  Motivated by the famous Waddington's epigenetic landscape metaphor in
developmental biology, biophysicists and applied mathematicians made different
proposals to realize this metaphor in a rationalized way. We adopt
comprehensive perspectives to systematically investigate three different but
closely related realizations in recent literature: namely the potential
landscape theory from the steady state distribution of stochastic differential
equations (SDEs), the quasi-potential from the large deviation theory, and the
construction through SDE decomposition and A-type integral.The connections
among these theories are established in this paper. We demonstrate that the
quasi-potential is the zero noise limit of the potential landscape. We also
show that the potential function in the third proposal coincides with the
quasi-potential. The most probable transition path by minimizing the
Onsager-Machlup or Freidlin-Wentzell action functional is discussed as well.
Furthermore, we compare the difference between local and global quasi-potential
through the exchange of limit order for time and noise amplitude. As a
consequence of such explorations, we arrive at the existence result for the SDE
decomposition while deny its uniqueness in general cases. It is also clarified
that the A-type integral is more appropriate to be applied to the decomposed
SDEs rather than the original one. Our results contribute to a better
understanding of existing landscape theories for biological systems.
