R: General rate Birth-Death simulation

bd.sim {paleobuddy}

R Documentation

General rate Birth-Death simulation

Description

Simulates a species birth-death process with general rates for any number of starting species. Allows for the speciation/extinction rate to be (1) a constant, (2) a function of time, (3) a function of time and;or an environmental variable, or (4) a vector of numbers representing a step function. Allows for constraining results on the number of species at the end of the simulation, either total or extant. The function can also take an optional shape argument to generate age-dependence on speciation and/or extinction, assuming a Weibull distribution as a model of age-dependence. Returns a sim object (see ?sim). It may return true extinction times or simply information on whether species lived after the maximum simulation time, depending on simulation settings.

Usage

bd.sim(
  n0,
  lambda,
  mu,
  tMax,
  lShape = NULL,
  mShape = NULL,
  envL = NULL,
  envM = NULL,
  lShifts = NULL,
  mShifts = NULL,
  nFinal = c(0, Inf),
  nExtant = c(0, Inf),
  trueExt = FALSE
)

Arguments

`n0`	Initial number of species. Usually 1, in which case the simulation will describe the full diversification of a monophyletic lineage. Note that when `lambda` is less than or equal to `mu`, many simulations will go extinct before speciating even once. One way of generating large sample sizes in this case is to increase `n0`, which will simulate the diversification of a paraphyletic group.
`lambda`	Speciation rate (events per species per million years) over time. It can be a `numeric` describing a constant rate, a `function(t)` describing the variation in speciation over time `t`, a `function(t, env)` describing the variation in speciation over time following both time AND an environmental variable (please see `envL` for details) or a `vector` containing rates that correspond to each rate between speciation rate shift times times (please see `lShifts`). Note that `lambda` should always be greater than or equal to zero.
`mu`	Similar to `lambda`, but for the extinction rate. Note: rates should be considered as running from `0` to `tMax`, as the simulation runs in that direction even though the function inverts speciation and extinction times before returning.
`tMax`	Ending time of simulation, in million years after the clade origin. Any species still living after `tMax` is considered extant, and any species that would be generated after `tMax` is not present in the return.
`lShape`	Shape parameter defining the degree of age-dependency in speciation rate. This will be equal to the shape parameter in a Weibull distribution: as a species' longevity increases (negative age-dependency). When larger than one, speciation rate will increase as a species' longevity increases (positive age-dependency). It may be a function of time, but see note below for caveats therein. Default is `NULL`, equivalent to an age-independent process. For `lShape != NULL` (including when equal to one), `lambda` will be considered a scale (= 1/rate), and `rexp.var` will draw a Weibull distribution instead of an exponential. This means Weibull(rate, 1) = Exponential(1/rate). Note that even when `lShape != NULL`, `lambda` may still be time-dependent.
`mShape`	Similar to `lShape`, but for the extinction rate. Note: Simulations with time-varying shape behave within theoretical expectations for most cases, but if shape is lower than 1 and varies too much (e.g. `0.5 + 0.5*t`), it can be biased for higher waiting times due to computational error. A degree of time dependence of the order of 0.01 events/my^2 are advisable. It might, although rarely, exhibit a small bias when using shape functions with abrupt time variations. In both cases, error is still quite low for the purposes of the package. Note: Shape must be greater than 0. We arbitrarily chose 0.01 as the minimum accepted value, so if shape is under 0.01 for any reasonable time in the simulation, it returns an error.
`envL`	A `data.frame` describing a time series that represents the variation of an environmental variable (e.g. CO2, temperature, available niches, etc) with time. The first column of this `data.frame` must be time, and the second column must be the values of the variable. This will be internally passed to the `make.rate` function, to create a speciation rate variation in time following the interaction between the environmental variable and the function. Note `paleobuddy` has two environmental data frames, `temp` and `co2`. One can check `RPANDA` for more examples, or use their own time series of a variable of interest
`envM`	Similar to `envL`, but for the extinction rate.
`lShifts`	Vector of rate shifts. First element must be the starting time for the simulation (`0` or `tMax`). It must have the same length as `lambda`. `c(0, x, tMax)` is equivalent to `c(tMax, tMax - x, 0)` for the purposes of `make.rate`.
`mShifts`	Similar to `mShifts`, but for the extinction rate.
`nFinal`	A `vector` of length `2`, indicating an interval of acceptable number of species at the end of the simulation. Default value is `c(0, Inf)`, so that any number of species (including zero, the extinction of the whole clade) is accepted. If different from default value, simulation will restart until the number of total species at `tMax` is in the `nFinal` interval. Note that `nFinal` must be a sensible vector. The function will error if its maximum is lower than `1`, or if its length is not `2`.
`nExtant`	A `vector` of length `2`, indicating an interval of acceptable number of extant species at the end of the simulation. Equal to `nFinal` in every respect except for that. Note: The function returns `NA` if it runs for more than `100000` iterations without fulfilling the requirements of `nFinal` and `nExtant`. Note: Using values other than the default for `nFinal` and `nExtant` will condition simulation results.
`trueExt`	A `logical` indicating whether the function should return true or truncated extinction times. When `TRUE`, time of extinction of extant species will be the true time, otherwise it will be `NA` if a species is alive at the end of the simulation. Note: This is interesting to use to test age-dependent extinction. Age-dependent speciation would require all speciation times (including the ones after extinction) to be recorded, so we do not attempt to add an option to account for that. Since age-dependent extinction and speciation use the same underlying process, however, if one is tested to satisfaction the other should also be in expectations.

Details

Please note while time runs from 0 to tMax in the simulation, it returns speciation/extinction times as tMax (origin of the group) to 0 (the "present" and end of simulation), so as to conform to other packages in the literature.

Value

A sim object, containing extinction times, speciation times, parent, and status information for each species in the simulation. See ?sim.

Author(s)

Bruno do Rosario Petrucci.

Examples


# we will showcase here some of the possible scenarios for diversification,
# touching on all the kinds of rates

###
# consider first the simplest regimen, constant speciation and extinction

# initial number of species
n0 <- 1

# maximum simulation time
tMax <- 40

# speciation
lambda <- 0.11

# extinction
mu <- 0.08

# set a seed
set.seed(1)

# run the simulation, making sure we have more than 1 species in the end
sim <- bd.sim(n0, lambda, mu, tMax, nFinal = c(2, Inf))

# we can plot the phylogeny to take a look
if (requireNamespace("ape", quietly = TRUE)) {
  phy <- make.phylo(sim)
  ape::plot.phylo(phy)
}

###
# now let us complicate speciation more, maybe a linear function

# initial number of species
n0 <- 1

# maximum simulation time
tMax <- 40

# make a vector for time
time <- seq(0, tMax, 0.1)

# speciation rate
lambda <- function(t) {
  return(0.05 + 0.005*t)
}

# extinction rate
mu <- 0.1

# set a seed
set.seed(4)

# run the simulation, making sure we have more than 1 species in the end
sim <- bd.sim(n0, lambda, mu, tMax, nFinal = c(2, Inf))

# we can plot the phylogeny to take a look
if (requireNamespace("ape", quietly = TRUE)) {
  # full phylogeny
  phy <- make.phylo(sim)
  ape::plot.phylo(phy)
}

###
# what if we want mu to be a step function?

# initial number of species
n0 <- 1

# maximum simulation time
tMax <- 40

# speciation
lambda <- function(t) {
  return(0.02 + 0.005*t)
}

# vector of extinction rates
mList <- c(0.09, 0.08, 0.1)

# vector of shift times. Note mShifts could be c(40, 20, 5) for identical 
# results
mShifts <- c(0, 20, 35)

# let us take a look at how make.rate will make it a step function
mu <- make.rate(mList, tMax = tMax, rateShifts = mShifts)

# and plot it
plot(seq(0, tMax, 0.1), rev(mu(seq(0, tMax, 0.1))), type = 'l',
     main = "Extintion rate as a step function", xlab = "Time (Mya)",
     ylab = "Rate (events/species/My)", xlim = c(tMax, 0))

# looking good, we will keep everything else the same

# a different way to define the same extinction function
mu <- function(t) {
  ifelse(t < 20, 0.09,
         ifelse(t < 35, 0.08, 0.1))
}

# set seed
set.seed(2)

# run the simulation
sim <- bd.sim(n0, lambda, mu, tMax, nFinal = c(2, Inf))
# we could instead have used mList and mShifts

# we can plot the phylogeny to take a look
if (requireNamespace("ape", quietly = TRUE)) {
  phy <- make.phylo(sim)
  ape::plot.phylo(phy)
}

###
# we can also supply a shape parameter to try age-dependent rates

# initial number of species
n0 <- 1

# maximum simulation time
tMax <- 40

# speciation - note that since this is a Weibull scale,
# the unites are my/events/lineage, not events/lineage/my
lambda <- 10

# speciation shape
lShape <- 2

# extinction
mu <- 0.08

# set seed
set.seed(4)

# run the simulation - note the message saying lambda is a scale
sim <- bd.sim(n0, lambda, mu, tMax, lShape = lShape, nFinal = c(2, Inf))

# we can plot the phylogeny to take a look
if (requireNamespace("ape", quietly = TRUE)) {
  phy <- make.phylo(sim)
  ape::plot.phylo(phy)
}

###
# scale can be a time-varying function

# initial number of species
n0 <- 1

# maximum simulation time
tMax <- 40

# speciation - note that since this is a Weibull scale,
# the unites are my/events/lineage, not events/lineage/my
lambda <- function(t) {
  return(2 + 0.25*t)
}

# speciation shape
lShape <- 2

# extinction
mu <- 0.2

# set seed
set.seed(1)

# run the simulation - note the message saying lambda is a scale
sim <- bd.sim(n0, lambda, mu, tMax, lShape = lShape, nFinal = c(2, Inf))

# we can plot the phylogeny to take a look
if (requireNamespace("ape", quietly = TRUE)) {
  phy <- make.phylo(sim)
  ape::plot.phylo(phy)
}

###
# and shape can also vary with time

# initial number of species
n0 <- 1

# maximum simulation time
tMax <- 40

# speciation - note that since this is a Weibull scale,
# the unites are my/events/lineage, not events/lineage/my
lambda <- function(t) {
  return(2 + 0.25*t)
}

# speciation shape
lShape <- function(t) {
  return(1 + 0.02*t)
}

# extinction
mu <- 0.2

# set seed
set.seed(4)

# run the simulation - note the message saying lambda is a scale
sim <- bd.sim(n0, lambda, mu, tMax, lShape = lShape, nFinal = c(2, Inf))

# we can plot the phylogeny to take a look
if (requireNamespace("ape", quietly = TRUE)) {
  phy <- make.phylo(sim)
  ape::plot.phylo(phy)
}
 
###
# finally, we can also have a rate dependent on an environmental variable,
# like temperature data

# get temperature data (see ?temp)
data(temp)

# initial number of species
n0 <- 1

# maximum simulation time
tMax <- 40

# speciation - a scale
lambda <- 10
# note the scale for the age-dependency could be a time-varying function

# speciation shape
lShape <- 2

# extinction, dependent on temperature exponentially
mu <- function(t, env) {
  return(0.1*exp(0.025*env))
}

# need a data frame describing the temperature at different times
envM <- temp

# by passing mu and envM to bd.sim, internally bd.sim will make mu into a
# function dependent only on time, using make.rate
mFunc <- make.rate(mu, tMax = tMax, envRate = envM)

# take a look at how the rate itself will be
plot(seq(0, tMax, 0.1), rev(mFunc(seq(0, tMax, 0.1))),
     main = "Extinction rate varying with temperature", xlab = "Time (Mya)",
     ylab = "Rate (events/species/My)", type = 'l', xlim = c(tMax, 0))

# set seed
set.seed(2)

# run the simulation
sim <- bd.sim(n0, lambda, mu, tMax, lShape = lShape, envM = envM,
              nFinal = c(2, Inf))

# we can plot the phylogeny to take a look
if (requireNamespace("ape", quietly = TRUE)) {
  phy <- make.phylo(sim)
  ape::plot.phylo(phy)
}

###
# one can mix and match all of these scenarios as they wish - age-dependency
# and constant rates, age-dependent and temperature-dependent rates, etc. 
# the only combination that is not allowed is a step function rate and 
# environmental data, but one can get around that as follows

# get the temperature data - see ?temp for information on the data set
data(temp)

# initial number of species
n0 <- 1

# maximum simulation time
tMax <- 40

# speciation - a step function of temperature built using ifelse()
# note that this creates two shifts for lambda, for a total of 3 values
# throughout the simulation
lambda <- function(t, env) {
  ifelse(t < 20, env,
         ifelse(t < 30, env / 4, env / 3))
}

# speciation shape
lShape <- 2

# environment variable to use - temperature
envL <- temp

# this is kind of a complicated scale, let us take a look

# make it a function of time
lFunc <- make.rate(lambda, tMax = tMax, envRate = envL)

# plot it
plot(seq(0, tMax, 0.1), rev(lFunc(seq(0, tMax, 0.1))),
     main = "Speciation scale varying with temperature", xlab = "Time (Mya)",
     ylab = "Scale (1/(events/species/My))", type = 'l', xlim = c(tMax, 0))

# extinction
mu <- 0.1

# maximum simulation time
tMax <- 40

# set seed
set.seed(1)


# run the simulation
sim <- bd.sim(n0, lambda, mu, tMax, lShape = lShape, envL = envL,
              nFinal = c(2, Inf))

# we can plot the phylogeny to take a look
if (requireNamespace("ape", quietly = TRUE)) {
  phy <- make.phylo(sim)
  ape::plot.phylo(phy)
}
time2 <- Sys.time()


# after presenting the possible models, we can consider how to
# create mixed models, where the dependency changes over time

###
# consider speciation that becomes environment dependent
# in the middle of the simulation

# get the temperature data - see ?temp for information on the data set
data(temp)

# initial number of species
n0 <- 1

# maximum simulation time
tMax <- 40

# time and temperature-dependent speciation
lambda <- function(t, temp) {
  return(
    ifelse(t < 20, 0.1 - 0.005*t,
           0.05 + 0.1*exp(0.02*temp))
  )
}

# extinction
mu <- 0.11

# set seed
set.seed(4)

# run simulation
sim <- bd.sim(n0, lambda, mu, tMax, envL = temp, nFinal = c(2, Inf))

# we can plot the phylogeny to take a look
if (requireNamespace("ape", quietly = TRUE)) {
  phy <- make.phylo(sim)
  ape::plot.phylo(phy)
}

###
# we can also change the environmental variable
# halfway into the simulation

# note below that for this scenario we need make.rate, which
# in general can aid users looking for more complex scenarios
# than those available directly with bd.sim arguments

# get the temperature data - see ?temp for information on the data set
data(temp)

# same for co2 data (and ?co2)
data(co2)

# initial number of species
n0 <- 1

# maximum simulation time
tMax <- 40

# speciation
lambda <- 0.1

# temperature-dependent extinction
m_t1 <- function(t, temp) {
  return(0.05 + 0.1*exp(0.02*temp))
}

# make first function
mu1 <- make.rate(m_t1, tMax = tMax, envRate = temp)

# co2-dependent extinction
m_t2 <- function(t, co2) {
  return(0.02 + 0.14*exp(0.01*co2))
}

# make second function
mu2 <- make.rate(m_t2, tMax = tMax, envRate = co2)

# final extinction function
mu <- function(t) {
  ifelse(t < 20, mu1(t), mu2(t))
}

# set seed
set.seed(3)

# run simulation
sim <- bd.sim(n0, lambda, mu, tMax, nFinal = c(2, Inf))

# we can plot the phylogeny to take a look
if (requireNamespace("ape", quietly = TRUE)) {
  phy <- make.phylo(sim)
  ape::plot.phylo(phy)
}

# note one can also use this mu1 mu2 workflow to create a rate
# dependent on more than one environmental variable, by decoupling
# the dependence of each in a different function and putting those
# together

###
# finally, one could create an extinction rate that turns age-dependent
# in the middle, by making shape time-dependent

# initial number of species
n0 <- 1

# maximum simulation time
tMax <- 40

# speciation
lambda <- 0.15

# extinction - note that since this is a Weibull scale,
# the unites are my/events/lineage, not events/lineage/my
mu <- function(t) {
  return(8 + 0.05*t)
}

# extinction shape
mShape <- function(t) {
  return(
    ifelse(t < 30, 1, 2)
  )
}

# set seed
set.seed(3)

# run simulation
sim <- bd.sim(n0, lambda, mu, tMax, mShape = mShape,
               nFinal = c(2, Inf))

# we can plot the phylogeny to take a look
if (requireNamespace("ape", quietly = TRUE)) {
  phy <- make.phylo(sim)
  ape::plot.phylo(phy)
}

###
# note nFinal has to be sensible
## Not run: 
# this would return an error, since it is virtually impossible to get 100
# species at a process with diversification rate -0.09 starting at n0 = 1
sim <- bd.sim(1, lambda = 0.01, mu = 1, tMax = 100, nFinal = c(100, Inf))

## End(Not run)

[Package paleobuddy version 1.0.0 Index]