Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 113

P 1 ≥ P lim

P 1 ≥ P lim

P 1 = P lim P 1 ≥ P lim

P 1 = P lim

P 1 ≤ P lim

R

δ δ δ δ δ δ

R u

R I

Figure 4.6. Probability distribution of the resource R . For a given δ the biodiversity

potential δ

remain unstressed for a sufficient fraction of time to avoid extinction (after D'Odorico
et al. , 2008 ).

is defined as the interval ( R l , R u ) in which all species with fitness range

stressed, and its dynamics are modeled by ( 4.6 b). The solution of the stochastic differ-
ential equation associated with these dynamics provides the probability distribution
p ( B ). It can be shown ( Camporeale and Ridolfi , 2006 ) that when

a

P 1 ≤ P lim = (4.7)

a + β

(with a a 1 ), the species goes extinct because B is zero with probability tending

to one. This condition is independent of the values of

= a 2 /

1 and 2 but depends on

the parameters and a of the dynamics. Relatively low values of P 1 correspond to

conditions in which the environmental variable remains too often outside the fitness
interval to allow for the survival of that species. Thus a species can survive only
when P 1

β

P lim . As shown in Fig. 4.6 for a given distribution of resources p ( R )and
fitness range

≥

, there are two limit positions [if p ( R ) is unimodal] along the R axis in
which the condition P 1

δ

P lim can be met. These two limit positions, R l ,and R u ,are
determined by the conditions

=

R l + δ R u

p ( R )d R = P lim , p ( R )d R = P lim . (4.8)

R l R u − δ

can survive when their fitness interval is contained
within the interval in that they remain unstressed for a sufficient fraction of time that
condition ( 4.7 ) is never met. Thus for a given distribution p ( R ) of the environmental
variable (Fig. 4.6 ) we can determine the interval ( R l , R u )onthe R axis in which

Species with fitness range δ