Environmental Engineering Reference
In-Depth Information
with abrupt shifts to an alternative state of
. Because the deterministic counterpart
of this system is monostable, the discontinuous transitions found in this model of
symbiotic interactions are purely noise induced. These phase transitions are often
defined as reentrant in that bistability appears as the noise amplitude (or the correlation
scale) exceeds a critical value, but it also disappears as another (larger) critical value
of a 0 is exceeded.
Similar results were obtained ( Mankin et al. , 2002 ) when the random driver was
a trichotomous Markov noise (i.e., three-state Markov noise). The same authors also
provided some examples of bistable systems with unidirectional transitions ( Sauga and
Mankin , 2005 ), i.e., with transitions that may occur in only one direction. Other studies
investigated the effect of noise on multivariate ecological systems in which random
forcing does not induce stochastic fluctuations in the carrying capacity as in Mankin
et al. ( 2002 , 2004 )and Sauga andMankin ( 2005 ) but in the interaction coefficients J i , j .
The following subsection shows the relation between random interspecies interactions
and the stability and complexity of ecological systems with a large number of species
( Gardner and Ashby , 1970 ; May , 1972 ).
A
4.7.3 Stability of multivariate ecological systems with random
interspecies interactions
We consider the case of an n -dimensional ecological dynamical systemwith n species
s
s n ). The temporal variability of each variable s i depends on the other
state variables and is expressed by a set of first-order differential equations:
d s i
d t =
=
( s 1
,
s 2
,...,
f i ( s 1 ,
s 2 ,...,
s n )
,
( i
=
1
,
2
,...,
n )
,
(4.65)
where f i ( s ) is a set of n functions, which can be in general nonlinear. The equilibrium
states s st =
( s 1 , st , s 2 , st ,..., s n , st ) of the system are obtained as solutions of the set of
equations
f i ( s st )
=
0( i
=
1
,
2
,...,
n )
.
(4.66)
The stability of an equilibrium state s st is typically assessed through a linear-
stability analysis, i.e., with respect to infinitesimal perturbations, which allow for
a Taylor expansion of ( 4.65 ) in the neighborhood of the equilibrium point (see
Box 4.3):
d s
d t =
As
,
(4.67)
where the matrix A is the Jacobian of ( 4.65 ), which is also known in ecology literature
as the community or interaction matrix (e.g., May , 1973 ) because its elements a i , j =
x j s = s st express the interactions between the state variables s i and s j (i.e., the
effect of s j on the dynamics of s i ). In the absence of interactions we have that a i , j
f i
/∂
=
0
Search WWH ::




Custom Search