  Cell-fate transition can be modeled by ordinary differential equations (ODEs)
which describe the behavior of several molecules in interaction, and for which
each stable equilibrium corresponds to a possible phenotype (or 'biological
trait'). In this paper, we focus on simple ODE systems modeling two molecules
which each negatively (or positively) regulate the other. It is well-known that
such models may lead to monostability or multistability, depending on the
selected parameters. However, extensive numerical simulations have led systems
biologists to conjecture that in the vast majority of cases, there cannot be
more than two stable points. Our main result is a proof of this conjecture.
More specifically, we provide a criterion ensuring at most bistability, which
is indeed satisfied by most commonly used functions. This includes Hill
functions, but also a wide family of convex and sigmoid functions. We also
determine which parameters lead to monostability, and which lead to
bistability, by developing a more general framework encompassing all our
results.
