21 Tues Nov 10: The evolution of reproduction effort in plants

DUE DATE: Nov 17

21.1 Required reading

Otto, Sarah P., and Troy Day. 2007. A Biologist's Guide to Mathematical Modeling in Ecology and Evolution, Princeton University Press. p494-p496. Link. The discussion of reproduction effort in plants continues after page 486, however, we haven't covered the necessary math background for this to be very easy going.

21.2 Objectives

We aim to understand: (1) the population dynamics, and then (2) the evolutionary dynamics.

21.3 The population dynamics

Consider equation 12.1:

\[\begin{equation} N(t+1) = N(t)\left(b e^{-\alpha N(t)} + p\right) \tag{21.1} \end{equation}\]

This equation can be solved using R. To write the code below, I returned to chapter 7 and copied, pasted, and modified the script.

# Parameter values
p <- 0.25
b <- 10
alpha <- 0.001
# Inital population size
Nt <- 1
# Preallocate the dataframe
df <- data.frame(time = 0, popn.size = Nt)
for(t in seq(1,50,1)){
  Nt1 <- Nt*(b*exp(-alpha*Nt) + p) # equation 12.1 from text
  new.result <- data.frame(time = t, popn.size = Nt1)
  df <- rbind(df, new.result)
  Nt <- Nt1 # update the value of N(t) for the next iteration of the loop.
}
# Make the plot
plot(df$time, df$popn.size, typ = "l", xlab = "time", ylab = "Population size")

21.4 Questions

  1. Write out all the parameters (\(b>0\), \(\alpha>0\), and \(0 \leq p \leq 1\)) in equation (21.1) and give their meanings. [1 mark]

  2. The variable in equation (21.1) is \(N(t)\). What does \(N(t)\) represent? [1 mark]

  3. If \(N(t) = 0\) what is the value of \(be^{-\alpha N(t)}\)? What is the value of \(be^{-\alpha N(t)}\) if \(N(t)\) is very, very large? [1 mark]

  4. Does equation (21.1) have density dependence in births, survivorship, or both? [1 mark]

  5. Hand-in a figure showing the population dynamics (\(N(t)\) on the y-axis and time, \(t\), on the x-axis). Remember to include a brief figure caption. Note that the population dynamics for the parameter values above show oscillations and convergence to equilibrium. [2 marks]

21.5 The evolutionary dynamics

# Parameter values
alpha <- 0.001
p0 <- 0.75
# Define the allele of the resident:
E = 0.5
# Define the allele of the mutant:
Em = 0.7
# Functions from top of p499 in text
b <- function(E){
  exp(E)
}
p <-function(E){
  p0 - E^2
}
# Initially the mutant is rare
Nm <- 100
# Initially resident population is at equilibrium (i.e. see just above equation 12.1.4 on p497 for the equation for equilibrium)
N <- (1/alpha)*(log(b(E)/(1-p(E))))
#Preallocate the dataframe
df <- data.frame(time = 0, res.popn.size = N, mut.popn.size = Nm)
for(t in seq(1,50,1)){
  N1 <- N*(b(E)*exp(-alpha*(N+Nm)) + p(E)) # equation 12.1.1a
  Nm1 <- Nm*(b(Em)*exp(-alpha*(N+Nm)) + p(Em)) # equation 12.1.1b
  new.result <- data.frame(time = t, res.popn.size = N1, mut.popn.size = Nm1)
  df <- rbind(df, new.result)
  N <- N1 # update the value of N(t) for the next iteration of the loop.
  Nm <- Nm1 # update the value of Nm(t) for the next iteration of the loop.
}
plot(df$time, df$res.popn.size, typ = "l", xlab = "time", ylab = "Population size", ylim = c(0,1500))
lines(df$time, df$mut.popn.size, lty = 2)

21.6 Questions

  1. In equations 12.1.1a and 12.1.1b (p496), what is the difference between \(N(t)\) and \(N_m(t)\)? [1 mark]

  2. In considering the evolutionary dynamics, \(b\) and \(p\) are assumed to be linked by reproductive effort, \(E\), such that \(b(E) = e^E\) and \(p(E) = p_0 - E^2\) (top of p499). Therefore, when \(E\) is small, \(b\) is small, but \(p\) is large, and visa versa when \(E\) is large. In evolutionary biology what name is given to this type of relationship between life history parameters? [1 mark].

  3. The default choices of the reproductive effort parameters \(E = 0.2\) and \(E_m = 0.7\) correspond to no invasion of the mutant. Choose a different value of the mutant reproductive effort parameter such that the mutant can invade. You may wish to look at equation 12.6c on p500. Hand in your figure with a figure caption. Be sure you know if the dashed line or the solid line corresponds to the number of individuals with the mutant allele, and provide this information in your figure caption. [3 marks]

  4. If \(E = 0.5\) and \(E_m = 0.7\), for the parameter values from the sample code, what are the values of \(b(E)\), \(b(E_m)\), \(p(E)\), and \(p(E_m)\)? Does the mutant or resident allele allocate more energy to reproduction? [1 mark]