Abstract
We investigate in this paper, the problem{∂u∂t(x,t)−d1Δxu(x,t)=−f(u(x,t))+∫0τh(a)∫ΩG(x,y,a)v(y,t−a)dyda,v(x,t)=g(u(x,t))+∫0τk(a)∫ΩG(x,y,a)v(y,t−a)dyda, with Neumann boundary conditions. Such a problem takes its origin in modelling the dynamics of quiescent and proliferating hematopoietic stem cells. Our aim is to provide a complete asymptotic analysis, to ensure global attractiveness and exponential stability of trivial and positive steady states. To this end, chiefly, a good combination of a suitable choice of Lyapunov functional with some monotony techniques are used. A comparison principle applied to a well chosen deformation of our problem, permits us to obtain the aim.
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