Analysis of a Model Arising from Invasion by Precursor and Differentiated Cells

We study the wave solutions for a degenerated reaction diffusion system arising from the invasion of cells. We show that there exists a family of waves for the wave speed larger than or equal a certain number and below which there are no monotonic wave solutions. We also investigate the monotonicity, uniqueness, and asymptotics of the waves. 1. Introduction In [1], the following coupled partial differential equation system was proposed to study the invasion by precursor and differentiated cells: where denotes the population densities of the precursor cells. The constant is the diffusion rate of the cell , which has proliferation rate , and is the carrying capacity of . The parameter measures the relative contribution that the differentiated cell with population density makes to the carrying capacity . The cell population density is limited by its carrying capacity and has a maximum differentiation rate . The model assumes that the differentiated cells do not have mobility. By letting (see [1]) and dropping the hat notation for convenience, system (1) is changed into where and . System (1) or (3) belongs to reaction diffusion systems of degenerate type, and such systems have attracted much attention in research fields such as epidemics and wound healing [2–4] as well as combustion and calcium wave problems [5–8]. However, system (3) differs from the above systems in the appearance of degenerate reaction terms. In fact, coupling with any consists of a constant solution of (3). This resembles the combustion wave equation considered in [9]; however, our method in proving the existence of the fronts of (3) differs from theirs. If the parameters satisfy then system (3) admits an additional equilibrium: representing the state that the spatial domain is successfully invaded. We also separate the equilibrium from the rest of the line of equilibria, . The unstable equilibrium represents the state before the invasion. We are interested in the existence of the wave solutions connecting with as time and space evolve from to . Setting , , , a traveling wave solution to (3) solves with boundary conditions: For the notational convenience, we further set and drop the bar on to have Numerical investigations [1] strongly suggest that system (10) and (9) admit traveling wave solutions for and . When the differentiated cell density does not affect the proliferation of the precursor cells, we have ; when the total cell population contributes to the proliferation carrying capacity, we have . Numerically, however, when , (8) may have nonmonotone traveling wave solutions and