Environmental Engineering Reference
In-Depth Information
MATERIALS AND METHODS
The following protocols apply to simulations of single and multi-niche communities
with density-dependent recruitment and density-independent loss of individuals. They
produce the outcomes illustrated in Figures 3-5 from input parameters specified at
the end of this section. The general model has species-specific vital rates; the intrin-
sically neutral and dominant-fugitive scenarios are special cases of this model, with
constrained parameter values.
The community occupies a homogenous environment represented by a matrix of
K
equally accessible habitat patches within a wider meta-community of
K
m
patches.
The dynamics of individual births and deaths are modeled at each time step by species-
specifi c probability
b
of each resident, immigrant, and individual of new invading
species producing a propagule, and species-specifi c probability
d
of death for each
patch resident. Recruitment to a patch is more or less suppressed from intrinsic rate
b
by the presence there of other species according to the value of α
ij
, the impact of spe-
cies
j
on species
i
relative to
i
on itself, where the intraspecifi c impact α
ii
= 1 always.
A patch can be occupied by only one individual of a species, and by only one species
unless all its resident α
ij
< 1. Conventional Lotka-Volterra competition is thus set in a
metapopulation context by equating individual births and deaths to local colonizations
and extinctions (following [24], consistent with [20]). A large closed metapopulation
comprising
S
species has rates of change for each species
i
in its abundance
n
i
of indi-
viduals (or equally of occupied patches) over time
t
approximated by:
⎛
⎜
⎞
⎟
−
dn
i
dt
S
∑
=
b
i
n
i
1
−
α
ij
n
j
K
d
i
n
i
.
(1)
j
=
1
This is the rate equation that also drives the dynamics of Figures 1 and 2, where
k
i
= (1-
d
i
/
b
i
)
K
. Co-existence of any two species to positive equilibrium
n
1
,
n
2
requires
them to have intrinsic differences such that
k
1
>α
12
k
2
and
k
2
>α
21
k
1
.
Each time-step in the simulation offers an opportunity for one individual of each
of two new species to attempt invasion (regardless of the size of the meta-commu-
nity). Each new species
i
has randomly set competitive impacts with respect to each
other resident species
j
, of α
ij
received and α
ji
imposed. It has randomly set
b
i
, and
an intrinsic lifetime reproduction
R
i
=
b
i
/
d
i
that is stratifi ed in direct proportion to its
dominance rank among residents, obtained from its ranked mean α-received minus
mean α-imposed. For example, an invader with higher dominance than all of three
resident species will have random
R
i
stratifi ed in the bottom quartile of set limits
R
min
to
R
max
.
Communities are thereby structured on a stochastic life-history trade-off between
competitive dominance and population growth capacity. This competition-growth
trade-off is a well-established feature of many real communities, which captures the
fundamental life-history principle of costly adaptations [11, 17, 21]. Its effect on the
community is to prevent escalations of growth capacity or competitive dominance
among the invading species. Neutral communities are a special case, with identical
values of
b
and
R
for all species and α = 1 for all.
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