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).