Environmental Engineering Reference
In-Depth Information
This model was used (
D'Odorico et al.
,
2006b
) to investigate the effect of stochastic
fire dynamics on tree-grass coexistence in mesic and subhumid savannas.
2
We assume
that - in the absence of disturbances - these savannas tend to be dominated by woody
plants (e.g.,
van Langevelde et al.
,
2003
). In these systems competition can at most
slow down the growth of tree biomass but it is unable to induce a stable state of tree-
grass coexistence. Thus the temporal dynamics of tree biomass can bemodeledwithout
explicitly accounting for interspecies competition. The relatively slow demographic
growth of woody vegetation enables grasses to use space and resources left available
by trees after fire occurrences. However, in periods between fire events, trees are able
to reclaim space and resources from grasses because no niche separation is assumed to
exist and trees are able to outcompete grasses. We characterize the state of the system
through the state variable
A
, representing the total woody biomass.
A
ranges between
0 and a maximum value
A
c
, the ecosystem
carrying capacity
, which depends on the
existing resources (i.e., water, nutrients, and light). We assume
A
c
to be constant in
time and normalize
A
with respect to
A
c
, i.e.,
A
=
A
c
. To simplify the notation
we drop the superscript from the normalized biomass and refer to
A
as the woody
biomass normalized between 0 and 1. As noted, the growth of
A
in periods between fire
occurrences is modeled with a logistic equation and the harvest rate is proportional to
the existing biomass. These dynamics are expressed by (
4.9
) with
f
(
A
)
A
/
=
aA
(1
−
A
)
and harvest rate proportional to
g
(
A
)
A
. The steady-state probability distribution
of
A
can be determined as described in Chapter 2 [see pdf (
2.34
)], leading to
=−
1
α
−
a
A
)
a
−
1
−
1
(1
p
(
A
)
=
CA
−
,
(4.11)
which is a beta distribution with normalization constant (e.g.,
Johnson et al.
,
1994
)
1
α
C
=
1
α
−
a
a
,
(4.12)
·
where
) is the gamma function.
A variety of dynamical behaviors emerge for different combinations of the param-
eters. Figure
4.9
shows the dependence of the shape of the distribution of
A
-which
is here studied through its modes
A
m
(see Chapter 3) - on the parameters of fire
and vegetation dynamics. In this figure, curves (a), (b), and (c) divide the parameter
space into five zones characterized by different dynamical behaviors. For relatively
low values of
(
λ/
a
, the mode of
A
is
A
m
=
1; thus the preferential state of the system
2
The dynamics of savanna ecosystems were investigated in the recent past with a variety of approaches (e.g.,
Sankaran
et al.
,
2004
), depending on whether (i) vegetation dynamics are studied in lumped (e.g.,
Walker et al.
,
1981
)or
spatially extended systems (e.g.,
Jeltsch et al.
,
1996
;
Rodriguez-Iturbe et al.
,
1999a
;
van Wijk and Rodriguez-Iturbe
,
2002
) and (ii) tree-grass coexistence results from competition (
Walter
,
1971
), demographic dynamics (e.g.,
Higgins
et al.
,
2000
), mixed competition-demographic processes (e.g.,
Anderies et al.
,
2002
;
van Langevelde et al.
,
2003
),
or landscape heterogeneities (e.g.,
Kim and Eltahir
,
2004
). In this subsection we use a simplistic stochastic model to
demonstrate the role of noise in the demographic dynamics resulting from fire-vegetation interactions in spatially
lumped systems.
Search WWH ::
Custom Search