The classical logistic-growth model (Kot 2001) 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:
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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