Nonlinear Analysis: Real World Applications www elsevier com locate na
30 Pages
Gain access to the library to view online
Learn more

Nonlinear Analysis: Real World Applications www elsevier com locate na


Gain access to the library to view online
Learn more
30 Pages


Nonlinear Analysis: Real World Applications 6 (2005) 337–366 Existence, positivity and stability for a nonlinear model of cellular proliferation Mostafa Adimy, Fabien Crauste? Laboratoire de Mathématiques Appliquées, FRE 2570, Université de Pau et des Pays de l'Adour, Avenue de l'Université, 64000 Pau, France Received 14 October 2003; accepted 2 September 2004 Abstract In this paper, we investigate a system of two nonlinear partial differential equations, arising from a model of cellular proliferation which describes the production of blood cells in the bone marrow. Due to cellular replication, the two partial differential equations exhibit a retardation of the maturation variable and a temporal delay depending on this maturity. We show that this model has a unique solution which is global under a classical Lipschitz condition. We also obtain the positivity of the solutions and the local and global stability of the trivial equilibrium. 2004 Elsevier Ltd. All rights reserved. Keywords: Nonlinear partial differential equation; Age-maturity structured model; Blood production system; Maturity-dependent delay; Positivity; Local and global stability 1. Introduction We analyze, in this paper, a mathematical model arising from the blood production system. It is based on a system proposed by Mackey and Rudnicki [19], in 1994, to describe the dynamics of hematopoietic stem cells in the bone marrow.

  • cytokinesis point

  • can also

  • cells just after

  • cellular proliferation

  • mature stem cells

  • partial differential equations

  • differential equation

  • maturity structured



Published by
Reads 26
Language English


Ex ,    b f  f u f M f,FbCu      ’  64  c 14Ob;Sb4

Ih , g fff  f u f hh  b h u f b  u , h   ff qu  b     g  h u   h h u hh  gb u    L hz    u  h   gb b f h  qubu Lhg  4 E
Kyw:u uu-g;uqff Mu-;P ;Lgb b
z,h ,h gfh I b   bMk[19],u1994,k b h  fh  hbTh  f Bu [T]hbhhhfk     g h GhT h-  ug     uu 
C g uh -  : f@uM- u ,fbu @uFC-u uf
148-118 $ -  f ghL E4   111 j491
Pfg h,hh fh  ,h -fk  f ugh  hh   h  g h  P b   ,  g  h Th  g h    qu  g  h u    h u ,  h   , h  h Th [19]Mbkb hz[1 ,18] 199, Cbb  [8,9]919,D [1]991-MjPu[j,4]u 1   I h  u , h uh  u h   g H,  h   g u   u u  qu f      []b,xF u   hh  h  u     , hh  h  u   [1I]M,k991    hh h  qu f        buhuh u  [f11,h1,] ,    g-u uu qu   h  g Th   h b  h f x ,  f h u  H,  h , h h fg ,  h  flux b h  I[],fghhh  bu g     h  u  b u  I h k,    h u h h g f    u f h   h  f , h  h h    u h h  g h  ghf  g=(m)h uT hh h bfg, [1,1]1uT1khT  hbu uh ,x[]h,uj-MP huj    b f -ffqu  h u    f h u b f x , uqu ,    b f h u Th   gz  f  I S ,    uu  B u g h h  h,   u uu ,hh fbf   g  h u     b I S ,  fi  g  g fu     f   fu  h   b u g  fix- h,  h gb u   I S 4,  b h   f h  u  b b [4b]bSI,bh  bu f h  ,  h   ,   u h bh
.y.C/Ny:Wc6(9) 33 –
Eh   hz,  h  h  , b  g    b h  f h  I  h     h     u x b  uu b m=1gfh  h C  h fa=uh ghhg  hg ,ab=fhhgg f  u  h fg   bb    hk  g, g ugh  h  g ha =hgh,f    h g ba=g +f  qu  hg hT u  I h h ,   h u hbfghgf S[h ,] u  h h u f    h   g u   h  f h  f   p(bt, m, a)n(t, m, a)  , h u   fg  h  ,hgh gu  hTqu  p+pa+(V(p)m)m= −(m)p,1 t nt+na+(V(m)n)= −((m)+(m, N (t, m)))n, m hV(m) h uN(t,m) f h g  h  u,b +∞ N (t, m)=n(t, m, a)a.   u  h hV fuuu  ff[,1] ,b(,1]  V ( fi)= mV)ss(= +∞fm(,1]. Smm1s/V (s), mh1< m,  h  qu f m1huh h um, h C   h   h      b u F x, f V (m)mph> p1, m h C    fi  u , hughu h ,huu [,1]g Th fu ub   uu