  With the growing number of discovered exoplanets, the Gaia concept finds its
second wind. The Gaia concept defines that the biosphere of an inhabited planet
regulates a planetary climate through feedback loops such that the planet
remains habitable. Crunching 'Gaia' puzzle has been a focus of intense
empirical research. Much less attention has been paid to the mathematical
realization of this concept. In this paper, we consider the stability of a
planetary climate system with the dynamic biosphere by linking a conceptual
climate model to a generic population dynamics model with random parameters. We
first show that the dynamics of the corresponding coupled system possesses
multiple timescales and hence falls into the class of slow-fast dynamics. We
then investigate the properties of a general dynamical system to which our
model belongs and prove that the feedbacks from the biosphere dynamics cannot
break the system's stability as long as the biodiversity is sufficiently high.
That may explain why the climate is apparently stable over long time intervals.
Interestingly, our coupled climate-biosphere system can lose its stability if
biodiversity decreases; in this case, the evolution of the biosphere under the
effect of random factors can lead to a global climate change.
