Noise-driven dynamical systems - Noise-induced Phenomena in the Environmental Sciences - page 12

represent random rainfall fluctuations that determine the occurrence of water-limited
conditions or of flooded or unflooded states, respectively. Thus the stochastic driver
determines the random alternation between stressed and unstressed conditions for the
ecosystem.
The two alternating dynamics of φ involve growth and decay and can be modeled

φ φ

by two functions, f 1 ( )and f 2 ( ), respectively,

f 1 (

φ ≥ θ .

)if q ( t ) (2 12a)

d d t =

,

f 2 ( φ )if q ( t ) <θ (2 . 12b)

whre f 1 ( 0. Equations ( 2.12 a) and (2.12b) are written assuming

that q is a resource, in that values of q exceeding the threshold are associated with
unstressed conditions (in the sense that

φ ) > 0and f 2 ( φ ) <

grows). However, the general results do not
change when the random driver is a stressor. In this case the conditions in ( 2.12 a)
and (2.12b) are reversed, i.e., growth or decay occurs when q is below or above the
threshold, respectively.
The class of processes defined by ( 2.12 a) and (2.12b) can be conveniently repre-
sented through a suitable dichotomous Markov process, thereby leading to a mech-
anistic usage of DMN. Thus the process is random and switches between two pos-
sible states: “success” (or “no stress”) when q is above the threshold or “failure”
(or “stress”) when q is below the threshold [see Fig. 2.2 (a)]. This is by definition
a dichotomous process. If we further suppose that q is uncorrelated, the driving
noise is the outcome of a Bernoulli trial with probability of success k 2 =

φ

1 − P Q ( θ ),

where P Q ( .

The residence time in the “above-threshold” state is then an integer number n 1
with a geometric probability distribution of p N 1 ( n 1 )

θ ) is the cumulative probability distribution of q , evaluated in q = θ

k n 1 − 1 = (1 − k 2 ), n 1 = 1 ,..., ∞ ,

2

with average k 2 ). Analogously, the residence time n 2 in the “below-

threshold” state is distributed as p N 2 ( n 2 )

n 1 = 1 / (1 −

k 2 ) n 1 − 1 k 2 , n 1 =

= (1 − 1 ,..., ∞ , with av-

erage k 2 . The DMN (in its mechanistic interpretation) is obtained as the

continuous-time approximation of this driving process [see Fig. 2.2 (b)]. In fact, in
continuous time the residence time in each state becomes exponentially distributed
(the exponential distribution is the continuous counterpart of the geometric distribu-
tion: e.g. Kendall and Stuart , 1977 ), which is a basic property of DMN (see Sub-
section 2.2.1 ).
The overall dynamics of the variable

n 1 = 1 /

φ can then be expressed by a stochastic

ξ

differential equation forced by DMN dn ( t ), assuming (constant) values 1 and 2

(see Fig. 2.1 ):

d d t =

f ( φ ) + g ( φ ) ξ dn ( t ) , (2.13)