  We develop a landscape-flux framework to investigate observed frequency
distributions of vegetation and the stability of these ecological systems under
fluctuations. The frequency distributions can characterize the
population-potential landscape related to the stability of ecological states.
We illustrate the practical utility of this approach by analyzing a
forest-savanna model. Savanna, and Forest states coexist under certain
conditions, consistent with past theoretical work and empirical observations.
However, a new Grassland state, unseen in the corresponding deterministic
model, emerges as an alternative quasi-stable state under fluctuations,
providing a novel theoretical basis for the appearance of widespread grasslands
in some empirical analyses. The ecological dynamics are determined by both the
population-potential landscape gradient and the steady-state probability flux.
The flux quantifies the net input/output to the ecological system and therefore
the degree of nonequilibriumness. Landscape and flux together determine the
transitions between stable states characterized by dominant paths and switching
rates. The intrinsic potential landscape admits a Lyapunov function, which
provides a quantitative measure of global stability. We find that the average
flux, entropy production rate, and free energy have significant changes near
bifurcations under both finite and zero fluctuation. These may provide both
dynamical and thermodynamic origins of the bifurcations. We identified the
variances in observed frequency time traces, fluctuations and time
irreversibility as kinematic measures for bifurcations. This new framework
opens the way to characterize ecological systems globally, to uncover how they
change among states, and to quantify the emergence of new quasi-stable states
under stochastic fluctuations.
