Evolution in random environment and structural instability

We consider stability and evolution of complex biological systems, in particular, genetic networks. We focus our attention on supporting of homeostasis in these systems with respect to fluctuations of an external medium (the problem is posed by M. Gromov and A. Carbone [32]). Using a measure of stochastic stability, we show that a generic system with fixed parameters is unstable, i.e., the probability to support homeostasis converges to zero as time $T\to\infty$. However, if we consider a population of unstable systems which are capable to evolve (change their parameters), then such a population can be stable as $T\to\infty$. This means that the probability to survive may be nonzero as $T\to\infty$. Evolution algorithms that provide stability of populations are not trivial. We show that the mathematical results on evolution algorithms are consistent with experimental data on genetic evolution.