Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 155

for i j . In this case the equilibrium point s st is stable if all the coefficients on the

diagonal of A are negative (i.e. a i , i

=

n ). We can set the time scale
of convergence to equilibrium by taking all the diagonal coefficients equal to

< 0, ( i = 1 , 2 ,...,

1, as
in May ( 1972 ). To study how random interactions among state variables (e.g., among
the populations of different species existing in the system) may affect the stability of
the system, we allow for the occurrence of random interactions between a fraction C
of pairs of state variables ( s i

−

,

s j ). To this end, we set the corresponding extradiagonal
elements a i , j of A equal to values drawn from a distribution of random numbers with
mean zero. Thus the interactions can be either positive or negative, and their intensity
is on average zero. The community matrix can be then expressed as

A = B − I , (4.68)

where I is the identity matrix and B is a random matrix. With probability C the
elements of B are nonzero, and their value is drawn from a zero-mean probability
distribution with standard deviation C the elements of

B are equal to zero. Known as connectance ( Gardner and Ashby , 1970 ), the parameter
C expresses the probability that any pair of state variables (e.g., populations) interacts.
Notice how, once the coefficients of A have been selected, the dynamics are com-
pletely deterministic and the stability of s st can be investigated with the standard
methods for deterministic systems. Thus the stability of s st requires that all the eigen-
values of A have negative real parts (see Box 4.3). Using the theory of random
matrices, May ( 1972 ) noticed that this condition is met when

σ , whereas with probability 1 −

1
√ nC ,

σ<

(4.69)

whereas the equilibrium point s st is unstable otherwise. The probability that the dy-
namics of randomly constructed communities are unstable (i.e., at least one eigenvalue
has a positive real part) increases with an increasing number of species. Known as
the complexity paradox , this result is consistent with earlier numerical simulations
by Gardner and Ashby ( 1970 ) and indicates that a stable multivariate system can be
destabilized by an increase either in the number of species or in the connectivity (or
both). This finding provided new insights on the relation between complexity and
stability in ecological systems (e.g., May , 1973 ; Pimm , 1984 ). Another interesting
result appearing in condition ( 4.69 ) is associated with a the randomness of the in-
teractions. In a system with a given connectance and number of species, an increase
in the variance of interspecies interaction may induce instability. The emergence of
instability with increasing values of

is clearly a noise-induced effect, though, as
noted, these dynamics are entirely deterministic once the random coefficients have
been selected. The applicability of these results to actual ecosystems has been often
challenged (e.g., De Angelis , 1975 ; Pimm , 1984 ) in that the randomness of the in-
teractions is not a realistic assumption for real-food Web models. Because in nature

σ