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 = r_{c}, which depends on the
time delay, the stationary solution is nonhomogeneous. For
values of r close to r_{c}, the solution dynamics exhibit critical
slowing down, similar to that seen in the neighborhood of a
phase transition. For 0 < r_{c} < r, the stationary solution is
uniformly zero, while for 0 < r < r_{c}, the stationary solution
is homogeneous and singular.
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