A cost–benefit model for plant–plant interactions: a density-series tool to detect facilitation

Abstract

Generally, only the net outcome of plant–plant interactions is measured in population and community ecology research, with few attempts to determine the relative importance of negative (competition) and positive (facilitation) interactions between subordinate species. Changes in the intensity of interactions along gradients, between life-stages, or with changing densities, and the use of selective removals enhance our capacity to infer positive and negative interactions. However, the most powerful examples at least in detecting facilitation typically involve measuring changes with or without a nurse-plant or benefactor species and often involve only a very limited numbers of species. In plant competition studies, however, greater number of species are commonly tested and density-dependent series are not an uncommon tool to test for net negative interactions. Here, we develop a cost–benefit model that can be used to comprehensively calculate the average expected net gain per individual at every point in a density series provided several response variables are recorded at each density. The utility of this model is demonstrated using both hypothetical data and several empirical data sets, and it is used to infer either both positive and negative net effects. Expected net gain can also serve as an accurate estimate of mean fitness per individual at a given density provided appropriate performance measures were recorded within the primary study. Within a single density series, both facilitation and competition can occur and were detectable using this method. This approach emphasizes the current view that both negative and positive interactions play a role in shaping plant communities. Furthermore, it is evident that facilitation can be detected using the manipulative density series typically associated with competition studies and not just using the typical target nurse-plant methodology. Finally, this method is a significant advance over the current practice of tallying up single responses within a study to estimate outcomes by providing a single, synthetic measure of the net gain or cost of interactions.

Appendix

A vignette for the ecological cost–benefit analyses from density series including the calculation of variance of products.

Description

Two functions were developed to conduct ecological cost–benefit analyses on plant data for density series. The equation (Eq. 1) was developed to integrate more than one estimate/measure of plant success, (e.g., fitness, survival, or emergence), represented as expected gain or W as fitness if there are three representative measures within a primary dataset; where W is the product of multiple plant estimates. Variance of W was based on Goodman (1960, 1962) and is calculated with Eq. 2 when there are two products in W and with Eq. 3 when three products are present in W. To calculate W and the Variance of W (Var.W), the mean and variance of the sample populations are required for all density levels. Other factors can also be included in the analysis including species, year, and population. The input data must be a matrix with two columns specified as “mean” and “se” to represent means and variance for plant estimates at the associated density.

Usage

W.2 (x, estimates)

W.3 (x, density, population, year)

Arguments

X—an object of class “data.frame”, with two columns labeled “mean” and “se”.

Estimates—a vector of the plant estimate or measure that is to be multiplied to derive W.

Density—a vector of the plant densities found within rows.

Population—a vector specifying a subset of observations to be aggregated by for W (e.g., species, population, survey method)

Year—an option vector to subset the observations further by year.

Value

W.2 and W.3 each returns an object of class “data.frame” with columns representing the original data except without columns for the estimate, mean, or se. The last two columns of the data frame represent W from Eq. 1, and Var(W) from Eqs. 2 or 3 depending on function used.

Functions

> W.2 <- function(x, estimates){

> est2 <- match(x[,estimates], levels(x[,estimates]))

> x1 <- subset(x, est2==1)

> y1 <- subset(x, est2==2)

> W <- x1$mean*y1$mean

> var.w <- (((x1$mean^2)*y1$se)+((y1$mean^2)*x1$se))-(x1$se*y1$se)

> W.2 <- cbind(x1[,1:(ncol(x1)-3)],W,var.w)

> return(W.2)

> }

>W.3 <- function(x, density, population, year){

>#calculate W

>if(missing(year)){

>W <- setNames(aggregate(x$mean, by=list(x[,density],x[,population]), prod), c(“Density”,”Population”,”W”))

>x[“var.w”] <- x$se/x$mean

>var.w <- setNames(aggregate(x$var.w, by=list(x[,density],data[,population]), sum), c(“Density”,”Population”,”Var.W”))

>var.w[,4] <- var.w[,4]*W[,4]

>metrics <- cbind(W,Var.w=var.w[,4])

>return(metrics)

>}

>else

>{

>W <- setNames(aggregate(x$mean, by=list(x[,density],x[,year],x[,population]), prod), c(“Density”,”Year”,”Population”,”W”))

>}

>#calculate variance

>x[“var.w”] <- x$se/x$mean

>var.w <- setNames(aggregate(x$var.w, by=list(x[,density],data[,year],data[,population]), sum), c(“Density”,”Year”,”Population”,”Var.W”))

>var.w[,4] <- var.w[,4]*W[,4]

>metrics <- cbind(W,Var.w=var.w[,4])

>return(metrics)

>}

Examples

## Lortie & Turkington 2002. J. Ecol.

data(lortie2002)

Cost.benefit <- W.2(lortie2002, estimates= “factor”)

Cost.benefit

data(lortie2002)

Cost.benefit <- W.3(lortie2002, density=“seed.density”, population=“Population”, year=“year”)

Cost.benefit