  When a system has more than one stable state, how can the stability of these
states be compared? This deceptively simple question has important consequences
for ecosystems, because systems with alternative stable states can undergo
dramatic regime shifts. The probability, frequency, duration, and dynamics of
these shifts will all depend on the relative stability of the stable states.
Unfortunately, the concept of stability in ecology has suffered from
substantial confusion and this is particularly problematic for systems where
stochastic perturbations can cause shifts between coexisting alternative stable
states. A useful way to visualize stable states in stochastic systems is with a
ball-in-cup diagram, in which the state of the system is represented as the
position of a ball rolling on a surface, and the random perturbations can push
the ball from one basin of attraction to another. The surface is determined by
a potential function, which provides a natural stability metric. However,
systems amenable to this representation, called gradient systems, are quite
rare. As a result, the potential function is not widely used and other
approaches based on linear stability analysis have become standard. Linear
stability analysis is designed for local analysis of deterministic systems and,
as we show, can produce a highly misleading picture of how the system will
behave under continual, stochastic perturbations. In this paper, we show how
the potential function can be generalized so that it can be applied broadly,
employing a concept from stochastic analysis called the quasi-potential. Using
three classic ecological models, we demonstrate that the quasi-potential
provides a useful way to quantify stability in stochastic systems.
