Environmental Engineering Reference
In-Depth Information
At each time step, new invaders and every resident each have species-specifi c
probability b of producing a propagule. Each propagule has small probability ν of
speciation (following [1]). The sample community additionally receives immigrant
propagules of its resident species that arrive from the wider meta-community in pro-
portion to their expected numbers out there ([ K m / K-1 ] n i ), assuming the same density
n i / K of each species i as in the sample community, and in proportion to their prob-
ability ( K / K m ) of landing within the sample community, and modifi ed by a dispersal
limitation parameter ω. In effect, for each resident species in the community, [(1- K /
K m ) n i ] 1-ω external residents each produce an immigrating propagule with probability
b i . Thus if K m >> K and ω = 0, a colonist is just as likely to be an immigrant from outside
as produced from within the sample community (no dispersal limitation, following
[1]). This likelihood reduces for ω > 0, and also for smaller K m . None of the propagules
generated within the sample community emigrate out into the meta-community, mak-
ing K a sink if smaller than K m (sensu [34]), or a closed community if equal to K m . The
simulation is thus conceptually equivalent to randomly assembled S -species systems
previously studied (e.g., [35]), except that it additionally accommodates a random drift
of invasions to sustain the dynamics of recruitment following deaths and extinctions.
Each propagule lands on a random patch within the sample community and es-
tablishes there only if (a) its species is not already present, and (b) it beats each prob-
ability α ij of repulsion by each other resident species j , and (c) it either beats the odds
on repulsion by all other propagules simultaneously attempting to colonize the patch,
or benefi ts from the random chance of being the fi rst arrival among them. Each pre-
established resident risks death with species-specifi c probability d i = b i / R i at each time
step. Each patch has probability X of a catastrophic hazard at each time step that ex-
tirpates all its occupants. The model thus captures the principles of stochastic niche
theory [21, 22] and pre-emptive advantage [20].
Each of the replicate communities contributing to distributions and relationships
in Figures 3-5 is represented by values averaged over time-steps 401-500, long after
the asymptote of species richness. For all graphs in Figures 3-5, meta-community car-
rying capacity K m = 10 6 , dispersal limitation parameter ω = 0.5, speciation probability
per resident propagation event ν = 10 −12 , two invasion attempts per time-step (setting
Hubbell's [1] fundamental biodiversity number θ~4 independently of K m ), probability
of catastrophe per patch X = 0.01. For neutral communities, all species take competi-
tion coeffi cients α = 1, individual intrinsic propagation probability b = 0.5, individual
intrinsic lifetime reproduction R = 1.5 (so lifespan R / b = 3); for Lotka-Volterra com-
munities, each species i takes random 0 ≤ α ij ≤ 1, random 0 ≤ b i ≤ 1, R i between 1.2 and
1.8 and proportional to dominance rank; dominant-fugitive communities are as Lotka-
Volterra except for random binary α ij = 0 or 1. All scenarios are thereby sampled from a
large meta-community with moderate dispersal limitation, low extrinsic mortality, and
suffi cient invasions to sustain a reasonably high asymptote of species richness from
the starting point of two species each occupying fi ve patches. Skew in the lognormal
distribution of species abundances (Figure 3) was measured for each replicate in its
dimensionless third moment about the mean, g 1 [36], and confi dence limits for the
sample of six values were tested against H 0 : g 1 = 0.
 
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