Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 112

Benedetti-Cecchi et al. ( 2006 )and Bertocci et al. ( 2007 ) recently found an increase
in the diversity of rock-shore populations of algae and invertebrates exposed to an
increase in the variance of environmental fluctuations (with constant mean). Using
a framework based on the theory of stochastic differential equations with dichoto-
mous noise, this subsection puts these experimental results into the broader context of
how the state of an ecosystem may benefit from moderate environmental variability
through the enhancement of biodiversity.
Following D'Odorico et al. ( 2008 ), we consider the case of a system in which all
species are controlled by the same environmental variable R (e.g., water, energy, light,
or nutrients). We assume that R is a positive random variable with gamma distribution
p ( R ) (though the use of other distributions would not alter the results presented in this
subsection), mean σ R . The species coexisting within the

ecosystem are from a pool determined by the ecosystem's biogeographical conditions
and evolutionary history. Each species is unstressed (i.e., able to live, grow, and
reproduce) when R remains within a certain fitness interval I δ ,or niche , whereas its
biomass decays when the environmental conditions keep R outside this interval. All
species have in general fitness intervals I δ with different amplitudes (or fitness ranges )
δ

R , and standard deviation

and different midpoint values. However, for the sake of simplicity we first assume
that all fitness intervals have the same amplitude and that no mutual interaction

(e.g., competition-facilitation) exists among species. We then consider more general
conditions. In the absence of interactions we can model the temporal dynamics of each
species independently of the others and assume that fluctuations in R determine the
switching between growth (unstressed conditions) and decay (stressed conditions) in
species biomass, depending on whether R falls within or outside of the fitness interval
I δ of that species. We use a linear decay and a logistic growth for the stressed and
unstressed conditions, respectively:

δ

a 1 B (

β − B )if R ∈ I δ (4 . 6a)

d B
d t =

,

− a 2 B otherwise (4 . 6b)

where B is the species' biomass, is the carrying capacity (i.e., the maximum

sustainable value of B ), and the coefficients a 1 and a 2 determine the decay and
growth rates, respectively.
Random fluctuations in R determine the switching between these two dynamics,
depending on whether R is contained within or falls outside the fitness window. Fol-
lowing the approach presented in the previous subsection the stochastic dynamics
resulting from the random switching between Eqs. ( 4.6 a) and ( 4.6 b) are modeled as
a dichotomous Markov process. With probability P 1 the environmental variable R is
contained within the fitness interval, where P 1

β

= R 0 + δ

R 0 p ( R )d R ,and R 0 is the lower
limit of the fitness interval I δ . In these conditions, the species is not stressed and
its growth is expressed by ( 4.6 a). Conversely, with probability 1 − P 1 the species is