PROPAGATING FRONTS, CHAOS AND MULTISTABILITY IN A CELL REPLICATION MODEL

Numerical solutions to a model equation that describes cell population dynamics are presented and analyzed. A distinctive feature of the mod el equation (a hyperbolic partial differential equation) is the presen ce of delayed arguments in the time (t) and maturation (x) variables d ue to the nonzero length of the cell cycle. This transport like equati on balances a Linear convection with a nonlinear, nonlocal, and delaye d reaction term. The linear convection term acts to impress the value of u(t,x=0) on the entire population while the death term acts to driv e the population to extinction. The rich phenomenology of solution beh aviour presented here arises from the nonlinear, nonlocal birth term. The existence of this kinetic nonlinearity accounts for the existence and propagation of soliton-like or front solutions, while the increasi ng effect of nonlocality and temporal delays acts to produce a fine pe riodic structure on the trailing part of the front, This nonlinear, no nlocal, and delayed kinetic term is also shown to be responsible for t he existence of a Hopf bifurcation and subsequent period doublings to apparent ''chaos'' along the characteristics of this hyperbolic partia l differential equation. In the time maturation plane, the combined ef fects of nonlinearity, nonlocality, and delays leads to solution behav iour exhibiting spatial chaos for certain parameter values. Although a nalytic results are not available for the system we have studied, cons istency and validation of the numerical results was achieved by using different numerical methods. A general conclusion of this work, of int erest for the understanding of any biological system modeled by a hype rbolic delayed partial differential equation, is that increasing the s patio-temporal delays will often lead to spatial complexity and irregu lar wave propagation. (C) 1996 American Institute of Physics.