0106-report.R

#' ---
#' title: "Comparing a simple growth model with a birth-death model"
#' author: "My name"
#' date: '`r format(Sys.Date(), "%B %d %Y")`'
#' output: html_document
#' ---
#'

#'
#' Load in the functions that do the work
library(RPiR)
source("0104-step-growth.R")
source("0105-step-birth-death.R")

#' We are going to compare two population dynamics models:
#'
#' 1. A simple growth model:
#'    $$N(t + 1) = \lambda \times N(t) + N(t)$$
#' 2. A birth death model:
#'    $$N(t + 1) = b \times N(t) - d \times N(t) + N(t)$$
#'

#' First we set up the simulation parameters for every experiment.

# Set the birth and death rates
birth.rate <- 0.2
death.rate <- 0.1

# Starting population size
initial.count <- 1

# And setting times
start.time <- 0
end.time <- 100

# The timesteps that the simulation will run through
timesteps <- seq(from = start.time + 1, to = end.time)

#'
#' ## Run the full $birth, death$ simulation

# Set up the population starting size (at the first timestep)
population.df1 <- data.frame(count = initial.count)

# Now we loop through the time itself (starting at the second timestep)
for (new.time in timesteps) {
  updated.population <-
    step_deterministic_birth_death(latest = tail(population.df1, 1),
                                   birth.rate = birth.rate,
                                   death.rate = death.rate)
  population.df1 <- rbind(population.df1, updated.population)
}
population.df1$time <- c(start.time, timesteps)

#+ initial, fig.width=5, fig.height=5, fig.align="center", fig.cap="Fig. 1: Birth-death model"
plot_populations(population.df1)

#'
#' ## Compare that to the $growth=birth$ simulation

# Set up the population starting size (at the first timestep)
population.df2 <- data.frame(count = initial.count)

# Now we loop through the time itself (starting at the second timestep)
for (new.time in timesteps) {
  updated.population <-
    step_deterministic_growth(latest = tail(population.df2, 1),
                              growth.rate = birth.rate)
  population.df2 <- rbind(population.df2, updated.population)
}
population.df2$time <- c(start.time, timesteps)

#+ birth, fig.width=5, fig.height=5, fig.align="center", fig.cap="Fig. 2: Birth only model"
plot_populations(population.df1, with.legend = FALSE)
plot_populations(population.df2, new.graph = FALSE, col = 2)
legend("topleft", inset = 0.02, legend = c("birth, death", "growth = birth"),
       col = c(1, 2), lty = c(1, 1))

#' Growth that only accounts for birth without death causes the population to
#' grow too fast (unsurprisingly).