Propagation of population pulses and fronts in a cell replication problem: Non-locality and dependence on the initial function

Abstract

Abstract Here we study the solution types in a simplified equation arising from a cell population dynamics model formulated as a non-linear first order partial differential equation for the cell density u(t, x) in the presence of non-local maturational (x) interactions. This can be considered as a reaction-convection equation in which the cell density is convected with maturation velocity r. The non-local action affects only the reaction (birth) rate. An initial function representing a pulse of small amplitude on an almost extinguished population may be dissipated, may produce multiple pulses, or may remain intact as it travels driven by reaction and convection, depending on the magnitude of the non-local action. Alternately, for localized initial perturbations the equation also has positive traveling front solutions. Positive fronts correspond to the invasion of the zero amplitude solution by a finite amplitude solution. For weak convection (r ⪡ 1, which is the only case treated here) reaction driven fronts arise if the localized initial perturbation acts at a non-zero maturation on the zero amplitude state. A classification arises according to whether the magnitude of the maturation “delay” is larger or smaller than a critical value, and scaling arguments are used to determine the critical delay. For delays larger than critical the reaction driven fronts are oscillatory, and correspond to the case for which the correlation length for non-local action is of the same order as the characteristic length of maturation population gradients that yield oscillatory birth rates. Simulation results are validated and interpreted using scaling arguments and solutions for the local equations.