In many tissues, such as skin or intestinal epithelial, stem cells lie in periodic local areas. In the areas, stem cells divide and differentiate without migration, whereas the differentiated cells migrate out of the local areas actively. Even though the previous model for stem cell patterns considered all the hydrodynamical couplings allowed by symmetry, it is still based on assumption that there was no spontaneous polarization in the homogeneous state. To extend the model, we derive a hydrodynamic model based on the Generalized Onsager Principle. In this model, the spontaneous polarity states, birth and death of cells, self-propelled motion of differentiated cells and ATP hydrolysis are all accounted for. Linear stability analysis is conducted to reveal the long-wave instability inherent in the neighborhood of the constant steady states. The spatial patterns due to instability are observed in numerical experiments. Especially, new spatial patterns with spontaneous polarity are observed. Comparison with previous models will be presented in this talk.
