Viability Kernel Algorithm for Shapes Equilibrium

Viability is a very important feature of dynamic systems under state constraints whose initial value problem does not ensure uniqueness of solutions. In this paper, we introduce an hybrid automaton to address the question of viability of a cellular tissue. This hybrid automaton couples two dynamical models: differential equations manage the energy of the system and morphological equations govern the growth of the tissue. The cells can proliferate when they have enough access to oxygen and nutrient to produce the energy, remain quiescent when this energy is between two levels, or die when this energy is too low. The constraint we choose is to maintain the number of cells of the tissue during a certain time horizon. We have shown that for all the 1029 2D-tissues of 16 cells with an associate genotype, only 5 are viable for this constraint in a long time horizon. Moreover, for all these tissues, they renew there cells periodically. These periodic shapes are like periodic limit cycles in the state space of shapes.