Dynamics of a Discrete Predator-Prey System with Beddington-DeAngelis Function Response

Transcription

1 Apped Mathematcs Pshed Onne Apr ( Dynamcs of a Dscrete Predator-Prey System wth Beddngton-DeAnges Fncton Response Qn Fang Xaopng L * Mey Cao Scence Coege Hnan Agrctra Unversty Changsha Chna Ema: * xp68@yahoocomcn Receved Ferary 7 ; revsed March 9 ; accepted March 6 STRACT Ths paper dscsses the dynamc ehavors of a dscrete predator-prey system wth Beddngton-DeAnges fncton response We frst show that nder some stae assmpton the system s permanent Frthermore y constrctng a stae Lyapnov fncton a sffcent condton whch garantee the goa attractvty of postve sotons of the system s estashed Keywords: Dscrete; Beddngton-DeAnges Fnctona Response; Permanence; Goa Attractvty Introdcton Snce the end of the 9th centry many oogca modes have een estashed to strate the evotonary of speces among them predator-prey modes attracted more and more attenton of oogsts and mathematcans There are many dfferent nds of predator-prey modes n the teratre In 975 Beddngton [] and DeAnges [] proposed the predator-prey system wth the Beddngton-DeAnges fnctona response as foows cy x xax mmxm3y fx y yd mmxm3y () Recenty L and Taech [3] proposed the foowng mode wth oth Beddngton-DeAnges fnctona response and densty dependent predator cy x xax mmxm3y fx y y d ey mmxm3y () and dscssed the dynamc ehavors of the mode On the other hand when the sze of the popaton s rarey sma or the popaton has non-overapng generaton the dscrete tme modes are more approprate than the contnos ones Dscrete tme modes can aso provde effcent comptatona modes of contnos * Correspondng athor modes for nmerca smatons In [4] Qn and L stded the dynamc ehavor of the foowng dscrete tme compettve system x n cnyn xnexp annxn yn yn f ( nxn ) ynexp dn enyn xn (3) In [5] W and L consdered the foowng dscrete tme predator-prey system wth hasse-varey type fnctona response x n cnyn xnexp annxn r mny n xn (4) yn f nxn ynexp dn r mny n xn some sffcent condtons for the permanence and goa attractvty of system (4) are otaned For more wor on ths drecton one cod refer to [6-4] Based on the aove dscsson n ths paper we consder the dscrete anaogos of () one can easy derve the dscrete anaoge of system () whch taes the form of Copyrght ScRes

2 39 Q B FANG ET AL cnyn xn xnexpannxn m n m n x n m3 n y n f nxn yn ynexpdnenyn mnmnxnm3nyn an In ths paper we aways assme that n cn dn en f n m n m n m 3 n are a postve onded seqences and a a n a n c cn c d d n d e en e f f n f f n sp f n f nf f n m m n m 3 Here for any onded seqence f nn nn From the vew pont of oogy we w focs or dscsson on the postve sotons of system (4) So t s assmed that the nta condtons of (4) are of the form x y (6) It s easy to see that the sotons of (4) wth the nta condton (5) are defned and reman postve for a N Permanence DEFINITION System (5) s sad to e permanent f there are postve constants r r R R sch that each postve soton xn yn of system (5) satsfes r m nf x n m sp x n R n n r m nf y n m sp y n R n n By Lemma we otan x n exp ( a n n x n (5) LEMMA [6] Assme that x n satsfes xn and xn xnexpan nxn for a n n a n n are postve seqences Then exp a m sp x n n LEMMA [6] Assme that x n satsfes exp m sp xn D and xn n an x n x n a n n x n n n n are postve seqences Then a expa D m nf xn n f LEMMA Assme that d hods then m for any postve soton xn yn of system (4) one has and m sp x n n exp a G f m sp yn G exp d n e m Proof Let x n y n e any postve soton of system (5) from the frst eqaton of (5) t foows that cn yn xn xnexpannxn m n m n x n m3 n y n m sp x n G exp a n Smary from the second eqaton of (5) t foows that f n xn yn ynexpdnenyn m n m n x n m3 n y n f n f ynexp dnenyn ynexp d e yn m n m Copyrght ScRes

3 Q B FANG ET AL 39 f Under the assmpton d y Lemma m we otan f m sp yng exp d n e m Ths competes the proof of Lemma LEMMA 4 Assme that h h Then for any postve soton xn yn of system (5) one has mnf xn g mspyn g n n h a c m 3 h d f g m m G m 3 G exp hexp h G h h e G g g e Proof Let x n y n e any postve soton of system (5) from the frst eqaton of (5) t foows that Under the assmpton cn yn xn xnexpannxn m n m n x n m3 n y n c n c xnexpan nxn xnexpa x n m3 n m3 x n exp h x n h By Lemma and Lemma we otan m nf x n n hexp h G g Smary from the second eqaton of (5) and Lemma t foows that f n xn y( n) ynexpdnenyn m n m n x n m3 n y n f g yn d e y n y n h e y n exp exp m mgm3g By Lemma and Lemma we have m nf y n n hexp h e G g e From Lemma and Lemma 4 we otan the foowng theorem THEOREM Assme that d f m a c m3 () f g d m m G m G 3 hod then system (5) s permanent () 3 Goa Attractvty Ths secton devotes to stdy the goa attractvty of the postve soton of system (5) * * DEFINITION 3 A postve soton x n y n of system (5) s sad to e goay attractve f each x n y n of (5) satsfes other postve soton * * m x n x n m y n y n n n THEOREM 3 In addton to () and () assme frther that there exst postve constants and sch that and mn cg m f m f G m3 G mm3g g 9 mm3gg 9 mmg g (3) Copyrght ScRes

4 39 Q B FANG ET AL f G m3 c m c G m e 3 G 3 9 mmg g 9 mmgg 3 9 mg g mn e (3) Then the postve soton of system (5)s goay attractve and 3 Proof From (3) and (3) there exsts an enogh sma postve constant mng g sch that 3 3 ac G m f m 3 G 9mm 3g g 9mm 3g g mn f G m 9 mm g g f G m3 c m e 3 G 9mm g g 9mm 3g g mn e c G m 9 m g g 3 3 For any postve sotons x y and x y of system (4) t foows from Lemma and Lemma 4 that m nf x g n m nf y g n m sp x G n m sp y G n (35) In vew of (35) for aove there exsts an nteger sch that for a n n n n V V V x x x x Let g x G n x n x [ x x ] n x n x c c By the mean vae theorem we have m y y m x y y c( ) g y G n n V x x A m m x m y 3 B m m x m y 3 From the frst eqaton of system (5) we have m y x x x x exp n x exp n x n x n x (37) es etween x and x It foows from (37) that 3 c m y x x V xx m m3 x x y c m y y c m x y y 9m m x x y y 9m x y y (33) (34) (36) Copyrght ScRes

5 Q B FANG ET AL 393 and so for Vmn xx G Let 3 c m G x x 9 mm 3g g 3 c m y y 9 mm 3gg 3 3 ) c m G y y 3 9 m g g n n V y y es etween (38) From the second eqaton of system (5) we have y y y y V V V n n n n n y n y e y( ) y n y n y f f f m3 x y y m x x m3 y x x By the mean vae theorem we have expn expn y n y n y y y y (39) y and y It foows from (39) that 3 f m3 x y y ( ) ( ) ( ) 9m m x y y 3 3 f m xx f m3 y xx m m x x y y 9m m x x y V e y y and so for V mn e e y y 3 3 f m G y y 3 G 9mm g g f m x x f m G x x 3 9 mm 3gg 9 mm g g (3) Now we defne a Lyapnov fncton as foows: Cacatng the dfference of V () aong the soton of V V V system (5) for t foows from (38) and (3) that V V V G 9 mm g g 3 3 c G m mn f m f G m 3 x x 3 9mm 3gg 9mm g g f G m c m c G m mn e e G 9[ mm( g )] ( g) 9[ mm3( g)( g )] 9[ m( g )] ( g) y y 3 3 Copyrght ScRes

6 394 Q B FANG ET AL It foows from (33) and (34) that V x x y y Smmatng oth sdes of the aove neqates from to we have V x x y y Whch mpes Then x x y y V x x y y Therefore That s m x x y y x m x x m y y Ths competes the proof of Theorem 3 REFERENCES [] J R Beddngton Mta Interference etween Parastes or Predators and Its Effect on Searchng Effcency Jorna of Anma Ecoogy Vo 44 No pp do:7/3866 [] D L DeAnges R A Godsten and R V O Ne A Mode for Trophc Interacton Ecoogy Vo 56 No pp do:7/93698 [3] H Y L and Y Taech Dynamcs of the Denst y De- System wth Beddngton-DeAn- pendent Predator-Prey ges Fnctona Response Jorna of Mathematca Anayss and Appcaton Vo 374 No 4 pp do:6/jjmaa89 [4] W J Qn Z J L and Y P Chen Permanence and Goa Staty of Postve Perodc Sotons of a Dscrete Compettve System Dscrete Dynamcs n Natre and Socety 9 Artce ID [5] R X W and Ln L Permanence and Goa Attractvty of Dscrete Predator-Prey System wth Hasse-Varey Type Fnctona Response Dscrete Dynamcs n Natre and Socety Appcatons Vo 99 No 4 pp [6] F Chen Permanence and Goa Staty of Nonatonomos Lota-Voterra System wth Predator Prey and Devatng Argments Apped Mathematcs and Comptaton Vo 73 No 6 pp 8- do:6/jamc5435 [7] F Chen Permanence and Goa Attractvty of a Dscrete Mtspeces Lota-Voterra Competton Predator- Prey Systems Apped Mathematcs and Comptaton Vo 8 No 6 pp 3- do:6/jamc636 [8] F Chen Permanence of a Dscrete n-speces Food-Chan System wth Tme Deays Apped Mathematcs and Comptaton Vo 8 No 7 pp do:6/jamc6779 [9] F Chen Permanence for the Dscrete Mtasm Mode wth Tme Deays Mathematca and Compter Modeng Vo 47 No pp do:6/jmcm7 [] YH Fan WT L Permanence for a Deayed Dscrete Rato-Dependent Predator-Prey System wth Hong Type Fnctona Response Jorna of Mathematca Anayss 9 Artce ID: 365 [] L Chen J X and Z L Permanence and Goa Attractvty of a Deayed Dscrete Predator-Prey System wth Genera Hong-Type Fnctona Response and Feedac Contros Dscrete Dynamcs n Natre and Socety 8 Artce ID 696 [] J Yang Dynamcs Behavors of a Dscrete Rato-Dependent Predator-Prey System wth Hong Type III Fnctona Response and Feedac Contros Dscrete Dynamcs n Natre and Socety Vo 8 Artce ID: [3] X L and W Yang Permanence of a Dscrete Predator- Prey Systems wth Beddngton-DeAnges Fnctona Response and Feedac Contros 8 Artce ID 4967 [4] M Fan and K Wang Perodc Sotons of a Dscrete Tme Nonatonomos Rato-Dependent Predator-Prey System Mathematca Compter Modeng Vo 35 No 9- pp do:6/s ()6-6 Copyrght ScRes