Contracting Quadratic Operators of Bisexual population

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1 App Math If Sc 9 No ( Apped Mathematcs & Iformato Sceces A Iteratoa Joura Cotractg Quadratc Operators of Bsexua popuato Nasr N Gahodjaev 1 ad Uygu U Jamov 12 1 Departmet of Computatoa ad Theoretca Sceces Facuty of SceceIIUM Kuata Maaysa 2 Isttute of Mathematcs Natoa Uversty of Uzbesta Tashet Uzbesta Receved: 15 Feb 2015 Revsed: 17 May 2015 Accepted: 18 May 2015 Pubshed oe: 1 Sep 2015 Abstract: I ths paper we fd a suffcet codto uder whch the operator of bsexua popuato s cotracto ad show that ths codto s ot ecessary Keywords: Quadratc stochastc operator fxed pottrajectory cotractg operators 1 Itroducto The acto of gees s mafested statstcay suffcety arge commutes of matchg dvduas (beogg to the same speces These commutes are caed popuatos [2] The popuato exsts ot oy space but aso tme e t has ts ow fe cyce The bass for ths pheomeo s reproducto by matg Matg a popuato ca be free or subject to certa restrctos The whoe popuato space ad tme comprses dscrete geeratos F 0 F 1 The geerato F 1 s the set of dvduas whose parets beog to the F geerato A state of a popuato s a dstrbuto of probabtes of the dfferet types of orgasms every geerato Type partto s caed dfferetato The smpest exampe s sex dfferetato I bsexua popuato ay d of dfferetato must agree wth the sex dfferetato e a the orgasms of oe type must beog to the same sex Thus t s possbe to spea of mae ad femae types The evouto (or dyamcs of a popuato comprses a determed chage of state the ext geeratos as a resut of reproductos ad seecto Ths evouto of a popuato ca be studed by a dyamca system (teratos of a quadratc stochastc operator The hstory of the quadratc stochastc operators ca be traced bac to the wor of S Bershte [1] For more tha 80 years ths theory has bee deveoped ad may papers were pubshed (see [1]-[7][10]-[17] Severa probems of physca ad boogca systems ead to ecessty of study the asymptotc behavor of the trajectores of quadratc stochastc operators Let E = {12m} By the (m 1 smpex we mea the set S m 1 ={x=(x 1 x m R m : x 0 m x = 1} Each eemet x S m 1 s a probabty measure o E ad so t may be ooed upo as the state of a boogca (physca ad so o system of m eemets A quadratc stochastc operator V : S m 1 S m 1 has the form V : x m = p j x x j (m (1 where p j coeffcet of heredty ad p j = p j 0 m p j = 1 ( j = 1m For a gve x (0 S m 1 the trajectory {x ( } =012 of x (0 uder the acto of QSO (1 s defed by x (1 = V(x ( where =012 Oe of the ma probems mathematca boogy s to study the asymptotc behavor of the trajectores There are may papers devoted to study of the evouto of the free popuato e to study of dyamca system geerated by quadratc stochastc operator (1 see eg Correspodg author e-ma: jamovu@yadexru Natura Sceces Pubshg Cor

2 2646 N N Gahodjaev U U Jamov: Cotractg Quadratc Operators of [3]-[16] I [15] a survey of theory quadratc stochastc operators s gve I ths paper we fd a codto uder whch the evoutoary operators of bsexua popuato s cotracto 2 Deftos I ths secto foowg [2] we descrbe the evouto operator of a bsexua popuato Assumg that the popuato s bsexua we suppose that the set of femaes ca be parttoed to ftey may dfferet types dexed by{1 2 } ad smary that the mae types are dexed by {12} The umber s caed the dmeso of the popuato The popuato s descrbed by ts state vector (xy S 1 S 1 the product of two ut smpexes R ad R respectvey Vectors x ad y are the probabty dstrbutos of the femaes ad maes over the possbe types: x 0 x = 1; y j 0 y j = 1 Deote S = S 1 S 1 We ca the partto to types heredtary f for each possbe state z = (xy S descrbg the curret geerato the state z = (x y S s uquey defed descrbg the ext geerato Ths meas that the assocato z z defed a map V : S S caed the evouto operator For ay pot z (0 S the sequece z (t = V(z (t 1 t = 12 s caed the trajectory of z (0 Let p ( j ad m j be hertace coeffcets defed as the probabty that a femae offsprg s type ad respectvey that a mae offsprg s type whe the pareta par s j( = 12; ad j = 12 We have p ( j 0 p ( j = 1 m j 0 j = 1 Let z = (x y be the state of the offsprg popuato at the brth stage Ths s obtaed from hertace coeffcets as x = p ( j x y j (1 y = j x y j (1 We see from (2 that for a bsexua popuato the evouto operator s a quadratc mappg of S to tsef But for free popuato the operator s quadratc mappg of the smpex to tsef gve by (1 I [89] a agebra of the bsexua popuato s defed as the foowg: Cosder {e 1 e } the caoca bass o R ad (2 dvde the bass as e ( = e ad e (m = e = 1 Itroduce or a mutpcato defed by e ( e (m j = e (m j e ( = 2( 1 e ( e ( = 0 ; p ( j e( j e(m ; e (m j e (m = 0 j = 1; (3 Thus the coeffcets of bsexua hertace s the structure costats of a agebra e a bear mappg of R R to R The geera formua for the mutpcato s the exteso of (3 by bearty e for zt R z=(xy= t=(uv= usg (3 we obta ( zt= 1 2 ( 1 2 p ( x e ( u e ( y j e (m j v j e (m j j (x v j u y j j (x v j u y j e ( e (m From (4 ad usg (2 the partcuar case that z=t e x=u ad y=v we obta zz=z 2 = ( ( p ( j x y j e ( j x y j e (m = W(z for ay z S Ths agebrac terpretato s very usefu For exampe a bsexua popuato state z = (xy s a equbrum (fxed pot precsey whe z s a dempotet eemet of the set S e z=z 2 The agebra B = B W geerated by the evouto operator W (see (2 s caed the evouto agebra of the bsexua popuato I [8] t was show that f z s a fxed pot the z R 0 R 1 where R η ={z=(xy : x = y j = η} η = 01 For smpex S=S 1 S 1 by taget space we get R 0 ={z=(xy : x = y j = 0} (4 Natura Sceces Pubshg Cor

3 App Math If Sc 9 No (2015 / wwwaturaspubshgcom/jourasasp Cotractg operators I operator theory a bouded operator X Y betwee ormed vector spaces X ad Y s sad to be a cotracto f ts operator orm W 1 A extrema exampe of a quadratc cotracto s the costat operator I ths case the coeffcets p ( j m j do ot deped o ad j Ths suggests that for a suffcety sma scatterg of coeffcet for every fxed the quadratc operator w be a cotracto Ths remar ca be expressed as a precse theorem The Lpschtz costat of a operator R R s L(W=sup z t W(z W(t z t where s some orm R If ths orm ca be chose so that L(W < 1 the W w be a strct cotracto ths orm wth the cosequeces: uque fxed pot covergece of a trajectores to ths pot expoeta rate of covergece Uess otherwse specfed we w use the 1 orm the bass e ( = e = 1 ad e (m = e = 1 defed as z = x y j for z=(xy= x e ( y j e (m j Lemma 31[2] Let be a covex dmesoa compact R F : be a smooth map The (for ay orm L(F max z d zf Lemma 32[2] Let a matrx A=(a j satsfes The a 1 = A R 0 = 1 2 max a 2 == j 1 j 2 a a j1 a j2 where A R 0 s restrcto operator A or 0 For each z B we have ear operator M z : B B defed by M z (t=zt Theorema 33 The foowg equaty hods for the Lpschtz s costat L(W 1 2 j( ( 1 j 2 j 1 j m 2 j j 1 j 2 ( ( j 1 j 2 j 1 m j 2 Proof For the operator W S the dervatve s d z W = 1 2 p ( 1 j1 y j p ( 1 j y j 1 j1 y j p ( j1 y j 1 j y j d z W = 2M z = 2 where M ( = M e ( p ( j y j j1 y j j y j p ( 11 x x M ( 2 ad p ( 1 x 11 x p ( 1 x M (m 1 x p ( x 1 x y M (m = M e (m x are mutpcato maps wth matrxes (p ( j respectvey( j j By Lemma 31 we have L(W=2max M z 2max z S By Lemma 32 M ( 2max the ad M (m M ( = j( ( 1 j 2 j 1 j m 2 j M (m = 1 2 j 1 j 2( ( j 1 j 2 j 1 m j 2 Coroary 34 A evoutoary operator (2 s a strct cotracto f 1 2 j( ( 1 j 2 j 1 j m 2 j j 1 j 2 ( ( j 1 j 2 j 1 m j 2 <1 (5 For evoutoary operators wth postve coeffcets there s a mutpcatve estmate of the dstace from the evoutoary operator to the costat oe Let µ f µ f p ( (W= max 1 2 j 1 j p ( 2 j ad et ζ(w equa to LHS of (5 Lemma 35 µ m µ m (W= max j 1 j 2 ζ(w 4 µ f 1 µ f 1 4 µm 1 µ m 1 j 1 j 2 Natura Sceces Pubshg Cor

4 2648 N N Gahodjaev U U Jamov: Cotractg Quadratc Operators of Proof If αβ > 0 ad µ = max( α β β α the obvousy Hece p ( 1 j ad respectvey α β = µ 1 (α β µ 1 2 j µ f 1 µ f 1 ( 1 j 2 j 1 j m 2 j µm 1 µ m 1 (m 1 j m 2 j p ( j 1 j 2 µ f 1 µ f 1 ( j 1 j 2 j 1 m j 2 µm 1 µ m 1 (m j 1 m j 2 It remas to sum these equates over ad respectvey over eepg md that p ( 1 j = p ( 2 j = j 1 = j 2 = 1 Coroary 36 L(W 4 µ f 1 µ f 1 4 µm 1 µ m 1 Coroary 37 If 7µ f µ m (µ f µ m < 9 the the evoutoary operator (2 s a strct cotracto Coroary 38 Let µ = max(µ f µ m The L(W 8 µ 1 µ 1 ad f µ < 7 9 the the evoutoary operator (2 s a strct cotracto Remar 39 For free ad bsexua popuatos the codtos of cotractty are essetay dfferet Let us gve severa exampes ad chec the codto of Coroary 34 Exampe 310 Cosder the operator x 1 = 3 7 x 1y x 1y x 2y x 2y 2 x 2 = 4 7 x 1y x 1y x 2y x 2y 2 y 1 = 4 7 x 1y x 1y x 2y x 2y 2 y 2 = 3 7 x 1y x 1y x 2y x 2y 2 (6 The coeffcets of the operator (6 are the foowg p ( 111 = 3 7 p ( 112 = = = = = = = = 4 7 m 121 = 1 2 m 211 = 1 2 m 221 = = 3 7 m 122 = 1 2 m 212 = 1 2 m 222 = 4 7 It s easy to chec that codto (5 fufed for (6 Ideed 1 2 j( ( 1 j 2 j 1 j m 2 j j 1 j 2 ( ( j 1 j 2 j 1 m j 2 = 4 7 Cosequety ths operator s a strct cotracto ad t has uque fxed pot ( Moreover ay trajectory of (6 coverges to the fxed pot The foowg exampe shows that the codto of Coroary 34 s ot fufed ad evoutoary operator has perodc trajectory Exampe 311 Cosder the operator x 1 = x 1y 1 x 2 = x 1y 2 x 2 y 1 = x (7 2y 2 y 2 = x 1 x 2 y 1 It easy to chec that operator (7 does ot satsfy the codto of Coroary 34 We rewrte the operator (7 the form x 1 = x 1y 1 y 1 =(1 x (8 1(1 y 1 Deote x = x ( 1 y = y ( 1 the from (8 we have x 1 = x y y 1 =(1 x (1 y Sce 0 x y x from the frst equato of (9 t foows that m x = x = 0 Ideed for z 0 t(s 1 S 1 ={z S 1 S 1 : x > 0y > 02} we get from (9 x 2 x 1 =(1 x ( 1 x 1 x (9 Natura Sceces Pubshg Cor

5 App Math If Sc 9 No (2015 / wwwaturaspubshgcom/jourasasp 2649 x 2 x = x 1 (1 x (x x 1 m x 2x = m x 1 (1 x (x x 1 (x 2 = 0 x = 0 Now cosder the operator x = xy x 2 y xy 2 x 2 y 2 W 2 : y = xy xy x 2 y xy 2 x 2 y 2 Ceary the operator W 2 has fxed pots (0y 0 y 1 The pot(0y s a sadde pot It s easy to chec that the set {(xy S 1 S 1 : x 1 = 0} s a varat subset for (7 Ay pot of the varat subset s perodc pot wth perod two for operator (7 So trajectory of the operator wth a ta pot from varat subset does ot coverge Thus operator (7 has a trajectory whch does o-coverge to the fxed pot( The foowg exampe shows that codto of Coroary 34 s suffcet but s ot ecessary Exampe 312 Cosder the operator wth coeffcets of hertace p ( 111 = = = 1 2 p ( 112 = = = = = = 0 m 121 = 1 2 m 211 = 0 m 221 = = 1 m 122 = 1 2 m 212 = 1 m 222 = 1 2 e the evouto operator has the form x 1 = 2 1x 2 x 2 = x 1 2 1x 2 y 1 = 1 2 y 2 y 2 = y 1 2 1y 2 (10 It easy to chec that operator (10 does ot satsfy the codto of Coroary j( ( 1 j 2 j 1 j m 2 j j 1 j 2 ( ( j 1 j 2 j 1 m j 2 =2>1 But ay trajectory of (10 coverges to ( Ideed from (10 we have x (1 1 = 1 2 (1 x( 1 We cosder foowg oe dmesoa dyamca system f(x= 1 2 (1 x It has uque fxed pot x = 1 3 ad decreasg o [01] Easy to chec that f (x= 1 2 ad f ( 3 1 = 2 1 < 1 therefore the fxed pot x= 1 3 s attractg We cam that ay trajectory of f(x coverges to the fxed pot x= 3 1 Ideed we have ad f (x= ( 1 m f 2 (x= m ( 1 2 ( 1 x x 2 2 = 1 3 ( 1 m f (x= m x 2 21 = 1 3 So for ay ta pot trajectory of (10 coverges to ( Acowedgemet The fa part of ths wor was doe at the Iteratoa Isamc Uversty of Maaysa (IIUM ad the secod author woud e to tha the IIUM for provdg faca support ad a factes The secod author aso thas Dr Masoor Saburov for usefu dscussos Refereces [1] SN Berste Uch Zaps NI Kaf Ur Otd Mat (1924 [2] YuI Lyubch Mathematca structures popuato geetcs Bomathematcs 22 Sprger-Verag Ber 1992 [3] RN Gahodzhaev Sbor: Mathematcs (1993 [4] RN Gahodzhaev Mathematca Notes (1994 [5] RN Gahodzhaev DB Eshmamatova Vadavaz Mat Zh (2006 [6] NN Gahodzhaev Doady Mathematcs (2001 [7] UA Rozov NB Shamsddov Stochastc Aayss ad Appcatos (2009 [8] M Ladra UA Rozov Joura of Agebra (2013 [9] M Ladra BA Omrov UA Rozov Lobachevs Joura of Mathematcs (2014 [10] UA Rozov UU Zhamov Mathematca Notes (2008 [11] UU Zhamov UA Rozov Sbor: Mathematcs (2009 Natura Sceces Pubshg Cor

6 2650 N N Gahodjaev U U Jamov: Cotractg Quadratc Operators of [12] UA Rozov UU Zhamov Uraa Mathematca Joura (2011 [13] UA Rozov A Zada Iteratoa Joura of Bomathematcs (2010 [14] Farruh Shahd O fte-dmesoa dsspatve quadratc stochastc operators Advaces Dfferece Equatos 272 (2013 [15] RN Gahodzhaev FM Muhamedov UA Rozov Ifte Dmesoa Aayss Quatum Probabty ad Reated Topcs (2011 [16] UU Zhamov RT Muhddov Uzbe Math Zh (2010 [17] UU Zhamov Do Acad Nau RUz (2010 Nasr Gahodjaev s Professor at Iteratoa Isamc Uversty Maaysa He has graduated the Tashet State Uversty (1971 He got PhD (1975 degree from the Isttute of Mathematcs Tashet ad Rehabtato Degree Doctor of Sceces Physcs ad Mathematcs (1991 from the Isttute for Low Temperature Physcs ad Egeerg Academy of Sceces of Urae Kharov He s ow for hs pubcatos attce modes of statstca mechacs ad oear dyamca systems Uygu Jamov was bor 1976 Buhara Uzbesta He graduated from Tashet State Uversty 1998 He got PhD degree Physcs ad Mathematcs Natoa Uversty of Uzbesta 2012 Hs research terests cude dyamca systems ad ergodc theory fuctoa aayss He has mportat resuts o o-voterra quadratc stochastc operators He has about 25 papers pubshed the eadg jouras Presety he s a seor scetfc feow the Isttute of Mathematcs at Natoa Uversty of UzbestaTashet Uzbesta Natura Sceces Pubshg Cor