in recent years , with the wide application of computers , difference system has become one of the important theoretical bases for computer , information system , engineering control , ecological balance , and so forth . as typical nonlinear difference equations the behavior of solutions of the system for rational difference equation has received extensive attention . in , ozban has investigated the periodicity of solutions of the system of difference equations : 
 ( 1)xn+1=1ynk ,  yn+1=ynxnmynmk ,  n=0,1,  . in , kurbanl et al . studied the behavior of the positive solutions of the system of difference equations : 
 ( 2)xn+1=xn1ynxn1 + 1 ,  yn+1=yn1xnyn1 + 1 ,  n=0,1,  . the periodicity of the positive solutions of the rational difference system 
 ( 3)xn+1=1yn ,  yn+1=ynxn1yn1 ,  n=0,1 ,  , 
 has been studied by inar in . in , ozban studied the behavior of the positive solutions of the system of difference equations 
 ( 4)xn = ayn3 ,  yn = byn3xnqynq ,  n=0,1,  . 
 for similar research on difference systems , we refer the reader to [ 5 , 6 ] and the references therein . in this paper , we investigate the behavior of positive solutions for the system of rational difference equations 
 ( 5)xn+1=axnyn2 ,  yn+1=bynxn1yn1 ,  n=0,1 ,  , 
 where a , b  [ 0 , +  ) and x1 , x0 , y1 , and y0  [ 0 , +  ) . before stating our main results , we state some definitions used in this paper . 
 definition 1 . a pair of sequences of positive real numbers { ( xn , yn)}n=0 that satisfies ( 5 ) a pair of sequences of positive real numbers { ( xn , yn)}n=0 that satisfies ( 5 ) is called a positive solution of ( 5 ) . 
 definition 2 . a solution { ( xn , yn)}n=0 of ( 5 ) is periodic , if there exists a positive integer t such that ( xn , yn ) = ( xn+t , yn+t ) , n = 0,1 ,  , and t is called a period . 
  a solution { ( xn , yn)}n=0 of ( 5 ) is periodic , if there exists a positive integer t such that ( xn , yn ) = ( xn+t , yn+t ) , first , we study the periodic nature of positive solution of system ( 5 ) . let x1 , x0 , y1 , and y0 be positive real numbers and let { ( xn , yn)}n=0 be a solution of system ( 5 ) . then for a = b , a , b  [ 0 , +  ) , all solutions of system ( 5 ) are periodic with period 3 . let x1 , x0 , y1 , and y0 be positive real numbers and let { ( xn , yn)}n=0 be a solution of system ( 5 ) . then for a = b , a , b  [ 0 , +  ) , all solutions of system ( 5 ) are periodic with period 3 . prooffor a , b > 0 , it can be seen easily that all solutions of ( 5 ) are positive . thus , by ( 5 ) we have the following equality : 
 ( 6)xn+1=axnyn2 ,  yn+1=bynxn1yn1 . repeating application of ( 5 ) yields 
 ( 7 )  xn+2=axn+1yn+12=xnxn12yn12b2 ,  yn+2=byn+1xnyn = b2xnxn1yn1 . 
 similarly , 
 ( 8)  xn+3=axn+2yn+22=ab2xn = xn ,  yn+3=byn+2xn+1yn+1=b2ayn = yn . b > 0 , it can be seen easily that all solutions of ( 5 ) are positive . thus , by ( 5 ) we have the following equality : 
 ( 6)xn+1=axnyn2 ,  yn+1=bynxn1yn1 . 
 repeating application of ( 5 ) yields 
 ( 7 )  xn+2=axn+1yn+12=xnxn12yn12b2 ,  yn+2=byn+1xnyn = b2xnxn1yn1 . 
 similarly , 
 ( 8)  xn+3=axn+2yn+22=ab2xn = xn ,  yn+3=byn+2xn+1yn+1=b2ayn = yn . let { ( xn , yn)}n=0 be a solution of system ( 5 ) with x1 = c , x0 = a , y1 = d , and y0 = b. then for a = b , where a and b are positive constants , all solutions of system ( 5 ) are 
 ( 9)x3n+1=aab2 ,  y3n+1=bbcd , x3n+2=ac2d2b2 ,  y3n+2=b2acd , x3n+3=a ,  y3n+3=b , n=0,1,  . let { ( xn , yn)}n=0 be a solution of system ( 5 ) with x1 = c , x0 = a , y1 = d , and y0 = b. then for a = b , where a and b are positive constants , all solutions of system ( 5 ) are 
 ( 9)x3n+1=aab2 ,  y3n+1=bbcd , x3n+2=ac2d2b2 ,  y3n+2=b2acd , x3n+3=a ,  y3n+3=b , n=0,1,  . prooffor a , b > 0 , it can be seen easily that all solutions of ( 5 ) are positive.for n = 0 we have 
 ( 10 )  x1=ax0y02=aab2 ,  y1=by0x1y1=bbcd ,  x2=ax1y12=ac2d2b2 ,  y2=by1x0y0=b2acd ,  x3=ax2y22=a ,  y3=by2x1y1=b . now suppose that n  z and that our assumption holds for n  1 . one will show that the result holds for n. from system ( 5 ) , we obtain 
 ( 11 )  x3n+1=ax3n1 + 3y3n1 + 32=aab2 ,  y3n+1=by3n1 + 3x3n1 + 2y3n1 + 2=bbcd . 
 then , 
 ( 12 )  x3n+2=ax3n+1y3n+12=ac2d2b2 ,  y3n+2=by3n+1x3n1 + 3y3n1 + 3=b2acd . 
 in particular , from theorem 3 , we get 
 ( 13 )  x3n+3=x0 + 3n+1=x0=a ,  y3n+3=y0 + 3n+1=y0=b . 
 therefore , the proof is complete . 
  for a , b > 0 , it can be seen easily that all solutions of ( 5 ) are positive . for n = 0 we have 
 ( 10 )  x1=ax0y02=aab2 ,  y1=by0x1y1=bbcd ,  x2=ax1y12=ac2d2b2 ,  y2=by1x0y0=b2acd ,  x3=ax2y22=a ,  y3=by2x1y1=b . now suppose that n  z and that our assumption holds for n  1 . one will show that the result holds for n. from system ( 5 ) , we obtain 
 ( 11 )  x3n+1=ax3n1 + 3y3n1 + 32=aab2 ,  y3n+1=by3n1 + 3x3n1 + 2y3n1 + 2=bbcd . then , 
 ( 12 )  x3n+2=ax3n+1y3n+12=ac2d2b2 ,  y3n+2=by3n+1x3n1 + 3y3n1 + 3=b2acd . 
 in particular , from theorem 3 , we get 
 ( 13 )  x3n+3=x0 + 3n+1=x0=a ,  y3n+3=y0 + 3n+1=y0=b . 
 therefore , the proof is complete . set a = 2.25 and b = 1.5 and x1 = 3 , x0 = 1 , y1 = 0.6 , and y0 = 1.6 . set a = 2.25 and b = 1.5 and x1 = 3 , x0 = 1 , y1 = 0.6 , and y0 = 1.6 . next , we consider the case that a  b , where a and b are positive constants . 
 theorem 6 let { ( xn , yn)}n=0 be an arbitrary positive solution of ( 5).(i)if a < b , then , for each integer n  0 , the subsequence { xn+3l}l=0  0 , and the subsequence { yn+3l}l=0  .(ii)if a > b , then , for each integer n  0 , the subsequence { xn+3l}l=0   , and the subsequence { yn+3l}l=0  0.(iii)xn+3yn+3 = xnyn , n = 0,1,  . let { ( xn , yn)}n=0 be an arbitrary positive solution of ( 5).(i)if a < b , then , for each integer n  0 , the subsequence { xn+3l}l=0  0 , and the subsequence { yn+3l}l=0  .(ii)if a > b , then , for each integer n  0 , the subsequence { xn+3l}l=0   , and the subsequence { yn+3l}l=0  0.(iii)xn+3yn+3 = xnyn , n = 0,1,  . if a < b , then , for each integer n  0 , the subsequence { xn+3l}l=0  0 , and the subsequence { yn+3l}l=0  . if a > b , then , for each integer n  0 , the subsequence { xn+3l}l=0   , and the subsequence { yn+3l}l=0  0 . proof(i ) for every fixed n , we will show that 
 ( 14)xn+3l = ab2lxn ,  l=0,1,2,  . assume that ( 14 ) holds for l  1 ; that is , 
 ( 15)xn+3l3=ab2l1xn ,  l=0,1,2,  . 
 for l , we have the following : 
 ( 16)xn+3l = xn+3 + 3l3=ab2l1xn+3=ab2l1ab2xn = ab2lxn . similarly , it can be obtained by induction that 
 ( 17)yn+3l = b2alyn ,  l=0,1,2,  . if a < b , we can get by ( 14 ) and ( 17 ) that 
 ( 18)xn+3ll=00 ,  yn+3ll=0. 
 ( ii ) if a > b , by ( 14 ) and ( 17 ) , 
 ( 19)xn+3ll=0 ,  yn+3ll=00 . ( iii ) from the proof of theorem 3 , we have 
 ( 20 )  xn+3=axn+2yn+22=ab2xn ,  yn+3=byn+2xn+1yn+1=b2ayn . 
 multiplying both sides , respectively , yields xn+3yn+3 = xnyn , n = 0,1,  . ( i ) for every fixed n , we will show that 
 ( 14)xn+3l = ab2lxn ,  l=0,1,2,  . assume that ( 14 ) holds for l  1 ; that is , 
 ( 15)xn+3l3=ab2l1xn ,  l=0,1,2,  . 
 for l , we have the following : 
 ( 16)xn+3l = xn+3 + 3l3=ab2l1xn+3=ab2l1ab2xn = ab2lxn . similarly , it can be obtained by induction that 
 ( 17)yn+3l = b2alyn ,  l=0,1,2,  . if a < b , we can get by ( 14 ) and ( 17 ) that 
 ( 18)xn+3ll=00 ,  yn+3ll=0. 
 ( ii ) if a > b , by ( 14 ) and ( 17 ) , 
 ( 19)xn+3ll=0 ,  yn+3ll=00 . ( iii ) from the proof of theorem 3 , we have 
 ( 20 )  xn+3=axn+2yn+22=ab2xn ,  yn+3=byn+2xn+1yn+1=b2ayn . 
 multiplying both sides , respectively , yields xn+3yn+3 = xnyn , n = 0,1,  . let x1 = 8 , x0 = 1 , y1 = 3 , and y0 = 7.(i)for a = 1 , b = 2 , and n = 2 , figure 2 and table 1 describe the behavior of the sequence { xn+3l}l=0 and { yn+3l}l=0 ( shown by the black spot at the top of every peak).(ii)for a = 2 , b = 1 , and n = 2 , figure 3 and table 2 describe the behavior of the sequence { xn+3l}l=0 and { yn+3l}l=0 ( shown by the black spot at the top of every peak ) . let x1 = 8 , x0 = 1 , y1 = 3 , and y0 = 7.(i)for a = 1 , b = 2 , and n = 2 , figure 2 and table 1 describe the behavior of the sequence { xn+3l}l=0 and { yn+3l}l=0 ( shown by the black spot at the top of every peak).(ii)for a = 2 , b = 1 , and n = 2 , figure 3 and table 2 describe the behavior of the sequence { xn+3l}l=0 and { yn+3l}l=0 ( shown by the black spot at the top of every peak ) . for a = 1 , b = 2 , and n = 2 , figure 2 and table 1 describe the behavior of the sequence { xn+3l}l=0 and { yn+3l}l=0 ( shown by the black spot at the top of every peak ) . for a = 2 , b = 1 , and n = 2 , figure 3 and table 2 describe the behavior of the sequence { xn+3l}l=0 and { yn+3l}l=0 ( shown by the black spot at the top of every peak ) .
our aim in this paper is to investigate the behavior of positive solutions for the following systems of rational difference equations : xn+1 = a / xnyn2 , and yn+1 = byn / xn1yn1 , n = 0,1 ,  , where x1 , x0 , y1 , and y0 are positive real numbers and a and b are positive constants .