Ecologists use different mathematical statistical models to explain how resources are partitioned. Here are some (very limited) examples of ecological theories and mechanisms that we can test by analyzing and modeling SADs:

Niche Preemption Model (Geometric Series)

• Assumes a dominant species takes a fixed fraction of total resources, the next takes the same fraction of the remainder, and so on.
• Common in species-poor or highly stressed environments.

Log-Normal & Log-Series Distributions

• Fisher’s Log-Series: Predicts a massive number of ultra-rare species. Often fits highly diverse tropical environments beautifully.
• Preston’s Log-Normal: Assumes abundances follow a normal distribution when log-transformed.

-Probability Modeling: We consider raw field counts as random variables or a random sample of the regional metacommunity. We test if this empirical data matches a specific theoretical probability density function.

-Quantifying Uncertainty: Even in very large samples like the BCI data, a sample rarely captures the whole “truth.” Thankfully, SAD analysis explicitly models sampling error. For example we can calculate “The Veil Line” to determine how many rare species we expect to have missed in our data collection. If a rare species is absent in the empirical data, we can gain insight as to whether it is truly absent from a regional species pool, or whether it was missed due to sample size restrictions.

-Model Selection Frameworks: Finding the model of “best-fit” is not about eyeballing to see which function best fits the data. Instead we calculate values like AIC (Akaike Information Criterion) to compare which model best fits the data. These metrics are considered superior to \(R^2\) values because they balance goodness-of-fit with model complexity. AIC, for example, will penalize extra complexity in a model unless it significantly improves the fit of the model to the data.

\[AIC=2k-2ln(\hat L)\]

Barro Colorado Island is a protected research area located on the Panama Canal. Biologists have been studying the island and testing ecological theory there for over 100 years. Within BCI, there is a 50-hectare plot where the tropical tree community has been censused repeatedly since 1982. We can find a snapshot of this data within the vegan package for R. Below we can see an example of the observed abundance data. In order to visualize this distribution we use the x-axis to sort the species by abundance rank or abundance classes. The y-axis represents the observed abundance.

radfit(species.totals) #Results of the best-fit analysis using the vegan package

RAD models, family poisson 
No. of species 225, total abundance 21457

           par1      par2     par3    Deviance AIC      BIC     
Null                                  10261.14 11387.97 11387.97
Preemption  0.034063                   3788.38  4917.21  4920.63
Lognormal   3.3569    1.5738            744.30  1875.13  1881.96
Zipf        0.14679  -0.94912          4335.50  5466.33  5473.16
Mandelbrot  17.014   -2.0064   15.048   988.02  2120.85  2131.10

veiledspec(pres.fit) #Tells us how many species we expect to have missed in sampling

Extrapolated     Observed       Veiled 
   235.40577    225.00000     10.40577 

fit.logseries <- fitsad(species.totals,sad = 'ls')
fit.lognormal <- fitsad(species.totals,sad = 'lnorm')
aic.ls <- AIC(fit.logseries)
aic.ln <- AIC(fit.lognormal)

cat("Fisher's Logseries AIC",aic.ls)

Fisher's Logseries AIC 2322.9

cat("Preston's Lognormal AIC",aic.ln)

Preston's Lognormal AIC 2318.025

Here we can see that Log-Series ‘wins’ with a lower AIC. Notice that the AIC value for the lognormal model calculated with the sads package differs from the one calculated with the vegan package on the previous slide. This is due to the different statistical frameworks used by each package. AIC values should never be compared between packages.

What it means to find the “best-fit” SAD

When we run the sads or vegan package to measure best-fit, we are asking What mathematical and biological laws govern the structure of this ecosystem or community?

1. Optimizing Parameters: The statistical software adjusts the shapes of different curves (like stretching a bell curve or steepening an exponential drop) until the distance between our real BCI data points and the theoretical line is minimized.
2. Decoding Ecological Mechanisms: A statistical fit acts as a proxy for biological processes. If a Log-Series fits best, it implies the community might be driven by neutral, random dispersal events. If a Log-Normal fits best, it implies that many independent, multiplicative ecological factors (like soil quality, sunlight, and moisture niches) are acting together to determine species abundance.

When we perform comparative analysis in this way we are engaging in hypothesis testing. Since the Neutral Theory Model posits that community structure is determined by random death and replacement events, it serves as a great \(H_0\) or null-hypothesis to test against other models.