Transitions and kinematics of reaction-convection fronts in a cell population model

Abstract

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 localizedd 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 ⪢ 1) the fronts are simple translations of the initial data, regardless of delay strength; (ii) for weak convection ( r ⪡ 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 τ b . It is found that the critical delay τ b corresponds to the Hopf bifircation of the reaction equation that is obtained in the absence of convection ( r = 0). For delays larger than τ 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.