A model about regulation on three division modes of stem cell.

We construct a multi-stage cell lineage model for cell division, apoptosis and movement. Cells are assumed to secrete and respond to negative feedback molecules which act as a control on the stem cell divisions (including self-renewal, asymmetrical cell division (ACD) and differentiation). The densities of cells and molecules are described by coupled reaction-diffusion partial differential equations, and the plane wavefront propagation speeds can be obtained analytically and verified numerically. It is found that with ACD the population and propagation of stem cells can be promoted but the negative regulation on self-renewal and differentiation will work slowly. Regulatory inhibition on differentiation will inversely increase stem cells but not affect the population and wave propagation of the cell lineage. While negative regulation on self-renewal and ACD will decrease the population of stem cells and slow down the propagation, and even drive stem cells to extinction. Moreover we find that inhibition on self-renewal has a strength advantage while inhibition on ACD has a range advantage to kill stem cells. Possible relations to model cancer development and therapy are also discussed.

Population dynamics and wave propagation in a Lotka-Volterra system with spatial diffusion.

We consider the competitive population dynamics of two species described by the Lotka-Volterra model in the presence of spatial diffusion. The model is described by the diffusion coefficient (d(α)) and proliferation rate (r(α)) of the species α (α = 1,2 is the species label). Propagating wave front solutions in one dimension are investigated analytically and by numerical solutions. It is found that the wave profiles and wave speeds are determined by the speed parameters, v(α) ≡ 2 sqrt [d(α)r(α)], of the two species, and the phase diagrams for various inter- and intracompetitive scenarios are determined. The steady wave front speeds are obtained analytically via nonlinear dynamics analysis and verified by numerical solutions. The effect of the intermediate stationary state is investigated and propagating wave profiles beyond the simple Fisher wave fronts are revealed. The wave front speed of a species can display abrupt increase as its speed parameter is increased. In particular for the case in which both species are aggressive, our results show that the speed parameter is the deciding factor that determines the ultimate surviving species, in contrast to the case without diffusion in which the final surviving species is decided by its initial population advantage. Possible relations to the biological relevance of modeling cancer development and wound healing are also discussed.