Oscillating spatial structures of a tissue.

We present a simple mathematical model for the self-controlled growth of a tissue giving rise to an oscillating tissue size under certain conditions. The control is brought about by two substances (two inhibitors or one inhibitor and one nutrient) which influence the cell kinetics locally. The inhibitors are produced by the tissue itself (whereas the nutrient comes from outside but is consumed by the tissue which produces the same effect). Both diffuse freely throughout the tissue, and thus realize a communication between different parts of the tissue. In any case the tissue approaches a self-maintaining space-time structure with properties depending on the parameters of proliferation, death and inhibiting control. We discuss the conditions for this structure not to be time-independent but oscillating.