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Volume 1,     Number 1,     Winter 1993

 

MULTISTABILITY AND
BOUNDARY LAYER DEVELOPMENT
IN A TRANSPORT EQUATION
WITH DELAYED ARGUMENTS
ALEJANDRO D. REY AND MICHAEL C. MACKEY

Abstract. Here we consider cell population dynamics in which there is simultaneous proliferation and maturation. The mathematical model for this process is derived, and results in a nonlinear first order partial differential equation for the cell density u(t,x) in which there is retardation in both temporal (t) and maturation (x) variables. Numerical analysis of a representative equation indicates that there are two classes of solution behavior depending on the initial function φ(x). If φ(0) > 0 there is a unique stationary solution. In this case the net effect of the time delay is to retard the dynamic approach to the stationary solution, while spatial delays modify the steady state distributions with respect to the maturation variable. Alternately, if φ(0) = 0, the stationary solutions display a multistability that depends on the maturation velocity r. For a critical value r = rc, which depends on the time delay, the stationary solution is nonhomogeneous. For values of r close to rc, the solution dynamics exhibit critical slowing down, similar to that seen in the neighborhood of a phase transition. For 0 < rc < r, the stationary solution is uniformly zero, while for 0 < r < rc, the stationary solution is homogeneous and singular.

 

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