Model
To involve the drivers as comprehensive as possible, four perspectives
form the heart of EGUS model. According to niche theories and
observations, first, each species should have a unique optimal niche.
The regular spacing of niches wherein niche differences exceed a
threshold, as in limiting similarity created by competitive exclusion
(Abrams 1983; Schwilk & Ackerly 2005; Gravel et al . 2006),
supports the coexistence of competitive winners which definitively and
stably contribute to the richness. Immigrant niches can exist in the
intervals among native species niches and can be created after the
competitive exclusion of natives (Schwilk & Ackerly 2005; Gravelet al . 2006). In this scenario, the regular spacing of niches
should be the regular spacing of optimal niches. If a species is
competitively excluded in the spaces with its optimal niche, the species
need not to be considered because it only can trigger a fluctuation of
richness. Second, most species perform well in moderate environments
(Rohde 1992; Rahbek 1995; Nogues-Bravo et al . 2008; Mellardet al . 2012; Mandal et al . 2018), and optimal niches
should gather at the moderate environmental range and should normally
distribute along stressful level. According to neutral theories and some
observations (Hubbell 2005; Gravel et al . 2006; Mellard et
al . 2012; Mandal et al . 2018), third, the averages of values of
birth rates, death rates, dispersal breaths and environmental capacities
of species are equivalent among different areas of an ecosystem and
species should be equivalent in niche breaths. Forth, the individual
birth, death and dispersal of a species should be random and thereby the
abundance should be random.
An area with a high EGUS exhibits a large environmental range, a high
environmental gradient, a low space size occupied by each niche and a
low environmental capacity of each species. Population extinction risks
increase with decreasing population sizes caused by declining
environmental capacities (Annette 2005; He 2012). Then
\(E_{\text{range}}=E_{\max}-E_{\min}\) (1)
\(E_{k}=E_{\min}+\frac{k}{K}E_{\text{range}}\) (2)
where Erangeis the environmental range of a random area in an ecosystem.Emin and Emax are the
environmental minimum and maximum of the area, respectively. The area
has K sections and K is a linear function ofErange . A section has an environmental value and
thereby is an optimal niche. Ek is the
environmental value for a random section k . J sites
distribute among K sections and section k hasJk sites. Each site is occupied by one individual
and the numbers of sites are same among sections due to the same
environmental capacity.
Because of the randomness of arrival time and abundance of species at
initial stages of community formations, the space with the optimal niche
of a species is not necessary to be totally occupied by the individuals
of this species. Thus, algal individuals should often randomly die due
to deterministic processes such as the mismatch between species
preferences and environments and competitive exclusions and
stochastically emigrate. The abundance of each species is random, and
therefore the decreases of richness or stabilization after individual
death or emigration have been considered. Further, if vacant sites
created by random death and emigration in section k are occupied
by immigrants from other sections or other areas but not native
offspring in section k , the richness in the area including
section k could profoundly change. Then based on previous studies
(Schwilk & Ackerly 2005; Kadmon & Allouche 2007; Allouchea et
al 2012; Bar-Massada 2015), the species composition in section kover time is:
\(N_{k,t+1}=\left\lfloor(1-m)N_{k,t}\right\rfloor+B_{k,t}+I_{k,t}\)(3)
where Nk ,t andNk ,t +1 are the abundances
of all species in section k at time t and t +1,
respectively. m is the mortality and emigrant rate, and therefore
is the proportion of vacant sites to the total sites in section kat time t . Because the species randomly distributes among areas
at initial community formation, the averages of death and emigrant rates
of species are equivalent among sections, m is same among
sections. The term in the bracket is a floor function, and the abundance
is integer. Bk ,t is the
abundance of native offspring from time t to time t +1.Ik ,t is the abundance of
immigrant species from time t to time t +1.
The potential proportion of vacant sites in section k being
occupied by immigrants is:
\(p_{k}=\frac{c}{c+r\left(\frac{J_{k}}{J}\right)}\) (4)
where c is immigration rate and thus is the proportion of
immigrant abundance in total abundance. Because random and equivalent
dispersal, c is same among sections. r is the birth rates
of native species in section k and thus is the proportion of
abundance of all new offspring in the abundance of all natives in
section k . Again, because the species randomly distributes, the
average probabilities of native births are same among sections, ris same.
Dispersal limitation and environmental filter are important stochastic
and deterministic processes posited by neutral and niche theories
(Hubbell 2005; Gravel et al . 2006; Levine & HilleRisLambers
2009), respectively. Then the survival rate of immigrants in sectionk is:
\(R_{s,k}=\frac{({\sum_{x=1}^{K}{D_{x,k}N_{s,x}}+\sum_{l=1}^{L}{D_{l,k}N_{s,l}})F}_{s,k}}{\sum_{s=1}^{S}{({\sum_{x=1}^{K}{D_{x,k}N_{s,x}}+\sum_{l=1}^{L}{D_{l,k}N_{s,l}})F}_{s,k}}}\)(5)
where Dx,k is
the
possibility of the arrival of a random species s from sectionx to section k , and x and k are in the same
area. Because the focus is local community and algal movability is high,
algal individuals can spread freely among sections in an area and the
value of Dx,k is set as 100%.Ns,x is the abundance of species s in
section x . L is the number of adjacent areas wherein
species may be able to arrive at the section k ,Dl,k is the possibility of the arrival of species
from another area l to section k (as calculated by
equation 6). Ns,l is the abundance of speciess in area l . Fs,k is the tolerance
of species s to the environmental value of section k (as
calculated by equation 7). S is the total species richness in the
simulated system.
Dispersal limitation: the possibility of arrival of an immigrant from
another area l to section k is:
\(D_{l,k}=e^{\frac{-\left(l-k\right)^{2}}{{2\theta}_{s}^{2}}}\)(6)
where θs is the dispersal breadth of speciess and is equivalent among species.
Environmental filter: species
tolerance to the environment of section k is :
\(F_{s,k}=e^{\frac{-\left(e_{s}-E_{k}\right)^{2}}{{2\sigma}_{s}^{2}}}\)(7)
where es is the niche optimum of species sand is different among species.ϭsis the niche breadth of species s and is equivalent among
species.
J was set as 300, and thus, the abundance of species in an area
was 300. The species richness (S ) in the simulated ecosystem was
set as 300. K = 100 × Erange . All species
had a same birth rate (r ) of 10, a same mortality rate (m )
of 0.25, a same immigration rate (c ) of 0.2, and a same niche
width (ϭ ) of 0.4. Moderate environmental values were set as
between 0 and 2, with values between 0 and 1 containing the highest
number of optimal niche.
Two simulation methods were used. In the first method, the areas were
divided into heterogeneous, transitional, and homogeneous according to
high EGUS, middle EGUS, and low EGUS. The environmental rangesErang s of these areas were 60%, 40%, and 20%
of the entire environmental range (0.01-10) of the simulated ecosystem,
and thus, the values of the range were 6, 4 and 2, respectively. The
environmental stress level of each area was varied (Table 1). In the
second method, the Erang exhibited seven levels
including 0.01 and 1 to 6, and the environmental minimumEmin of each Erang was a
random value between 0 and 3. Thus, the simulations should cover the
possible combinations of environmental gradient levels and stressful
levels and thereby should cover almost all natural cases such as areas
exhibit a high stress with a large environmental range and thus a high
environmental gradient, areas exhibit a high stress with a narrow range
and thus a low gradient, and areas exhibit a low stress with various
ranges and gradients.
The species in each area were randomly sampled when the simulations
started to run. The model was run over 1,000 time steps, such that
species richness stabilized by the end of the run. The replications of
the first and second method were 10 and 30, respectively. ANOVA followed
by Tukey’s HSD test was employed to analyze the effects of environmental
stress levels and environmental range on richness in the results of the
first method. Negative binomial regression was employed to analyze the
relationship between EGUS and species richness in the results of the
second method. The significance was 0.05. The model and results were
evaluated in the R programming system (Seattle, WA: MathSoft, Inc).