In EcoEvoEducation/EcoEvoApps: What the package does (short line)
library(shiny)
library(shinyapps)
#devtools::install_github("EcoEvoEducation/EcoEvoApps")
Introduction
This tutorial walks you through the logic and math behind coexistence of two annual plants via the storage effect (see here for Wikipedia description). The basic idea behind the storage effect is that two species (or more) can coexist in fluctuating environment because no one species "performs" best under all environmental conditions. So, species partition their niche space in time rather than in space or resources. There are three conditions that must be met for the storage effect to operate:
-
Buffered population growth: the species must have a way to avoid population declines during "bad" years. Many species achieve this by having long life spans with adult stages that are relatively unaffected by "bad" conditions. For example, adult trees tend to not be affected by single-year droughts or grass fires. So even all the seedlings in a population are killed by fire or drought, the adults remain to buffer the population until conditions improve.
-
Species-specific responses to environmental conditions: species must have non-identical responses to environmental conditions so that "good" and "bad" years are partitioned among species (to some extent). For example, annual plant species tend to have unique optimal temperatures for germination so that in any given year one species will have relatively high germination compared to other species. The storage effect can operate so long as the correlation between species' environmental responses are not perfectly correlated. But the strength of the storage effect increases as their correlation decreases.
-
Covariance between environment and competition: basically, the idea here is that competition must be stronger in a year that is good for a species compared to a year that is bad. In most cases this occurs, in part, when condition 2 is satisfied because it partitions intraspecific and interspecific competition into different years. So when a spceies experiences a good year, intraspecific competition must be more strongly negative than in a bad year to stabilize coexistence. This tends to happen because in good years the population increases and with it competition since the overall competitive effect is $\alpha N$.
Mathematical example
The storage effect is most easily described mathematically by a two-species annual plant model where the species compete for shared resources but have species-specific germination rates that depend on environmental conditions. In the model below, $N$ is the number of seeds in the seedbank at time $t$. All other parameters are described in the table below the model equations.
$$
N_{1,t+1} = s_{1}[1-g_{1}(E_{t})]N_{1,t} + \frac{\lambda_{1}g_{1}(E_{t})N_{1,t}}{1+ \alpha_{11}g_{1}(E_{t})N_{1,t} + \alpha_{12}g_{2}(E_{t})N_{2,t}} \
N_{2,t+1} = s_{2}[1-g_{2}(E_{t})]N_{2,t} + \frac{\lambda_{2}g_{2}(E_{t})N_{2,t}}{1+ \alpha_{21}g_{2}(E_{t})N_{2,t} + \alpha_{22}g_{2}(E_{t})N_{2,t}}
$$
| Parameter | Definition |
|---|---|
| $s$ | survival of seeds in seedbank (proportion) |
| $g$ | fraction of seeds that germinate (dependent on $E_{t}$) |
| $E_{t}$ | environmental driver of germination rate ($g$; see description below) |
| $\lambda$ | per capita fecundity (number of seeds produced per individual plant) |
| $\alpha$ | per capital competitive effect ($\alpha_{ii}$ is intraspecific; $\alpha_{ij}$ is interspecific) |
Key to having the storage effect operate in this annual plant model with two species is that each species has unperfectly correlated responses to the environment. In this model we include this environmental dependence throught the germination term $g$ that is dependent on the environment $E$ (this is written mathematically as $g(E_{t})$). So, first we need to simulate a randomly fluctuating environment. To do this we can make random draws from a multivariate normal distribution with means of 0 and variance-covariance structure of
$$
\begin{align}
\begin{bmatrix}
\sigma^2_{E} & \rho\sigma^2_{E} \
\rho\sigma^2_{E} & \sigma^2_{E}
\end{bmatrix}
\end{align}
$$
where $\sigma^2_{E}$ is the variance of the environment and $\rho$ is the correlation between the two species' germination rates. The value of $E$ returned from the random draw can be converted to a germination rate ($g(E_{t})$) in the range 0-1 by using an inverse logit transformation: $g(E) = e^E/(1+e^E)$.
Implementing the model in R
To implement the storage effect model in R we can make a function called get_new_n that takes parameters we define and the number of current seeds in the seedbank ($N_{t}$) to calculate the expected numnber of seeds at the next time step ($N_{t+1}$). Here we go.
get_new_n <- function(params, N1, N2){
# Define parameters from the params list
s1 <- params$s1
s2 <- params$s2
g1 <- params$g1
g2 <- params$g2
lambda1 <- params$lambda1
lambda2 <- params$lambda2
alpha11 <- params$alpha11
alpha12 <- params$alpha12
alpha22 <- params$alpha22
alpha21 <- params$alpha21
# Simulate the model one time step
newN1 <- s1*(1-g1)*N1 + ((lambda1*g1*N1) / (1 + alpha11*g1*N1 + alpha12*g2*N2))
newN2 <- s2*(1-g2)*N2 + ((lambda2*g2*N2) / (1 + alpha21*g1*N1 + alpha22*g2*N2))
# Collect the output and return the values
newNs <- c(newN1, newN2)
return(newNs)
}
Let's go through this line-by-line. The very first line says we want to create a function called get_new_n with params, N1, and N2 as required inputs. We then have a bunch lines that just take the parameters fed in by params and redefine them as simpler variables for the equation (e.g., s1 <- params$s1 is just setting s1 equal to whatever was fed in via the params list for s1). Then we actually get to simulating the model one time step. Notice how the R code version perfectly mirrors the equations we presented above. Here they are together for species 1 ($N_{1}$):
$$
N_{1,t+1} = s_{1}[1-g_{1}(E_{t})]N_{1,t} + \frac{\lambda_{1}g_{1}(E_{t})N_{1,t}}{1+ \alpha_{11}g_{1}(E_{t})N_{1,t} + \alpha_{12}g_{2}(E_{t})N_{2,t}}
$$
newN1 <- s1*(1-g1)*N1 + ((lambda1*g1*N1) / (1 + alpha11*g1*N1 + alpha12*g2*N2))
The get_new_n function only calculates $N_{t+1}$, so to run this model for several time steps we need to embed the function in a loop and redefine $N$ and $g$ as we go to make it dynamic. Why does $g$ vary through time rather then being fixed? For the storage effect to operate we need $g_{1}(E_{t})$ to covary with $g_{2}(E_{t})$ through time so that each species has unique germination rates each year due to environmental fluctuations (defined by $E$). To simulate these values we'll define another function (get_g_series) that creates a series of germination rates (a series as long as the simulation we wish to run) based on the method we described above of making random draws from a multivariate normal distribution. Here's how we do this in R:
get_g_series <- function(sigE, rho, nTime){
varcov <- matrix(c(sigE, rho*sigE, rho*sigE, sigE), 2, 2) # variance-covariance matrix
e <- rmvnorm(n = nTime, mean = c(0,0), sigma = varcov) # make 'n' random draws
g <- exp(e) / (1+exp(e)) # inverse-logit transformation to convert to rates
return(g)
}
So we created a function called get_g_series that requires sigE (the environmental variance), rho (the correlation between species' germination rates), and nTime (the number of time steps for which we will simulate the model). Together, the get_new_n and get_g_series define the core of the model. All that's left is to define the non-varying parameters and the initial seed bank conditions.
params <- list() # initialize an empty list to hold parameters
# set all global parameters
params$s1 <- 0.5 #spp 1 seedling survival
params$s2 <- 0.5 #spp 2 seedling survival
params$lambda1 <- 100 #spp 1 per capita fecundity
params$lambda2 <- 100 #spp 2 per capita fecundity
params$alpha11 <- 1 #spp 1 per capita competitive effect on itself
params$alpha12 <- 1 #spp 2 per capita competitive effect on spp 1
params$alpha21 <- 1 #spp 2 per capita competitive effect on itself
params$alpha22 <- 1 #spp 1 per capita competitive effect on spp 2
# set parameters for germination rate series
sigE <- 2 #environmental variance
rho <- 0 #correlation between species' germination rates
# set simulation length and initial seed bank numbers
timeSim <- 100 #number of years to simulate
initialN <- 100 #initial seed abundance in the seed bank (for both species)
Then all that's left is to simulate the model through time. We do this by first getting a series of germination rates via the get_g_series function and then embedding the get_new_n function within a loop to update seed bank abundance based on that year's germination rates, the previous year's seed abundance, and the model parameters.
Nsave <- matrix(nrow=timeSim, ncol=2)
Nsave[1,] <- initialN
for(t in 2:timeSim){
# Set seed abundances to previous year's abundance
N1 <- Nsave[t-1,1]
N2 <- Nsave[t-1,2]
# Set germination rates from the germination rate vectors
params$g1 <- gVec1[t]
params$g2 <- gVec2[t]
# Call the model function
newN <- get_new_n(params = params, N1 = N1, N2 = N2)
# Save the new seed abundance values
Nsave[t,1] <- newN[1]
Nsave[t,2] <- newN[2]
}
Play with the model parameters
#library(shiny)
#library(shinyapps)
library(ggplot2)
library(mvtnorm)
library(EcoEvoApps)
shinyAppDir(
system.file("examples/03_StorageEffect", package="EcoEvoApps"),
options=list(
width="100%", height=550
)
)
EcoEvoEducation/EcoEvoApps documentation built on May 6, 2019, 3:10 p.m.
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library(shiny) library(shinyapps) #devtools::install_github("EcoEvoEducation/EcoEvoApps")
Introduction
This tutorial walks you through the logic and math behind coexistence of two annual plants via the storage effect (see here for Wikipedia description). The basic idea behind the storage effect is that two species (or more) can coexist in fluctuating environment because no one species "performs" best under all environmental conditions. So, species partition their niche space in time rather than in space or resources. There are three conditions that must be met for the storage effect to operate:
-
Buffered population growth: the species must have a way to avoid population declines during "bad" years. Many species achieve this by having long life spans with adult stages that are relatively unaffected by "bad" conditions. For example, adult trees tend to not be affected by single-year droughts or grass fires. So even all the seedlings in a population are killed by fire or drought, the adults remain to buffer the population until conditions improve.
-
Species-specific responses to environmental conditions: species must have non-identical responses to environmental conditions so that "good" and "bad" years are partitioned among species (to some extent). For example, annual plant species tend to have unique optimal temperatures for germination so that in any given year one species will have relatively high germination compared to other species. The storage effect can operate so long as the correlation between species' environmental responses are not perfectly correlated. But the strength of the storage effect increases as their correlation decreases.
-
Covariance between environment and competition: basically, the idea here is that competition must be stronger in a year that is good for a species compared to a year that is bad. In most cases this occurs, in part, when condition 2 is satisfied because it partitions intraspecific and interspecific competition into different years. So when a spceies experiences a good year, intraspecific competition must be more strongly negative than in a bad year to stabilize coexistence. This tends to happen because in good years the population increases and with it competition since the overall competitive effect is $\alpha N$.
Mathematical example
The storage effect is most easily described mathematically by a two-species annual plant model where the species compete for shared resources but have species-specific germination rates that depend on environmental conditions. In the model below, $N$ is the number of seeds in the seedbank at time $t$. All other parameters are described in the table below the model equations.
$$ N_{1,t+1} = s_{1}[1-g_{1}(E_{t})]N_{1,t} + \frac{\lambda_{1}g_{1}(E_{t})N_{1,t}}{1+ \alpha_{11}g_{1}(E_{t})N_{1,t} + \alpha_{12}g_{2}(E_{t})N_{2,t}} \ N_{2,t+1} = s_{2}[1-g_{2}(E_{t})]N_{2,t} + \frac{\lambda_{2}g_{2}(E_{t})N_{2,t}}{1+ \alpha_{21}g_{2}(E_{t})N_{2,t} + \alpha_{22}g_{2}(E_{t})N_{2,t}} $$
| Parameter | Definition | |---|---| | $s$ | survival of seeds in seedbank (proportion) | | $g$ | fraction of seeds that germinate (dependent on $E_{t}$) | | $E_{t}$ | environmental driver of germination rate ($g$; see description below) | | $\lambda$ | per capita fecundity (number of seeds produced per individual plant) | | $\alpha$ | per capital competitive effect ($\alpha_{ii}$ is intraspecific; $\alpha_{ij}$ is interspecific) |
Key to having the storage effect operate in this annual plant model with two species is that each species has unperfectly correlated responses to the environment. In this model we include this environmental dependence throught the germination term $g$ that is dependent on the environment $E$ (this is written mathematically as $g(E_{t})$). So, first we need to simulate a randomly fluctuating environment. To do this we can make random draws from a multivariate normal distribution with means of 0 and variance-covariance structure of
$$ \begin{align} \begin{bmatrix} \sigma^2_{E} & \rho\sigma^2_{E} \ \rho\sigma^2_{E} & \sigma^2_{E} \end{bmatrix} \end{align} $$
where $\sigma^2_{E}$ is the variance of the environment and $\rho$ is the correlation between the two species' germination rates. The value of $E$ returned from the random draw can be converted to a germination rate ($g(E_{t})$) in the range 0-1 by using an inverse logit transformation: $g(E) = e^E/(1+e^E)$.
Implementing the model in R
To implement the storage effect model in R we can make a function called get_new_n that takes parameters we define and the number of current seeds in the seedbank ($N_{t}$) to calculate the expected numnber of seeds at the next time step ($N_{t+1}$). Here we go.
get_new_n <- function(params, N1, N2){ # Define parameters from the params list s1 <- params$s1 s2 <- params$s2 g1 <- params$g1 g2 <- params$g2 lambda1 <- params$lambda1 lambda2 <- params$lambda2 alpha11 <- params$alpha11 alpha12 <- params$alpha12 alpha22 <- params$alpha22 alpha21 <- params$alpha21 # Simulate the model one time step newN1 <- s1*(1-g1)*N1 + ((lambda1*g1*N1) / (1 + alpha11*g1*N1 + alpha12*g2*N2)) newN2 <- s2*(1-g2)*N2 + ((lambda2*g2*N2) / (1 + alpha21*g1*N1 + alpha22*g2*N2)) # Collect the output and return the values newNs <- c(newN1, newN2) return(newNs) }
Let's go through this line-by-line. The very first line says we want to create a function called get_new_n with params, N1, and N2 as required inputs. We then have a bunch lines that just take the parameters fed in by params and redefine them as simpler variables for the equation (e.g., s1 <- params$s1 is just setting s1 equal to whatever was fed in via the params list for s1). Then we actually get to simulating the model one time step. Notice how the R code version perfectly mirrors the equations we presented above. Here they are together for species 1 ($N_{1}$):
$$ N_{1,t+1} = s_{1}[1-g_{1}(E_{t})]N_{1,t} + \frac{\lambda_{1}g_{1}(E_{t})N_{1,t}}{1+ \alpha_{11}g_{1}(E_{t})N_{1,t} + \alpha_{12}g_{2}(E_{t})N_{2,t}} $$
newN1 <- s1*(1-g1)*N1 + ((lambda1*g1*N1) / (1 + alpha11*g1*N1 + alpha12*g2*N2))
The get_new_n function only calculates $N_{t+1}$, so to run this model for several time steps we need to embed the function in a loop and redefine $N$ and $g$ as we go to make it dynamic. Why does $g$ vary through time rather then being fixed? For the storage effect to operate we need $g_{1}(E_{t})$ to covary with $g_{2}(E_{t})$ through time so that each species has unique germination rates each year due to environmental fluctuations (defined by $E$). To simulate these values we'll define another function (get_g_series) that creates a series of germination rates (a series as long as the simulation we wish to run) based on the method we described above of making random draws from a multivariate normal distribution. Here's how we do this in R:
get_g_series <- function(sigE, rho, nTime){ varcov <- matrix(c(sigE, rho*sigE, rho*sigE, sigE), 2, 2) # variance-covariance matrix e <- rmvnorm(n = nTime, mean = c(0,0), sigma = varcov) # make 'n' random draws g <- exp(e) / (1+exp(e)) # inverse-logit transformation to convert to rates return(g) }
So we created a function called get_g_series that requires sigE (the environmental variance), rho (the correlation between species' germination rates), and nTime (the number of time steps for which we will simulate the model). Together, the get_new_n and get_g_series define the core of the model. All that's left is to define the non-varying parameters and the initial seed bank conditions.
params <- list() # initialize an empty list to hold parameters # set all global parameters params$s1 <- 0.5 #spp 1 seedling survival params$s2 <- 0.5 #spp 2 seedling survival params$lambda1 <- 100 #spp 1 per capita fecundity params$lambda2 <- 100 #spp 2 per capita fecundity params$alpha11 <- 1 #spp 1 per capita competitive effect on itself params$alpha12 <- 1 #spp 2 per capita competitive effect on spp 1 params$alpha21 <- 1 #spp 2 per capita competitive effect on itself params$alpha22 <- 1 #spp 1 per capita competitive effect on spp 2 # set parameters for germination rate series sigE <- 2 #environmental variance rho <- 0 #correlation between species' germination rates # set simulation length and initial seed bank numbers timeSim <- 100 #number of years to simulate initialN <- 100 #initial seed abundance in the seed bank (for both species)
Then all that's left is to simulate the model through time. We do this by first getting a series of germination rates via the get_g_series function and then embedding the get_new_n function within a loop to update seed bank abundance based on that year's germination rates, the previous year's seed abundance, and the model parameters.
Nsave <- matrix(nrow=timeSim, ncol=2) Nsave[1,] <- initialN for(t in 2:timeSim){ # Set seed abundances to previous year's abundance N1 <- Nsave[t-1,1] N2 <- Nsave[t-1,2] # Set germination rates from the germination rate vectors params$g1 <- gVec1[t] params$g2 <- gVec2[t] # Call the model function newN <- get_new_n(params = params, N1 = N1, N2 = N2) # Save the new seed abundance values Nsave[t,1] <- newN[1] Nsave[t,2] <- newN[2] }
Play with the model parameters
#library(shiny) #library(shinyapps) library(ggplot2) library(mvtnorm) library(EcoEvoApps) shinyAppDir( system.file("examples/03_StorageEffect", package="EcoEvoApps"), options=list( width="100%", height=550 ) )
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