TRANSITIONS AND KINEMATICS OF REACTION-CONVECTION FRONTS IN A CELL-POPULATION MODEL

Here we consider cell population dynamics in which there is a simultaneous proliferation and maturation. The mathematical model of this process is formulated as a nonlinear first order partial differential equation for the cell density u(t, x) in which there is retardation (delay) in the temporal (t) variable. Thus we consider a transient reaction-convection equation in which the cell density is convected with maturation velocity r. For localized initial perturbations the equation has positive and negative traveling front solutions. Positive fronts correspond to the invasion of the zero amplitude solution by a finite amplitude solution, and negative fronts correspond to the reversed case. Three classes of fronts are found according to the strength of the convection velocity; (i) For strong convection (r much-greater-than 1) the fronts are simple translations of the initial data, regardless of delay strength; (ii) for weak convection (r much-less-than 1) two types of fronts exist: (a) reaction-convection fronts arise if the localized initial perturbation acts at a non-zero maturation on the zero amplitude state, and (b) convection fronts arise if the localized initial perturbation acts at zero maturation on a finite amplitude state. For weak convection (cases (ii)a and (ii)b)a further classification arises according to whether the magnitude of the temporal delay is larger or smaller than a critical value tau(b). It is found that the critical delay tau(b) corresponds to the Hopf bifurcation of the reaction equation that is obtained in the absence of convection (r = 0). For delays larger than tau(b) the convective and reaction-convection fronts are oscillatory. In addition all the reaction-convection fronts reverse their direction of motion, undergo a positive to negative transition, and display non-uniform kinematics. Simulation results are validated and interpreted using solutions for the unretarded equations, and by local stability analysis.