EXTINCTION IN NONAUTONOMOUS COMPETITIVE LOTKA-VOLTERRA SYSTEMS

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Vome 124, Nmber 12, December 1996, Pages S ) EXTINCTION IN NONAUTONOMOUS COMPETITIVE LOTKA-VOLTERRA SYSTEMS FRANCISCO MONTES DE OCA AND MARY LOU ZEEMAN Commcaed by Lda Kee) Absrac. I s we kow ha for he wo speces aoomos compeve Loka-Voerra mode wh o fxed po he ope posve qadra, oe of he speces s drve o exco, whs he oher popao sabses a s ow carryg capacy. I hs paper we prove a geerasao of hs res o oaoomos sysems of arbrary fe dmeso. Tha s, for he speces oaoomos compeve Loka-Voerra mode, we exhb smpe agebrac crera o he parameers whch garaee ha a b oe of he speces s drve o exco. The resrco of he sysem o he remag axs s a oaoomos ogsc eqao, whch has a qe soo ) ha s srcy posve ad boded for a me; see Coema Mah. Bosc ), ) ad Ahmad Proc. Amer. Mah. Soc ), ). We prove addo ha a soos of he -dmesoa sysem wh srcy posve a codos are asympoc o ). 1. Irodco Ths paper coes Zeema [15] a aemp o geerase cassca ress abo Loka-Voerra sysems he dreco sggesed by Ahmad [3], Gopasamy [7] ad ohers [5, 13]. Cosder a commy of may compeg speces modeed by he oaoomos Loka-Voerra sysem 1.1) x ) =x ) b ) a j )x j ), =1,...,, where x ) s he popao sze of he h speces a me, adx deoes dx d. Each k-dmesoa coordae sbspace of R s vara der sysem 1.1), k {1,...,}), ad we adop he rado of resrcg aeo o he cosed posve coe R +. We deoe he ope posve coe by R +, ad ca a vecor x posve f x R +, srcy posve f x R +.Gvex, y R,wewrex y o deoe ha x y) s posve. Receved by he edors March 21, Mahemacs Sbjec Cassfcao. Prmary 34C35, 92D25; Secodary 34A26. Key words ad phrases. Loka-Voerra, oaoomos, Lapov, compeo, exco. The frs ahor was sppored par by he Dvso of Mahemacs ad Sascs a he Uversy of Texas a Sa Aoo. The secod ahor was sppored par by he Offce of Research Deveopme a he Uversy of Texas a Sa Aoo c 1996 Amerca Mahemaca Socey Lcese or copyrgh resrcos may appy o redsrbo; see hp://

2 3678 FRANCISCO MONTES DE OCA AND MARY LOU ZEEMAN The ma compeo bewee he speces dcaes ha a j ) > 0 for a j, ad for a. I addo we assme hrogho ha for a ad j, a j ) adb ) are coos fcos, boded above ad beow by srcy posve reas. Ths, whe we cosder sysem 1.1) resrced o he h coordae axs, we have he oaoomos ogsc eqao 1.2) x ) =x )b ) a )x )). I s we kow ha a aoomos ogsc eqao ẋ = xb ax) wh a, b > 0 has a goba aracor o R + a he carryg capacy x = b a. The combed ress of Ahmad [3] ad Coema [6], saed as Lemmas 1.1 ad 1.2 beow, show ha he oaoomos eqao 1.2) he roe of he gobay aracg carryg capacy of he aoomos eqao s payed by a we defed caoca soo x ) o whch a oher soos coverge. Lemma 1.1 Ahmad, Coema). Eqao 1.2) has a qe soo x ) whch s boded above ad beow by srcy posve reas for a. We ca x he caoca soo of eqao 1.2). Lemma 1.2 Coema). If ),v) are soos of 1.2), he) v)) 0 as. Ths ),v) x )as. I s a cassca res ha for a wo speces aoomos compeve Loka- Voerra mode wh o fxed po he ope posve coe R 2 +,oeofhe speces s drve o exco, whs he oher popao sabses a s ow carryg capacy. I [3], Ahmad proves a aaogos res for oaoomos wo-dmesoa compeve Loka-Voerra sysems. Tha s, der he assmpo ha each of he coeffce fcos s coos ad boded above ad beow by srcy posve mbers, he gves smpe agebrac crera der whch here s o coexsece of he wo speces. Oe of he speces s drve o exco, whs he oher speces sabses a he caoca soo of he ogsc eqao o ha axs. Ahmad ad Lazer [4], ad Zeema [15] geerase he cassca res a dffere dreco: o aoomos compeve Loka-Voerra sysems of arbrary fe dmeso. Ahmad ad Lazer fd agebrac crera der whch oe of he speces s drve o exco, whs he remag 1) speces coexs saby. Zeema fds agebrac crera der whch 1) of he speces are drve o exco, whs he remag speces sabses a s ow carryg capacy. I hs paper we mprove ad geerase he ress of [15] o he case of oaoomos compeve Loka-Voerra sysems of arbrary fe dmeso. See aso [5, 11] for frher geerasaos of hs work whch brdge he gap bewee he ress of Ahmad ad Lazer, ad hose of hese ahors. I seco 2 we sae or ma res Theorem 2.1) ad compare wh he ma res [15]. I seco 3 we gve a geomerc erpreao of or agebrac hypoheses, ad se hs o gve a geomerc skech of he proof. We make he proof rgoros secos 4-6. Lcese or copyrgh resrcos may appy o redsrbo; see hp://

3 NONAUTONOMOUS COMPETITIVE LOTKA-VOLTERRA SYSTEMS Saeme of res Reca ha we assme hrogho ha for a ad j, a j ) adb ) are coos fcos, boded above ad beow by srcy posve reas. To fx oao, e a j =f a j ), a j =sp a j ), b =f b ), ad b =sp b ). Noe ha he aoomos case, a j = a j = a j) for a. I seco 3 we gve a geomerc erpreao of eqaes 2.1) of he foowg heorem, whch shod hep o rave he sbscrps. Theorem 2.1. Gve sysem 1.1), sppose ha k >1, k <k j k, a < b 2.1) k kj a. k j The every rajecory wh a codo R + s asympoc o x 1. I oher words, for a srcy posve a codos, speces x 2,...,x are drve o exco, whs speces x 1 sabses a he qe boded soo x 1 of he ogsc eqao o he x 1 -axs. We prove Theorem 2.1 secos 4, 5 ad 6, provg he exco of speces x 2,...,x seco 5 Theorem 5.1), ad he covergece of rajecores o x 1 seco 6 Theorem 6.1). Aowg for reabeg of he axes, we have: Coroary 2.2. If here s a permao φ of he dces {1,...,} afer whch sysem 1.1) sasfes eqaes 2.1), he every rajecory wh a codo R + s asympoc o x φ 1 1) der he orga sysem. The foowg coroares reae Theorem 2.1 o he ress Ahmad [3] ad Zeema [15]. Coroary 2.3. Gve sysem 1.1), sppose ha k >1, j k, a < b 1 kj a. 1j The every rajecory wh a codo R + s asympoc o x 1. Proof. Coroary 2.3 foows drecy from Theorem 2.1 by seg k =1foreach k. Coroary 2.4. Gve sysem 1.1), sppose ha b j a jj < b 1 a 1j j >1, ad b k b j a jj b k > b k a kj k >j. The every rajecory wh a codo R + s asympoc o x 1. Proof. b k a kj < b j a jj b j a jj < b 1 a 1j k >j, Lcese or copyrgh resrcos may appy o redsrbo; see hp://

4 3680 FRANCISCO MONTES DE OCA AND MARY LOU ZEEMAN ad whe j = k b k a kj = b j a jj < b 1 a. 1j Hece Coroary 2.4 foows from Coroary 2.3. Remark 2.5. I s cear from hs proof ha he hypoheses of Coroares 2.4 ad 2.6 may be reaxed o perm oe of he ses of eqaes o be weak eqaes. I he wo-dmesoa case, hs agrees wh he ress of Ahmad [3]. Coroary 2.6 foows drecy from Coroary 2.4. We cde o show how Theorem 2.1 mproves he ma heorem [15], whch s gve by appyg Coroary 2.6 o he aoomos case. Coroary 2.6. Gve sysem 1.1), sppose ha b j a jj < b a j <j, ad b j a jj > b a j >j. The every rajecory wh a codo R + s asympoc o x 1. Remark 2.7. I s eresg o compare he reave sreghs of he hypoheses of Theorem 2.1 hrogh Coroary 2.6, by cosderg he appcao of each o hree-dmesoa aoomos compeve Loka-Voerra sysems. These sysems were sded [14], ad cassfed o 33 ope eqvaece casses caed ce casses. I [15] was show ha sysems ce cass 1 are precsey hose sasfyg he hypoheses of Coroary 2.6, ad ha hs res s far from beg sharp, sce ce casses 2,3,7 ad 8 aso coss of sysems whch a b oe speces are drve o exco. I s sraghforward o verfy ha permg permao of he axes) he hypoheses of Coroary 2.4 are sasfed by sysems ce casses 1 ad 3; he hypoheses of Coroary 2.3 are sasfed by sysems ce casses 1, 3 ad 7; ad he hypoheses of Theorem 2.1 are sasfed by sysems ce casses 1, 2, 3 ad 7. Ths we see ha ahogh he ma res hs paper s cosderaby sroger ha ha [15], eve for he aoomos case, s s o sharp, as does o appy o he sysems ce cass Geomerc erpreao of he eqaes The aoomos case. Cosder he aoomos compeve Loka-Voerra sysem 3.1) x = x b a j x j, =1,...,, sasfyg he eqaes b k k >1, k <k j k, < b 3.2) k. a kj a k j Sysem 3.1) resrced o he posve x 1 -axs s a aoomos ogsc eqao wh a goba aracor a he carryg capacy b1 a 11. Theorem 2.1 es s ha he Lcese or copyrgh resrcos may appy o redsrbo; see hp://

5 NONAUTONOMOUS COMPETITIVE LOTKA-VOLTERRA SYSTEMS 3681 x 2 x 2 a) a 12 a 12 N 1 a 12 a a N 2 a 22 N 1 N 2 b) a 21 a 11 x 1 x 1 a 21 a 21 a 11 a 11 Fgre 1. Exampe of ces of a) a aoomos, ad b) a oaoomos wo-dmesoa compeve Loka-Voerra sysem sasfyg eqaes 2.1). po b1 a 11, 0,...,0) s fac a goba aracor o R + for he f sysem 3.1). Tha s, each of speces x 2,...,x s drve o exco. We proceed o rasae eqaes 3.2) o geomerc properes of sysem 3.1), ad o skech he proof of Theorem 2.1 for he aoomos case, o show how hese eqaes dcvey ead o he exco of each x k for k>1. A deaed proof of Theorem 2.1 s gve secos 5 ad 6. The h ce of sysem 3.1) s he se R + o whch ẋ =0. Isgveby {x =0} N where N s he hyperpae b = a jx j, whch has posve axa erceps b a j. See Fgre 1a) for a wo-dmesoa exampe. Ieqaes 3.2) gve a para orderg bewee he axa erceps of he hyperpaes N aog each axs, from whch we ca dedce o-erseco properes of he ces R +. A geomerc aayss of smar o-erseco properes was sed o prove he ress [15]. The o-erseco properes correspodg o eqaes 3.2) are cosderaby weaker ha hose sed [15], b everheess ead o he same cocso. For exampe, cosder he hyperpaes N ad N. By eqaes 3.2), b < b 3.3), j. a j a j Hece N ad N are dsjo R +. Moreover, N es erey above N, meag ha N s coaed he boded compoe of R + \ N. The foowg emma s proved he frs haf of he proof of Theorem 5.1 repacg by j he proof). Geomerc Lemma 3.1. Gve sysem 3.1), fhereexs, j sch ha N es erey above N j, he speces x j s drve o exco. Ths x s drve o exco by eqaes 3.3). We ow resrc aeo o he sbspace H 1 o whch x = 0, ad we cosder he resrco of he Lcese or copyrgh resrcos may appy o redsrbo; see hp://

6 3682 FRANCISCO MONTES DE OCA AND MARY LOU ZEEMAN hyperpaes N 1) ad N 1) o hs sbspace. By eqaes 3.2), b 1) a 1)j < b 1) a 1) j j 1), so N 1) es erey above N 1) H 1) +, ad hece x 1) s drve o exco he sbsysem correspodg o x =0. Smary, for each r>1, N r es erey above N r he r-dmesoa sbspace H r o whch x r+1),...,x vash, ad hece x r s drve o exco he correspodg sbsysem. Ths eqaes 3.2) ead dcvey o he exco of x k for each k>1, ad hece we prove Theorem 6.1 ha R 1 s gobay aracg o R +. The oaoomos case. Now cosder he oaoomos sysem 1.1) sasfyg eqaes 2.1). Defe he pper sysem of sysem 1.1) o be he aoomos sysem x = x b a j x j, =1,...,, ad defe he ower sysem by x = x b a j x j, =1,...,. The for each, he pper sysem has h ce N, as defed above ad whch mees he x j -axs a b. Smary, he ower sysem has h ce N a,whch j mees he x j axs a b a. j We se hese pper ad ower ces o defe a hckeed ce N for sysem 1.1) by N = {x R + : y N, z N y x z} I oher words, N s he rego R + bewee N ad N. See Fgre 1b) for a wo-dmesoa exampe. For each fxed, he h ce N ) of he aoomos sysem wh coeffces a j ),b ) mees he x j -axs a b) a.now, j) b a < b ) j a j ) < b a j =1,...,, j ad hece N ) N. I oher words, he h ce of he aoomos sysem correspodg o each fxed s coaed he h hckeed ce of he oaoomos sysem. Ieqaes 2.1) ca ow be erpreed, drec aaogy wh he aoomos case, as o-erseco properes of he hckeed ces. For exampe: N es erey above N, ad hece hckeed) N es erey above hckeed) N. Moreover, he proof of Theorem 5.1 shows ha he geomerc emma Lemma 3.1) hods for he oaoomos case as we as he aoomos case, ad hs he o-erseco properes ead o he exco of speces x 2,...,x js as before. Lcese or copyrgh resrcos may appy o redsrbo; see hp://

7 NONAUTONOMOUS COMPETITIVE LOTKA-VOLTERRA SYSTEMS A compac aracg rego There are may ways o fd a compac aracg rego for sysem 1.1). Lemma 4.1 ses a parcary smpe compac aracg rego o gve coarse bods o he compoes of a soo o sysem 1.1). These bods are eeded for he esmao secos 5 ad 6. A more decae ad opma compac aracg rego for sysem 1.1) s fod [16]. Lemma 4.1. If x) s a soo of sysem 1.1) wh a codo R +, he here exs r, δ > 0 ad T R sch ha for a >T, x ) δ ad 0 <x ) r, =1,...,. =1 Proof. I s cear ha he ope ad cosed posve coes are vara der sysem 1.1). Now choose { } { } b r 2max :, j =1,...,, δ 1 2 m b :, j =1,..., ad defe a j a j { } S = x R + : δ x ) r. We sha show ha S s a gobay aracg posvey vara compac se for R + \{0}.Thefx) s a soo of sysem 1.1) wh a codo R +, here exss T R sch ha for a >T,x) S, ad he cocso foows. Cosder he fco L : R + R + defed by Lx) = =1 x. The L = 1,...,1), ad he dervave of L aog rajecores of sysem 1.1) s gve by L = L.ẋ = ẋ = x ) b ) a j )x j ). =1 =1 I s easy o see ha L s Lapov-ke osde S. Tha s, by or choce of δ, f x R + \{0}sasfes =1 x δ he ẋ s o-egave for each, adsrcy posve for a eas oe vae of. So a each x sasfyg 0 <Lx) δ, Ls srcy posve. Smary, for each x sasfyg Lx) r, L s srcy egave. Hece [δ, r] s a compac aracg se for L aog each rajecory of sysem 1.1), ad hs S s a compac aracg se for he fow of sysem 1.1) o R + \{0}. =1 5. Exco of x 2,...,x Theorem 5.1. If sysem 1.1) sasfes eqaes 2.1) ad x) s a soo of sysem 1.1) wh x 0 ) R + for some 0, he for a =2,...,, a) b) x ) 0 as, ad 0 x )d <. Remark 5.2. Le H k deoe he k-dmesoa sbspace o whch x k+1,...,x a vash. Geerasg he mehod [15], cocso a) of Theorem 5.1 ca be proved by dcvey appyg he Lapov fcos V k x) =x b k k x b k k Lcese or copyrgh resrcos may appy o redsrbo; see hp://

8 3684 FRANCISCO MONTES DE OCA AND MARY LOU ZEEMAN o each H k. However, we sha eed cocso b) o prove ha soos are asympoc o x 1, so we se he dea behd he Lapov fcos V k o deveop a egrao proof here. Proof of Theorem 5.1. Le x) be a soo of sysem 1.1) wh x 0 ) R + for some 0. By Lemma 4.1 we may assme ha x 0 ) S. We prove Theorem 5.1 by dco. Frs we show ha cocsos a) ad b) hod for x ). Le = gve by eqaes 2.1). By defo ẋ ) x ) b a j x j) ad ẋ ) x ) b a j x j), so d d x b ) )x b ) ẋ = b ) x ) ) b ) ẋ ) x ) b a j b a ) j xj ) { max j b a j b a } j x j ). Hece, for > 0 d d x b ) { )x b ) max j b a j b j} a δ<δ for some rea δ < 0, by Lemma 4.1 ad eqaes 2.1). Iegrag hs eqao we have ad so for > 0 x b ) )x b ) <δ 0 ) 0 x b )<Ce δ 0), where C = xb 0 ) x b 0 ) rb. Ths x ) <K e ε 0), > 0, where ε = δ < 0adK b >0. Cocsos a) ad b) for x foow drecy. We ow prove ha for 1 <r<, x r 0as der he assmpo ha for r<j, x j 0as. The mehod s esseay he same as ha sed above for x.nowe= r gve by eqaes 2.1). The ẋ ) x ) b a j x j) ad ẋ r ) x r ) b r a rj x j), Lcese or copyrgh resrcos may appy o redsrbo; see hp://

9 NONAUTONOMOUS COMPETITIVE LOTKA-VOLTERRA SYSTEMS 3685 so d d x b r ) )x b r ) ẋr = b ) x r ) = r ) b r ) ẋ ) x ) b r a j ) b a rj xj ) b r a j ) b a rj xj )+ j=r+1 b r a j ) b a rj xj ). By eqaes 2.1) each erm he frs smmao s srcy egave. The eqaes do o gve s coro over he sg of he secod smmao. Isead we se he assmpo ha x j 0as,forj>r, as foows. Frsy, oe ha for sffcey arge r x j) > δ 2,whereδs gve by Lemma 4.1. Secody, choose ν>0sch ha ν< max j r b r a j b a rj )δ 2,ad oe ha for sffcey arge j=r+1 b r a j b a rj )x j) <ν.thshereexss r Rsch ha for > r d ) d x b r )x b r ) < max j r b r a j b a rj) δ 2 + ν = δ r < 0. Iegrag hs eqao we have ) x b r )x b r ) <δ r r ) r ad so for > r xr)<kre εr r), where ε r = δr b < 0adK r >0. Cocsos a) ad b) foow drecy. 6. Covergece o x 1 Reca from seco 1 ha x 1 s he caoca soo o he oaoomos ogsc eqao obaed by resrcg sysem 1.1) o he x 1 -axs. Theorem 6.1. If sysem 1.1) sasfes eqaes 2.1) ad x) s a soo of sysem 1.1) wh x 0 ) R + for some 0,hex 1 ) x 1 )as. Proof. By Lemma 4.1, we may assme ha x 0 ) S R +. The for a > 0, x) Sad x 1 ) s boded above ad beow by posve cosas. Le 1 ) be a soo of he oaoomos ogsc eqao 1.2) sch ha 1 0 ) x 1 0 ). The 1 ) >x 1 ) for a > 0 see Ahmad [1], Lemma 2.8 or Teo ad Avarez [12], Proposo 2.1), ad 1 ) s boded Lemmas 1.1 ad 1.2). We ow foow a echqe smar o ha sed he proof of he prevos heorem o compare 1 ) adx 1 )as : d x ) 1) d 1 ) = ẋ1) x 1 ) 1) 1 ) = a 11 ) 1 ) x 1 )) a 1j )x j ), j=2 Lcese or copyrgh resrcos may appy o redsrbo; see hp://

10 3686 FRANCISCO MONTES DE OCA AND MARY LOU ZEEMAN so 1 ) x 1 ) 1 d a x ) 1) + a 1j )x j ). 11 d 1 ) j=2 Iegrag hs eqay we have 1 ) x 1 )) d 1 x 1) 0 a 11 1 ) + a 1j )x j )d 0 j=2 0 1 ) x1 ) 1 0 ) a + a 1j x j )d 11 1 )x 1 0 ) j=2 0 < K <, where K s some cosa depede of, scex 1 ), 1 ) are boded by posve cosas, ad for j>1, x j )d < Theorem 5.1). Ths 0 1 ) x 1 )) d <, 0 ad so x 1 ) 1 ) as, sce 1 ) x 1 ) s a o-egave dffereabe fco sch ha 1 ) ẋ 1 ) s boded o [ 0, ). Moreover, by Lemma ) x 1 ) as, ad hece x 1 ) x 1 ) as. Theorem 2.1 s ow a coroary of Theorems 5.1 ad 6.1. Ackowedgme We wod ke o hak Shar Ahmad for rodcg s o hs probem ad for hepf dscssos. Refereces [1] S. Ahmad. Covergece ad Umae Bods of Soos of he Noaoomos Voerra- Loka Compeo Eqaos, J. Mah. Aa. App ), MR 89a:92032 [2] S. Ahmad. O Amos Perodc Soos of he Compeg Speces Probems, Proc. Amer. Mah. Soc ), MR 89f:92055 [3] S. Ahmad. O he Noaoomos Voerra-Loka Compeo Eqaos, Proc. Amer. Mah. Soc ), MR 93c:34109 [4] S.AhmadadA.C.Lazer.Oe Speces Exco a Aoomos Compeo Mode, Proc. Frs Word Cogress Noear Aayss, Waer DeGryer, Ber, [5] S. Ahmad ad A. C. Lazer O he Noaoomos N-Compeg Speces Probem, App. Aa. 1995) To appear. [6] B. D. Coema. Noaoomos Logsc Eqaos as Modes of he Adjsme of Popaos o Evromea Chage, Mah. Bosc ), MR 80f:92012 [7] K. Gopasamy. Gobay Asympoc Saby a Perodc Loka-Voerra Sysem, J. Mah. Aa. App ), MR 86f:34094 Lcese or copyrgh resrcos may appy o redsrbo; see hp://

11 NONAUTONOMOUS COMPETITIVE LOTKA-VOLTERRA SYSTEMS 3687 [8] M. W. Hrsch. Sysems of Dfferea Eqaos ha are Compeve or Cooperave. III: Compeg Speces, Noeary ), MR 90d:58070 [9] J. Hofbaer ad K. Sgmd. The Theory of Evoo ad Dyamca Sysems. Cambrdge Uv. Press, Cambrdge, MR 91h:92019 [10] R. M. May. Saby ad Compexy Mode Ecosysems. Prceo Uv. Press, Prceo, NJ, [11] F. Moes de Oca ad M. L. Zeema. Baacg Srvva ad Exco Noaoomos Compeve Loka-Voerra Sysems, J. Mah. Aa. App ), MR 96c:92017 [12] A. Teo ad C. Avarez. A Dffere Cosderao abo he Gobay Asympocay Sabe Soo of he Perodc -Compeg Speces Probem, J. Mah. Aa. App ), MR 93d:34080 [13] A. Teo. O he Asympoc Behavor of some Popao Modes, J. Mah. Aa. App ), MR 93g:92027 [14] M. L. Zeema. Hopf Bfrcaos Compeve Three-Dmesoa Loka-Voerra Sysems, Dyamcs Saby Sysems ), MR 94j:34044 [15] M. L. Zeema. Exco Compeve Loka-Voerra Sysems, Proc. Amer. Mah. Soc ), MR 95c:92019 [16] M. L. Zeema. Thckeed Carryg Smpces Noaoomos Compeve Loka-Voerra Sysems, To appear. Uversdad Ceroccdea, Lsadro Avarado, Barqsmeo, Veezea Dvso of Mahemacs ad Sascs, Uversy of Texas a Sa Aoo, Sa Aoo, Texas E-ma address: zeema@rger.cs.sa.ed Lcese or copyrgh resrcos may appy o redsrbo; see hp://