An Indian Journal FULL PAPER. Trade Science Inc. A stage-structured model of a single-species with density-dependent and birth pulses ABSTRACT

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1 [Typ x] [Typ x] [Typ x] ISSN : Volum 1 Issu 24 BioTchnology 214 An Indian Journal FULL PAPE BTAIJ, 1(24), 214 [ ] A sag-srucurd modl of a singl-spcis wih dnsiy-dpndn and birh pulss LI Wnh Collg of mahmaics and saisics, Norhas Prolum Univrsiy, Daqing , (CHINA) xiongdi163@163.com ABSTACT This papr considrs a sag-srucurd modl of a singl-spcis wih dnsiy-dpndn and birh pulss. Using h sroboscopic map, w ransform h mahmaical modl ino discr dynamical sysm. Furhrmor, h local sabiliy of h quilibrium is provd. KEYWODS Dnsiy-dpndn; Birh puls; Sag srucur; Local sabiliy. Trad Scinc Inc.

2 15198 A sag-srucurd modl of a singl-spcis wih dnsiy-dpndn and birh pulss BTAIJ, 1(24) 214 INTODUCTION In naur world, many populaions hav h diffrn characrisics in diffrn growh procss. For xampl, hy hav wo lif sags, immaur and maur. Thy hav diffrn cological characrisics and dynamic bhaviors in a diffrn sag, jus as carpillars will procd hrough h diffrn phass and bcom burflis finally. Many rsarchrs hav akn up h dp sudy of h sag-srucurd mahmaical modls [1,2]. Mos of hs modls suppos ha birh ra of adul populaion is coninuous,bu h rproducion of many animals has obvious sasonal and insananous characrisics. Tha is, hy brd onc vry onc in a whil. By us of impulsiv diffrnial quaion hory, w can dscrib h disconinuous dynamic phnomna [3,4,5]. This papr considrs a sag-srucurd modl of a singl-spcis wih dnsiy-dpndn and birh pulss. Using h sroboscopic map, w ransform h mahmaical modl ino discr dynamical sysm. Furhrmor, h local sabiliy of h boundary quilibrium and posiiv quilibrium is provd. Th sag-srucurd Modl of a Singl-spcis L h singl-spcis growh modl wih sag is as follows N & () = BNN ( ) N Whr posiiv d is h dah ra consan. B( N) N is birh ra. B( N ) saisfis h following assumpions. (1) BN ( ) > (2) B( N ) is a coninuous diffrniabl funcion and B ( N) <. (3) B( + ) < d < B() Whr N is a posiiv numbr. W can discovr B( N) = b N from biological liraur. W suppos ha h populaion has wo lif sags, immaur and maur. x() and y () rprsn h numbrs of immaur and maur rspcivly. Adul populaion has h abiliy o rproduc and h growh ra of i is qual o convrsion ra from lavr o adul minus dah ra. Juvnil populaion dos no hav h abiliy o rproduc and h growh ra of i is qual o rproduciv ra of adul populaion minus dah ra of juvnil populaion and convrsion ra from lavr o adul. And w suppos ha h dah ra of adul populaion is qual o h dah ra of juvnil populaion. Thn, w g h following modl. x () = b y() x() sx() (1) y () = sx() dy() sb Thorm1 Th sysm (1) has a boundary quilibrium A (,). Whn = 1 dd ( + >, w can g h uniqu and posiiv quilibrium B( x, y ).Whr d bs s bs x = ln, y = ln s + d d ( d + s ) s + d d ( d + s ) Thorm2 For h sysm (1), whn < 1, h quilibrium A is locally asympoically sabl; whn > 1, h quilibrium B is locally asympoically sabl. Proof L P( x) = b y() x() sx(), Qxy (, ) = sx ( ) dy ( ), W hav P = b y() d s, P = [ x( ) + y( )] [ x( ) + y( )] b y() + b, Q = s, Q = Thn h Jacobi marix is J = Q Q

3 BTAIJ, 1(24) 214 LI Wnh So h characrisic polynomial is λ 2 Q Q Q λe J = = λ ( + ) λ+ Q Q λ For h quilibrium A, h characrisic polynomial is 2 λ + (2 d + λ+ ( d + d sb=. Whn < 1, i has wo ngaiv ignvalus, so h quilibrium A is locally asympoically sabl. Whn > 1, h quilibrium A is unsabl. For h quilibrium B, h characrisic polynomial is 2 λ + ( k + 2 d + λ+ k( d + =. dd ( + Whr k = y. Whn > 1, i has wo ngaiv ignvalus, so h quilibrium B is s locally asympoically sabl. Th modl of a singl-spcis wih birh pulss W suppos adul populaion can no brd a any im, bu in h form of puls propagaion. Tha is,adul populaion only brd in im of mm, + 1, m+ 2, L. W can sablish h following impulsiv diffrnial quaions. x () =x() sx(), m< < m+ 1 y () = sx() dy(), (2) + x( m ) = x( m ) + b y( m ) Th manings of s and d ar h sam as in h sysm (1). L N () = x () + y () (3) W solv h firs quaion of (2) and hav + ( d+ ( m) x () = xm ( ), m< < m+ 1 (4) Adding h firs quaion o h scond quaion of h quaions (2),w ingra h rsul and obain + ( m) N () = Nm ( ), m< < m+ 1 (5) (5) minus (4) lavs h h following formula ( m) + + s( m) y () = [ ym ( ) + xm ( )(1 )], m< < m+ 1 (6) In im of mm, + 1, m+ 2, L,w can g h diffrnc quaion as follows d ( d+ ( x( m ) + y( m )) d + s + x(( m+ 1) ) = x( m ) + b [ y( m ) + (1 ) x( m )] (7) + s + d + y(( m+ 1) ) = (1 ) x( m ) + y( m ) Thorm3 L diffrnc quaions hav h boundary quilibrium A (,).Whn x s b (1 ) = > 1,hy hav h posiiv quilibrium ( d+ (1 )(1 ) Whr d ln B( x, y ). s s 1 (1 ) d 1 =, y = ln = ln 1 ( d+ ( d+ 1 1 Lmma1 (Jury inqualiis ) Suppos ha h linar sysm of diffrnc quaions is X m+ 1 = WX m,hn h sysm is sabl if h spcral radius of W is lss han 1.Tha is, W Saisfy h following condiions. 1 rw + dw > 1 rw + dw > (9) 1 dw > (8)

4 152 A sag-srucurd modl of a singl-spcis wih dnsiy-dpndn and birh pulss BTAIJ, 1(24) 214 Thorm4 For h sysm (2), whn < 1, h quilibrium A is locally asympoically sabl; whn > 1, h quilibrium B is locally asympoically sabl. Proof For h quilibrium A (,), h linar approximaion marix is s + b (1 ) b W = s (1 ) By calculaing, w g ( d+ s 1 rw + d W = (1 )(1 ) b (1 ) s 1+ rw + d W = (1+ )(1 + ) + b (1 ) > (2d 1 dw = 1 + > So whn < 1, h quilibrium A is locally asympoically sabl. For h quilibrium B( x, y ), h linar approximaion marix is A B W = s (1 ) Whr ( d+ 1 (1 s A b y ) 1 = +, B = b (1 y ) ( d+ 1 (1 s ) rw = A + = + b y + s ( d+ 1 s d W = A B (1 ) = [ b y ] Whn > 1,w subsiu ino Jury inqualiis and can rciv b ( d+ 1 rw + d W = y (1 ) > rw + d W = (1+ )(1 + ) (1 )(1 )(1 ) > 1 (2d+ ( d+ 1 1 dw = 1 + b y > So whn > 1, h posiiv quilibrium B is asympoically sabl. CONCLUSIONS This papr considrs a sag-srucurd modl of a singl-spcis wih dnsiy-dpndn and birh pulss.firs,w sudy a sag-srucurd mahmaical modl of a singl-spcis and obain h quilibrium(thorm 1) and h local sabiliy(thorm 2).Scond,w ransform h mahmaical modl ino discr dynamical sysm and acquir h quilibrium(thorm 3) by using h sroboscopic map.furhrmor,h local sabiliy of h boundary quilibrium and posiiv quilibrium is provd. ACKNOWLEDGEMENTS I would lik o hank h rfrs and h dior for hir valuabl suggsions. EFEENCES [1] Aillo W G, Frdman H I. A Tim Dlay of Singl-spcis Growh wih Sag Srucur. Mah Biosci, pg.11,pg (199). [2] Aillo W G, Frdman H I, Wu J. Analysis of a Modl prsning Sag Srucurd Populaion Growh wih Sa- Dpndn Tim Dlay. SIAM Appl. Mah, 52, pg (199).

5 BTAIJ, 1(24) 214 LI Wnh 1521 [3] Zhang S W, CHEN L S. Sag-Srucurd Modl of a Singl-spcis wih Dnsiy-Dpndn Birh Pulss. Sysm scinc and mah scinc, 26,6, pg (26). [4] YU S M. Singl Populaion Birh Puls Sag-Srucur Modl. Mahmaics in Pracic and Thory,36,4,pg (26). [5] LI Q Y, ZHANG F Q, WANG W J. Singl-spcis modl wih impulsiv birh and conracpion conrol. Journal of Shandong Univrsiy. 49,6, pg.85-9 (214).

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