Pearl-Verhulst Logistic Growth model
In GillespieSSA2: Gillespie's Stochastic Simulation Algorithm for Impatient People

set.seed(1)
knitr::opts_chunk$set(fig.width = 6, fig.height = 4)
if("package:GillespieSSA" %in% search()) detach("package:GillespieSSA", unload=TRUE)

The classical logistic-growth model [@Kot2001] assumes that the growth of a population decreases with increasing population size and is given by the following equation,

$$ \frac{dN}{dt} = rN \times \left(1 - \frac{N}{K}\right) $$ where $N$ is the number (density) of indviduals at time $t$, $K$ is the carrying capacity of the population, $r$ is the intrinsic growth rate of the population.

This model consists of two reactions, birth and death, whose propensity functions are defined as:

$a_1(x) = bN$
$a_2(x) = (d + (b - d) \times N / K) \times N$

where $b$ is the per capita birth rate and $d$ is the per capita death rate.

Assuming $b=2$, $d=1$, $K=1000$ and $X(0)=(500)$, we can define the following parameters:

library(GillespieSSA2)
sim_name <- "Pearl-Verhulst Logistic Growth model"
params <- c(b = 2, d = 1, K = 1000)
final_time <- 10
initial_state <- c(N = 500)

The reactions (each consisting of a propensity function and a state change vector) can be defined as:

reactions <- list(
  reaction("b * N", c(N = +1)),
  reaction("(d + (b - d) * N / K) * N", c(N = -1))
)

Run simulations with the Exact method

set.seed(1)
out <- ssa(
  initial_state = initial_state,
  reactions = reactions,
  params = params,
  final_time = final_time,
  method = ssa_exact(),
  sim_name = sim_name
) 
plot_ssa(out)

Run simulations with the Explict tau-leap method

set.seed(1)
out <- ssa(
  initial_state = initial_state,
  reactions = reactions,
  params = params,
  final_time = final_time,
  method = ssa_etl(tau = .03),
  sim_name = sim_name
) 
plot_ssa(out)

Run simulations with the Binomial tau-leap method

set.seed(1)
out <- ssa(
  initial_state = initial_state,
  reactions = reactions,
  params = params,
  final_time = final_time,
  method = ssa_btl(mean_firings = 5),
  sim_name = sim_name
) 
plot_ssa(out)