The logistic growth model is given by `dN/dt = rN(1-N/K)`

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. We assume `r=b-d`

where `b`

is the per capita p.c. birth rate and `d`

is the p.c. death rate.

This model consists of two reaction channels,

```
N ---b---> N + N
N ---d'---> 0
```

where `d'=d+(b-d)N/K`

. The propensity functions are `a_1=bN`

and `a_2=d'N`

.

Define 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)
```

Define reactions

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
)
autoplot.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
)
autoplot.ssa(out)
```

Run simulations with the Binomial tau-leap method