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)

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