Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 100

Box 4.1: Properties of bistable deterministic systems

Bistable systems are by definition characterized by the existence of two alternative
stable states. Ecosystem bistability is often induced by positive feedbacks ( Wilson and
Agnew , 1992 ) between the state of the system and the dynamics of its limiting resources
( Walker et al. , 1981 ; Walker and Noy-Meir , 1982 ; Rietkerk and van de Koppel , 1997 ;
D'Odorico et al. , 2007a ; DeLonge et al. , 2008 ) or disturbance regime ( Anderies et al. ,
2002 ; D'Odorico et al. , 2006b ; Ridolfi et al. , 2006 ).
The dynamics of these systems are typically visualized by use of plots of the potential
function, which resembles the vertical section of a landscape (Fig. B4.1-1) with two
valleys, s 1 and s 2 , separated by a “hill ridge,” u . In this landscape a ball would roll down
toward either one of the two valley bottoms (i.e., stable states in this gravitational-field
analog), depending on the initial condition, i.e., on whether the ball (i.e., state of the
system) is initially located to the right or the left of the hilltop (unstable state u ). The
two stable states are also known as attractors , and each of the two “valleys” is the
attraction basin of the corresponding stable state.

u

s

s

Figure B4.1-1. Potentials associated with (a) bistable and (b) monostable
dynamics.

Bistable deterministic systems [Fig. B4.1-1(a)] have dynamical properties that are
profoundly different from those of their monostable counterparts [Fig. B4.1-1(b)].