# 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)


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GillespieSSA2 documentation built on Jan. 24, 2023, 1:10 a.m.