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 12 k 2 and k 2 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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