DYNAMICS OF A POPULATION SUBJECT TO IMPULSE TYPE RANDOM LOSS

Transcription

1 PROCEEDINGS OF THE LATVIAN ACADEMY OF SCIENCES. Scion B Vol. 7 (7 No. 4 (79 pp DOI:.55/prol-7-5 DYNAMICS OF A POPULATION SUBJECT TO IMPULSE TYPE RANDOM LOSS Jvgòij Crov nd Kârli Ðduri # Dprmn of Tory of Probbiliy nd Mmicl Siic Rig Tcnicl Univriy Kïíu Sr. Rig LV-658 LATVIA # Corrponding uor rli.duri@ru.lv Communicd by Jurij Mrurjv W prn quliiv populion grow nlyi pproc uing Prl logiic populion grow diffrnil quion for populion wi inniy of bir ionry iz K o innc of f mll rndom populion iz xrcion proporionl o n rndom im momn n wr i mll poiiv prmr. Auming inrvl n r n indpndn idniclly xponnilly diribud rndom vribl wi prmr / nd n r indpndn idniclly diribud poiiv rndom vribl wi mn nd vrinc b w nly populion dynmic nd populion ympoic bviour. W propo probbiliic limi orm bd ocic pproximion lgorim for quliiv nlyi of bov modl on ny fini im inrvl. A fir w driv linr diffrnil quion for mmicl xpcion { x(} of populion grow nd ocic Io diffrnil quion for normlid dviion ({x(} x( /. Auming diffrnc =c i ufficinly mll w driv ocic diffrnil quion for cld populion grow in cclrd im Kx( / nd prov undr condiion c < ( +b populion dippr wi probbiliy on orwi diribuion of cld populion iz wi incring im nd o Gmm-diribuion (q wi p = c/( +b nd cl = ( +b /c. Ky word: populion dynmic ocic quion diffuion pproximion. INTRODUCTION: THE MODEL AND ASSUMPTIONS T logiic grow modl i mo populr on in populion dynmic nd conidr bir nd d r ypicl o pci wll ffc of diffrn mcnim of conumpion of rourc on dynmic of biologicl ym. For quliiv nlyi of bviour of n coym ordinry diffrnil quion (Oo nd Dy 995: dx d x ( K x x ( i commonly ud wr prmr dcrib grow r i.. friliy nd prmr drmin inniy of xrcion of pcimn from coym ( n xmpl inniy of fiing in c of fi populion in firi wic w u r nd furr n illuriv xmpl cnrio. Prmr K corrpond o ionry populion iz in c of n bnc of fiing i.. ionry oluion of quion ( if =.A K dcrib r of rducion of populion iz i cofficin my b conidrd d r of populion. L u no ionry oluion of quion ( x K( diffr from K nd oluion of ( nd xponnilly o x only if >. If i inquliy i no ru oluion of ( nd xponnilly o zro. Ti mn populion di ou undr uc fiing rgim. If fiing r i qul o bir r oluion of ( lo nd o zro lbi no wi n xponnil vlociy. Ti mn in ordr for fiing o b uinbl i i ncry for bir r o b grr n fiing r. I ould b nod i i no ju iz of populion drmin fiing rgim. Uing fini diffrnc mod w moun of fiing im ould b pproximly qul o x(. Howvr i my b c nir volum of populion x( my no b uibl for fiing im nd w cn only rly on vrg vlu of fiing x(. Somim i i poibl o u diffrn fiing rgy: wi for im priod wic gnrlly ping i rndom im dpndn vribl nd only n rmov ncry moun of populion. Trfor drminiic firy rgi do no llow u o prdic dynmic of populion iz nd/or o drmin fiing mod pcilly if. In i c conrucion nd xplorion of mmicl modl 98 Proc. Lvin Acd. Sci. Scion B Vol. 7 (7 No. 4. Ununicd Downlod D 3/6/8 7:3 PM

2 of coym by uing modrn mod of ocic nlyi i don for xmpl in Prjnu (98 nd Fldmn nd Rouggrdn (975 m o b mor produciv. Hr w lo u uc ocic pproc for quliiv nlyi of grow of populion wic in bnc of fiing i ubjc o logiic quion ( wi =. W um prmr dcrib prnc of inrpcific compiion i ufficinly mll nd w inroduc mll poiiv prmr > wic will b ud mll prmr for ocic pproximion of nlying dynmicl ym. Trfor w um : in bnc of fiing populion grow i dcribd by logiic quion dx d x ( K x ( wi prmr of bir r ; AVERAGING AND NORMALISED DEVIATIONS { x ( } dfind by quion ( (3 (4 cn b provd o ify Mrov propry (Dynin 965. For ny boundd coninuou funcion {( vx x nd ny condiionl xpcion of vx ( ( providd pr of rjcory { x ( } i nown dpnd only on { x ( : } { v( x( x( { v( x( / x( } Ti prmi (Dynin 965 o nly probbiliic propri of ny rjcory{ x ( } uing w infiniiml opror L( (Dynin 965 wic i dfind by following formul for ny ufficinly moo funcion v(x (Dynin 965: (((: L v x lim [{( vx( x ( x} vx (] (5 By dfiniion for mll >: only individul wo v rcd crin lvl of muriy r uibl for fiing nd fiing occur rndom im momn { N} nd im inrvl { N} r mll muully indpndn nd xponnilly diribud: {v( x( x( x } vx ( x( K x v ( x o( nd {v( x( vx ( x( x } (6 P( volum uibl for fiing x ( im momn lo r lo rndom ir rmovl from populion i innnou nd rfor im momn rjcory x (i diconinuou: (3 x( x( x( (4 wr r indpndn idniclly diribud poiiv rndom vribl wi xpcion = nd vrinc =. A furr on w ll limi if r w u ub-indx for nlying ocic proc by prviouly inroducd prmr. In bov formul x ( mn lf nd id limi. L u no umpion of xponnil diribuion of im bwn ny wo diconinuii of rjcori ugg probbiliy crcriic of dynmic ym ny fixd im momn i complly drmind by i prn vlu im nd do no dpnd on prviou of modl i.. gurn Mrov propry of rndom proc. Ti m i poibl o dvnc mod propod in Aniimov (995 nd Korolyu nd Limnio (5 of diffuion pproximion for wicing Mrov proc nd conruc ocic pproximion procdur for impul yp diffrnil quion wi f Mrov impul wicing ( (4. T ruling diffrnil quion for mn vlu of populion iz nd ocic diffrnil quion for dviion on mn prmi u o dcrib probbiliy crcriic of populion dynmic. {v( x x vx ( } o ( (7 Combining quion (6 nd (7 nd conidring formul for jump condiion (3 w my rwri quion for w infiniiml opror in following form: L( v( x: lim [ { vx ( ( x ( x} vx ( ] x( K x v( x (( v x x v( x O( E ( x K x v( x xv( x b v ( x o ( O ( [( K x] xv( x O( (8 A provd in Trov (993 if vlu i ufficinly lrg bviour of proc{ x ( } on ny fini im inrvl [T] my b pproximd wi oluion of following ordinry diffrnil quion: d x ( ( x ( K x ( d (9 o wic w will rfr vrg quion. Ti mn for ny x ( nd T> r xi uc numbr M for ny poiiv w my wri inquliy {up x ( x(} M ( T wr { x ( } i oluion of quion (9 wi iniil condiion x( x (. T quion (9 wo ionry oluion: x nd x K( /. T Mrov dynmicl ym (3 (4 lo rivil oluion. If n on cn prov ny oluion of quion (9 wi poiiv Proc. Lvin Acd. Sci. Scion B Vol. 7 (7 No. 4. Ununicd Downlod D 3/6/8 7:3 PM 99

3 iniil condiion nd o ionry poin x. Trfor i bn provd in Trov ( if i ufficinly mll poiiv numbr ny rjcory of Mrov dynmicl ym (3 (4 wi poiiv iniil condiion nd o zro wi probbiliy on. If n ionry poin x i unbl wr ionry poin x i xponnilly bl. Applying formul ( w cn confirm for mll poiiv nd ufficinly lrg populion dcribd by Mrov dynmicl ym (3 (4 will b locd in -nigbourood of poin x K(. T proximiy of populion iz o i ionry poin my b imd (Trov 993; o n ccurcy wiin O( by normlid dviion: y x ( x( (: ( Ti proc lo po Mrov propri nd for ufficinly moo funcion v(y nd infiniiml w cn pproxim: [ { vy ( ( } vx ( ] y ( y y( K x ( x ( v( y o( [ { vy ( ( vy ( ( } y y [ v( y ( y ( x( ( v ( y( x( ] o( Trfor w infiniiml opror L ( of Mrov proc { y ( } my b dcompod o n ccurcy wiin O( follow: L((: v y lim [ { vy ( ( y ( y} vx ( ] y( K x ( x ( v( y O( lim (v(y y x ( v( y [( K x( ] yv( y ( b v( y( x( O ( ( Ti mn normlid dviion ( my b pproximd by oluion of ocic diffrnil quion dy( [( K x ( ] y( d x ( ( b dw( (3 wi iniil condiion y (. If iniil populion iz i qul o x ( x n quion (3 mor impl form: dy( yd ( K ( b dw( (4 T diffrnc of ny wo oluion y ( y ( ( y ( ( ifi n ordinry diffrnil quion dy ( ( yd ( nd rfor my prnd in xponnil form y ( y( xp{(. Bid undr condiion w my conruc oluion of quion (4 in form of Io ocic ingrl: y ( K ( b xp ( dw ( (5 By dfiniion y ( i Guin rndom vribl wi zro mn nd i no dpndn on vrinc D( y K ( ( b /. Trfor undr condiion Mrov proc dfind by i ocic diffrnil quion po n rgodic propry du o ny oluion y( of quion (3 bing prnd in form y ( y ( {( y y ( } wr cond rm xponnilly nd o zro bu fir rm i no dpndn on diribuion of. Hnc for ufficinly mll vlu of if iniil populion iz i wiin dinc from poin x K( undr condiion w my b ur populion do no lv -nigbourood of poin x nd populion iz my b pproximd by Guin rndom vribl wi prmr: K K ( ( b THE DIFFUSION APPROXIMATION Hr w nly bviour of populion undr umpion diffrnc i proporionl o mll prmr i c c. Uing i umpion quion (9 i no pplicbl bcu lim L( v( x for ny x> nd ny ufficinly moo funcion v(x. Bid quilibrium poin x K( i proporionl o nd w my no ru o. Howvr for lrg populion iz K of ordr w v no infiniiml ionr nd nlyi of dynmicl ym ( (3 (4 i poibl. To bviour of rndom proc { x ( } undr bov umpion w will cl im nd populion iz: X (: ( K x ( /. Ting ino ccoun wicing im { N} for i proc r dfind by qulii = N w cn rwri quion ( (3 nd (4 for c N follow: : d d X c X X ( ( (( ( (6 P( X (7 ( X ( ( (8 To civ diffuion pproximion (Trov for Mrov proc{ X ( } w v o pply w infini- 3 Proc. Lvin Acd. Sci. Scion B Vol. 7 (7 No. 4. Ununicd Downlod D 3/6/8 7:3 PM

4 iml opror L( i dfind by i proc o ufficinly moo funcion v(x: L( [ vx ( ]: lim [ { ( ( X ( X} ( ] EvX V X ( c X( X v ( X E( vx ( ( X vx ( X cx ( c X v ( X Xv X b ( X v ( X o( nd o p o limi : b lim L( ( X X( cx v( X X v ( X Ti mn diffuion pproximion {( X } for proc X(: ( K x( / form of Io ocic diffrnil quion dx X( c X d ( b Xdw( (9 To minimi numbr of prmr in furr clculion w will opr wi proc z ( c X ( i- fying quion dz cz( z d zdw( ( wr ( b. In ordr o b bl o conclud populion will no dippr for ufficinly long im w v o nly ympoic biliy condiion for rivil oluion of quion (. Hnc poin i n borbing boundry for ny rjcory ifying i quion. Trfor pplying mod propod in Trov (993 w cn u funcion vz (z wr Lypunov funcion for globl ympoic biliy nlyi of rivil oluion. By pplying Io formul o proc vz (( w cn driv following quion: dv(( z v( z( { cz( ( z( d ( zdw( } z ( v(( z d z ({ cz(( z( dz( dw(} z ( ( z ( d c( v( z(( z( dv( z( dw( Applying mmicl xpcion procdur o quion bov nd ing ino ccoun by dfiniion inf z ( w cn wri quion d { vz ( (} d c( {( z( v( z( } c( { v( z( } nd rfor by Gronwll lmm w cn driv inquliy { z ( } { z ( xp{ c ( } for ny. Ifc ( b w cn co uc poiiv numbr c ( nd n (Trov for ny iniil populion iz populion will dcr ympoiclly. Now l c ( b. Prforming ubiuion z ( Y ( in quion ( w cn wri linr non-omognou ocic quion for Y (: dy( ( c Y( d Y( dw( cd ( ( ( T diffrnc of ny wo oluion Y ( Y ( Y ( of quion ( ifi ordinry ocic Io diffrnil quion dy ( ( c Y ( d Y ( dw( ( T oluion of i quion wi iniil condiion Y ( Y form of ocic xponn Y ( Y xp c w( w ( Y xp c (3 I i wll nown (Doob 953 P w lim ( Trfor undr umpion c diffrnc of ny wo oluion of quion ( will xponnilly nd o zro. Bid undr umpion bov r xi rndom proc c ( [ w( w( ] Y ( c d (4 ifi quion (. T diribuion funcion for i proc FY ( P ( Y ( Y (5 do no dpnd on bcu for ny c w w ( ( ( ( d c ( ( w( w( c ( u ( w( w( u d nd w ( wu ( m diribuion w( w(u. Trfor FY ( F ( Y. Any oluion Y ~ ( of quion ( my b prnd um Y ~ (} = Y ( + { ~ Y ( Y ( wr cond rm ifi ocic Io ordinry diffrnil quion nd rfor form of d Proc. Lvin Acd. Sci. Scion B Vol. 7 (7 No. 4. Ununicd Downlod D 3/6/8 7:3 PM 3

5 ocic xponn (. Undr umpion c i xponn nd o zro nd o infiniy. Ti mn quion ( dfin rgodic Mrov proc (Dynin 965 nd lim P(( Y Y/ Y( Y F ( Y (6 for ny iniil condiion Y ( = Y. Trfor if c n ing ino ccoun quliy z ( Y ( w my driv formul lim P( z ( z/ z( z lim P( Y ( z / Y( Y (7 F ( z : F( z (8 L u dno p(z dniy funcion corrponding o F(z. By dfiniion of ionry diribuion of Mrov proc dfind by quion ( (Dynin 965 w cn loo for dniy funcion p(x poiiv oluion of cond ordr diffrnil quion d z dz. Ti oluion n xpo- ifying condiion nnil form c pz ( c d pz ( { dz cz ( z p (} z (9 c pzdz ( c { c z} z ( dz (3 No our nlyzd Mrov proc i dfind by formul X ( c( z (. Trfor i proc lo po- rgodic propry wi limi diribuion F ( x lim P( c( z( x F( x / c X Diffrniing i funcion by z nd ing ino ccoun noion ( b w cn conclud undr condiion c( b populion iz nd o rndom vribl Gmm-diribuion wi p c nd cl b i.. wi dniy funcion ( b c ( b b px ( x c ( b c ( b b (3 No w umd diffrnc bwn populion incr.g. bir r nd xrcion.g. fiing inniy i ufficinly mll c nd ionry populion iz K i of ordr wr i mll poiiv prmr. Uing i umpion w cn m following concluion on ympoic bviour of dcribd by Mrov dynmicl ym ( (3 (4 populion grow x (: r xi uc poiiv numbr for ny ( nd ny iniil condiion x ( : if ( b n P lim x ( ; if c b lim P Kx ( x F( x wr F(x i Gmm-diribuion wi dniy (3. ( n REFERENCES Aniimov V. V. (995. Swicing proc: Avrging principl diffuion pproximion nd pplicion. Ac Appl. M. 4 ( Doob J. L. (953. Socic Proc. Jon Willy & Son Nw Yor. 654 pp. Dynin E. B. (965. Mrov Proc. Brlin Nw Jry pp. Fldmn M. W. Rouggrdn J. (975. A populion ionry diribuion nd cnc of xincion in ocic nvironmn wi rmr on ory of pci pcing. Tor. Popul. Biol. 9 ( Korolyu V. S. Limnio N. (5. Diffuion pproximion of voluionry ym wi quilibrium in ympoic pli p pc. Tor. Probb. M. Si Oo S. P. Dy T. (995. A Biologi Guid o Mmicl Modling in Ecology nd Evoluion. Princon Univriy Pr Nw Jry. Prjnu P. (98. Tim-dpndn oluion of Logiic Modl for Populion Grow in Rndom Environmn. J. Appl. Prob. 9 ( Trov Y. (J. Crov (993. Avrging in Dynmicl Sym wi Mrov Impul Prurbion. Brmn Univriy FRG 9 Rp. Nr. 8 4 pp. Trov Y. (J. Crov (. Aympoic mod for biliy nlyi Mrov impul dynmicl ym. Nonlinr Dyn. Sy. Tory ( 3 5. Rcivd Novmbr 6 Accpd in finl form April 7 POPULÂCIJAS DINAMIKA IEVÇROJOT TÂS ÎPATÒU NEJAUÐUS IMPULSVEIDA ZUDUMUS Izmnojo pr pmu Pçrl loìiio populâciju dinmi modli râ vi populâcij piugum vliîv nlîz pi nocîjumim njuðo li momno lîdzinoði nlil populâcij îpòu i biþi i izòm no â dzîvon. Ðâd populâcij îpòu rcij noi pimçrm zivju populâcijâ ur zivju nozvj rzulââ njuðo li momno no populâcij dzîvon i izòm njuð i indivîdu lîdz r o bûii imçjo populâcij dinmiu un â lîdzvr biliâi. Râ nlizç populâcij lilum impoi pidâvâ vrbûîbu orij robþorçmâ blî vliîvâ nlîz oiâ proimâcij lgorim pvïîgm glîgm li inrvâlm. Anlizç populâcij vidçjo lilumu proð linâr difrnciâlvinâdojum un Io oii difrnciâlvinâdojum pr normlizçâ novirz no vidçjâ. Ari prmru nocîjumi pi urim populâcij onvrìç uz îpòu iu uru pr gmm dlîjum r ðjâ râ prçíinâim prmrim vi prçjâ gdîjumâ gndrîz droði izmir. 3 Proc. Lvin Acd. Sci. Scion B Vol. 7 (7 No. 4. Ununicd Downlod D 3/6/8 7:3 PM