RN+ . Then there exists a uniquesolution u∗ = (u∗1, . . . , u∗N ) of system (LV) defined on R and the followingstatements (I)–(III) hold:(I) 0 < inft∈R u∗i (t) ≤ supt∈R u∗i (t) <∞ for 1 ≤ i ≤ N ;(II) limt→∞(ui(t)− u∗i (t)) = 0 for 1 ≤ i ≤ N ;(III) If ai(t), bij(t), 1 ≤ i, j ≤ N , are periodic with period T > 0, then u∗is T -periodic.In [2] to see property (I) only the conditioninft∈Rai(t)−∑j 6=i bij(t)(aj/bjj)Mbii(t)> 0, 1 ≤ i ≤ N, (1.6)4 Kunihiko Taniguchiis required. Note that (1.6) is weaker condition than (1.5). In [1], [6]–[8]and [13] results related to properties (I)–(III) have been obtained under theconditionaiL >∑j 6=ibijMajMbjjL, 1 ≤ i ≤ N,that is a stronger condition than (1.5).To state our previous result in [12] we introduce the notation and thesymbols: For a continuous and bounded function g on R, we setA[g, t1, t2] =1t2 − t1∫ t2t1g(s)ds,where t1 < t2. We define the upper average M [g] and the lower averagem[g], respectively, byM [g] = lims→∞sup{A[g, t1, t2] | t2 − t1 ≥ s}; andm[g] = lims→∞inf{A[g, t1, t2] | t2 − t1 ≥ s}.For i = 1, . . . , N , we putf˜i(x) = fii(x, x), x ∈ R+.By (1.3), f˜i, i = 1, . . . , N , has the inverse function f˜−1i : R+ → R+. ForR > 0 and δ > 0, δ < R, we define two constants C∗(δ, R) and C∗(δ, R),respectively, byC∗(δ, R) = max{Dkfij(x, y) | 1 ≤ i, j ≤ N, k = 1, 2, (x, y) ∈ [δ, R]2},C∗(δ, R) = min{D2fij(x, y) | 1 ≤ i, j ≤ N, (x, y) ∈ [δ, R]2}.Let R > 0 and δ > 0. For system (GLV) we introduce the conditionm[ai] >C∗(δ, R)C∗(δ, R)∑j 6=ibijMM [aj ]bjjL, 1 ≤ i ≤ N. (GA)Condition (GA) is a generalization of the average condition that S. Ahmadand A. C. Lazer [3, 4] supposed for system (LV):m[ai] >∑j 6=ibijMM [aj ]bjjL, 1 ≤ i ≤ N. (A)Global asymptotic stability for a nonautonomous Lotka-Volterra competition system 5Remark 1.1. When bii, 1 ≤ i ≤ N , is positive constants, it is clear that(1.5) implies condition (A).Conditions (A) and (GA) have played an important role in studyingsystems (LV) and (GLV) (see [3]–[5] and [12]). For example, in [12] wehave shown that condition (GA) and some additional conditions implylimt→∞(ui(t) − vi(t)) = 0, 1 ≤ i ≤ N , where u = (u1, . . . , uN ) andv = (v1, . . . , vN ) are arbitrary solutions of system (GLV) (see [12], The-orem 2.3). Therefore when system (GLV) satisfies condition (GA) andsimilar condition to (1.6), we naturally expect that (I)–(III) hold for sys-tem (GLV). In this paper we will show that this conjecture is true; thefollowing is the main result of this paper.Theorem 1.2. Let conditions (1.1), (1.2) and (1.3) hold . Suppose that forsome δ > 0 and R satisfyingR > max{f˜−1i ((ai/bii)M ) | 1 ≤ i ≤ N}, (1.7)condition (GA) and the conditioninft∈Rai(t)−∑j 6=i fij(R, R)bij(t)bii(t)> max1≤j≤N{fjj(δ, δ)}, 1 ≤ i ≤ N (1.8)hold. Then there exists a unique solution u∗ = (u∗1, . . . , u∗N ) of sys-tem (GLV) defined on R such that (I)–(III) hold for any solution u =(u1, . . . , uN ) of system (GLV) with u(t0) ∈ RN+ .We give an example of system (GLV) for which above conditions hold(see [12, Example 1.2]).Example 1.3. We consider the following competition system for two species:u′1 = u1[(cos t+ 7)− (sin t+ 7) ·( u12+ 1)u1 −{110(sin t+ 1)}· u1u2]u′2 = u2[(cos t+ 9)−{16(sin t+ 1)}·3u2u1u2 + 1− (sin t+ 9) ·( u23+ 1)u2],where f11(x, y) = (y/2 + 1)y, f12(x, y) = xy, f21(x, y) = 3xy/(x+ 1) andf22(x, y) = (y/3 + 1)y.6 Kunihiko TaniguchiIn [12] we have shown that for δ = 9/20 and R = 1, this system satisfiesconditions (GA) and (1.8). So this system has only one periodic solutionu∗ satisfying (I)–(III).The paper is organized as follows. In Section 2 we present preparatoryresults that are employed in proving Theorem 1.2. The proof of Theorem1.2 are given in Section 3.2. Preliminary resultsIn this section we give preliminary results to prove Theorem 1.2. Theproofs of Proposition 2.1 and Theorem 2.3 are based on [12].Proposition 2.1. Let u = (u1, . . . , uN ) and v = (v1, . . . , vN ) be solutionsof (GLV). Suppose that there exist constants A, B > 0 and T = Tu, v ≥ t0such that for j = 1, . . . , N and t ≥ T ,A ≤ uj(t), vj(t) ≤ B. (2.1)Suppose moreover that for system (GLV) there exist positive constantsα1, . . . , αN such that for j = 1, . . . , N ,lim inft→∞αjbjj(t)− C∗(A, B)C∗(A, B)∑i6=jαibij(t) > 0. (2.2)Then there exist some constants T˜ ≥ T , C = CA, B > 0 and γ = γA, B > 0such that for t ≥ T˜ ,N∑i=1| ui(t)− vi(t) |≤(N∑i=1| ui(T˜ )− vi(T˜ ) |)Ce−γ(t−T˜ ). (2.3)Proof. Firstly, by the mean value theorem, there exist 0 < wij < 1, 1 ≤i, j ≤ N , satisfying for i, j = 1, . . . , N ,fij(ui, uj)− fij(vi, vj)=(ui − vi)D1fij(wijui + zijvi, wijuj + zijvj)+ (uj − vj)D2fij(wijui + zijvi, wijuj + zijvj), (2.4)Global asymptotic stability for a nonautonomous Lotka-Volterra competition system 7where zij = 1− wij , 1 ≤ i, j ≤ N . By (2.1), we haveC∗(A, B) ≤ Dkfij(wijui + zijvi, wijuj + zijvj) ≤ C∗(A, B) (2.5)for k = 1, 2 and i, j = 1, . . . , N . By (2.5), we can have (2.3) as in theproof of [12, Proposition 1].When condition (GA) holds, we can reduce (2.2) to a simpler one (see[12, Proposition 2]).Proposition 2.2. Suppose that condition (GA) holds with δ = A andR = B; that ism[ai] >C∗(A, B)C∗(A, B)∑j 6=ibijMM [aj ]bjjL, 1 ≤ i ≤ N.Then there exist some positive constants α1, . . . , αN such that for j =1, . . . , N ,αjbjjL −C∗(A, B)C∗(A, B)∑i6=jαibijM > 0.Therefore, if condition (GA) holds, so does (2.2) with A = δ and B = R.By Proposition 2.1 and 2.2, the following theorem holds (see [12, Theo-rem 2]):Theorem 2.3. Let conditions (1.1), (1.2) and (1.3) hold . Suppose that forsome δ > 0 and R > 0 satisfying (1.7), condition (GA) and the conditionlim inft→∞ai(t)−∑j 6=i fij(R, R)bij(t)bii(t)> max1≤j≤N{fjj(δ, δ)},1 ≤ i ≤ N, (2.6)hold . Then for solutions u = (u1, . . . , uN ) and v = (v1, . . . , vN ) of system(GLV) with u(t0), v(t0) ∈ RN+ , the following statements (i) and (ii) hold :(i) 0 < lim inft→∞ ui(t) ≤ lim supt→∞ ui(t) <∞, i = 1, . . . , N ;(ii) limt→∞(ui(t)− vi(t)) = 0, i = 1, . . . , N .8 Kunihiko TaniguchiRemark 2.4. We note that (1.8) implies (2.6). Therefore, by Theorem2.3, in order to prove Theorem 1.2, it is sufficient to prove the existence ofa unique solution of system (GLV) satisfying (I) and (III).Lemma 2.5. Let conditions (1.1), (1.2) and (1.3) hold . Let u = (u1, . . . , uN )and v = (v1, . . . , vN ) be solutions of system (GLV) defined on (−∞, T ].Suppose that there exist constants A, B > 0 satisfyingA ≤ uj(t), vj(t) ≤ B (2.7)for j = 1, . . . , N and t ≤ T . Suppose moreover that for system (GLV)there exist positive constants α1, . . . , αN satisfyinginft≤Tαjbjj(t)− C∗(A, B)C∗(A, B)∑i6=jαibij(t) > 0 (2.8)for j = 1, . . . , N . Then u ≡ v.Proof. Step 1. Firstly, from (2.8), there exists some ε > 0 such that forj = 1, . . . , N and t ≤ T ,αjbjj(t)−C∗(A, B)C∗(A, B)∑i6=jαibij(t) ≥ ε. (2.9)Next letθ(t) =N∑i=1∣∣∣∣ log(ui(t)vi(t)) ∣∣∣∣ , t ≤ T.Then we can claim the following:Claim. There exists some η > 0 such that for t < T ,θ(T ) ≤ θ(t)− η∫ TtN∑i=1| ui(s)− vi(s) | ds. (2.10)In fact, since | log(ui(t)/vi(t)) |, 1 ≤ i ≤ N , is absolutely continuous onGlobal asymptotic stability for a nonautonomous Lotka-Volterra competition system 9every finite interval I ⊂ (−∞, T ], for almost t ≤ T ,θ′(t) =N∑i=1αi[u′iui−v′ivi]sgn(ui − vi)=N∑i=1αi− N∑j=1bij(t)(fij(ui, uj)− fij(vi, vj)) sgn(ui − vi)=N∑j=1[− αjbjj(t)(f˜j(uj)− f˜j(vj))sgn(uj − vj)−∑i6=jαibij(t)(fij(ui, uj)− fij(vi, vj))sgn(ui − vi)],where u = u(t), v = v(t). By (2.4), we haveθ′(t) =N∑j=1[− αjbjj(t)(D1fjj(wjjuj + zjjvj , wjjuj + zjjvj)+D2fjj(wjjuj + zjjvj , wjjuj + zjjvj))(uj − vj)sgn(uj − vj)−∑i6=jαibij(t)D1fij(wijui + zijvi, wijuj + zijvj)(ui − vi)sgn(ui − vi)−∑i6=jαibij(t)D2fij(wijui + zijvi, wijuj + zijvj)(uj − vj)sgn(ui − vi)]≤N∑j=1−C∗(A, B)αjbjj(t) | uj − vj | +∑i6=jC∗(A, B)αibij(t) | uj − vj | .By (2.9), we haveθ′(t) ≤ −εC∗(A, B)N∑j=1| uj − vj | a.e. t ≤ T.Hence, putting η = εC∗(A, B), we can obtain (2.10).Step 2. By (2.7), there exists some C > 0 such that θ(t) ≤ C, t ≤ T .Since (2.10) can be rewritten asη∫ Tt| ui(s)− vi(s) | ds ≤ θ(t)− θ(T ) ≤ θ(t) ≤ C10 Kunihiko Taniguchifor t < T and i = 1, . . . , N , we find∫ T−∞| ui(s)− vi(s) | ds <∞.Here letm(t) = max{| ui(t)− vi(t) | | 1 ≤ i ≤ N}.Since m(t) ≤| u1(t)− v1(t) | + · · ·+ | uN (t) + vN (t) |, we have∫ T−∞m(s)ds <∞and so we may assume that lim inft→−∞m(t) = 0. Thus there exists somesequence {tn}∞n=1 ⊂ (−∞, T ] such thattn → −∞ and m(tn)→ 0 as n→∞.Since for i = 1, . . . , N ,∣∣∣∣ ui(tn)vi(tn) − 1∣∣∣∣ =∣∣∣∣ ui(tn)− vi(tn)vi(tn)∣∣∣∣ ≤ m(tn)A ,it follows that θ(tn)→ 0 as n→∞. Furthermore, since θ is nonincreasingby (2.10),0 ≤ θ(T ) ≤ θ(tn).Hence θ(T ) = 0, that is, u(T ) = v(T ) and u ≡ v.Remark 2.6. By Proposition 2.2, we note that condition (GA) implies(2.8).The following lemma has been employed in [11, 12]. However, its proofhas not been given explicitly therein. Therefore we give the sketch of theproof.Lemma 2.7. Let conditions (1.1), (1.2) and (1.3) hold . Let u = (u1, . . . , uN )be a solution of system (GLV) with u(t0) ∈ RN+ . Suppose that for someδ > 0 and R satisfying (1.7), (1.8) hold . Then the following statements (i)and (ii) hold :Global asymptotic stability for a nonautonomous Lotka-Volterra competition system11(i) For 1 ≤ i ≤ N and t ≥ t0,ui(t) ≤ max{ui(t0), f˜−1i ((ai/bii)M )};(ii) For 1 ≤ i ≤ N and t ≥ t0,ui(t) ≥ min{ui(t0), δ}if 0 < ui(t0) < f˜−1i ((ai/bii)M ), 1 ≤ i ≤ N .Proof. (i) If there exist some T and i ∈ {1, . . . , N} such that f˜i(ui(T )) >(ai/bii)M , thenu′i(T ) ≤ ui(T )[ai(T )− bii(T )f˜i(ui(T ))] < 0.This proves (i).(ii) From (i), it follows that for i = 1, . . . , N and t ≥ t0,ui(t) ≤ R.Therefore, by (1.8), if there exist some T and i ∈ {1, . . . , N} such thatui(T ) ≤ δ, thenu′i(T ) ≥ ui(T )ai(T )−∑j 6=ibij(T )fij(R, R)− bii(T )f˜i(ui(T )) > 0.This proves (ii).Remark 2.8. Suppose that for some δ > 0 and R satisfying (1.7), (1.8)hold. By Lemma 2.7, if u is a solution of system (GLV) such that fori = 1, . . . , N ,δ < ui(t0) < f˜−1i ((ai/bii)M ),then for i = 1, . . . , N and t ≥ t0,δ ≤ ui(t) ≤ f˜−1i ((ai/bii)M ).Note that the proof of Theorem 1.2 is essentially based on this simpleconsideration.12 Kunihiko Taniguchi3. Proof of theorem 1.2In this section we prove Theorem 1.2 by employing the results in theprevious section.Proof of Theorem 1.2. (I) Step 1. Form = 1, 2, . . ., let um = (um1, . . . , umN )be a solution of system (GLV) such that δ < umi(−m) < f˜−1i ((ai/bii)M ),1 ≤ i ≤ N . By Lemma 2.7, it follows that for i = 1, . . . , N and t ≥ −m,δ ≤ umi(t) ≤ f˜−1i ((ai/bii)M ). (3.1)Step 2. We put I1 = [−1, 1]. By (3.1), a sequence {u′m} are uniformlybounded on I1. Therefore, by Arzela`-Ascoli theorem, there exists a subse-quence {um, 1} ⊂ {um} such that {um, 1} converges to a solution of system(GLV) uniformly on I1For n = 1, 2, . . ., we put In = [−n, n]. Similarly to the above ar-gument, {um, 1} has a subsequence {um, 2} that converges to a solutionof system (GLV) uniformly on I2. More generally {um, n−1} has a subse-quence {um, n} that converges to a solution of system (GLV) uniformly onIn. Therefore the diagonal sequence {um, m} converges to a solution u∗ ofsystem (GLV) uniformly on every finite interval I ⊂ R.Moreover, since for i = 1, . . . , N and t ∈ R,δ ≤ u∗i (t) ≤ f˜−1i ((ai/bii)M ),it follows from Proposition 2.2 and Lemma 2.5 that u∗ is the unique solutionsatisfying property (I).(II) This is a direct consequence of Theorem 2.3.(III) From (I), u∗(t) and u∗(t+ T ) satisfy (I). Since ai(t), bij(t), 1 ≤ i, j ≤N , are periodic with period T > 0, it follows that for i = 1, . . . , N ,u∗′i (t+ T ) = u∗i (t+ T )ai(t+ T )− N∑j=1bij(t+ T )fij(u∗i (t+ T ), u∗j (t+ T ))= u∗i (t+ T )ai(t)− N∑j=1bij(t)fij(u∗i (t+ T ), u∗j (t+ T )) .Hence, by Proposition 2.2 and Lemma 2.5, u∗(t) ≡ u∗(t+ T ), that is, u∗ isT -periodic.Global asymptotic stability for a nonautonomous Lotka-Volterra competition system13Acknowledgement. The author would like to express his appreciation forProfessors Toshitaka Nagai of Hiroshima University and Hiroyuki Usami ofGifu University for their useful comments and suggestions. The authorwould like to express his appreciation also for the referee for his accuratecomments.References[1] S. Ahmad, A. C. Lazer, On almost periodic solutions of the competingspecies problem, Proc. Amer. Math. Soc. 102 (1988) 855–861.[2] S. Ahmad, A. C. Lazer, On the nonautonomous N -competing speciesproblems, Appl. Anal 57 (1995) 309–323.[3] S. Ahmad, A. C. Lazer, Average conditions for global asymptotic sta-bility in a nonautonomous Lotka-Volterra system, Nonlinear Anal. 40(2000) 37–49.[4] S. Ahmad, A. C. Lazer, Average growth and extinction in a competitiveLotka-Volterra system, Nonlinear Anal. 62 (2005) 545–557.[5] S. Ahmad, A. C. Lazer, Average growth and total permanence in a com-petitive Lotka-Volterra system, Ann. Mat. Pura Appl. (4) 185 (2006)S47–S67.[6] C. Alvarez, A. C. Lazer, An application of topological degree to theperiodic competing species problem, J. Austral. Math. Soc. Ser. B28(1986) 202–219.[7] K. Gopalsamy, Global asymptotic stability in a periodic Lotka-Volterrasystem, J. Austral. Math. Soc. Ser B 27 (1985) 66–72.[8] K. Gopalsamy, Global asymptotic stability in an almost-periodic Lotka-Volterra system, J. Austral. Math. Soc. Ser B 27 (1986) 346–360.[9] K. Gopalsamy, Global asymptotic stability in a generalized Lotka-Volterra system, Int. J. Systems Sci., 17 (1986) 447–451.14 Kunihiko Taniguchi[10] J. Hofbauer, K. Sigmund, Evolutionary Games and Population Dy-namics, Cambridge University Press, New York, 1998.[11] K. Taniguchi, Asymptotic property of solutions of nonautonomousLotka-Volterra model for N -competing species, Differ. Equ. Appl. 2(2010) 447–464.[12] K. Taniguchi, Permanence and global asymptotic stability for a gen-eralized nonautonomous Lotka-Volterra competition system, HiroshimaMath. J. 42 (2012) 189–208.[13] A. Tineo, C. Alvarez, A different consideration about the globallyasymptotically stable solution of the periodic n-competing species prob-lem, J. Math. Anal. Appl. 159 (1991) 44–60.Kunihiko TaniguchiFukuoka Prefectural Kokura Nishi High School5-7-1, Simoitozu, Kokurakita-ku, Kitakyushu-shi, Fukuoka 803-0846, JAPANe-mail: kunihiko.taniguchi@gmail.com(Received June 10, 2013)

Global asymptotic stability for a nonautonomous Lotka-Volterra competition system

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Global asymptotic stability for a nonautonomous Lotka-Volterra competition system

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