set.seed(1) knitr::opts_chunk$set(fig.width = 8, fig.height = 6)

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`

.

Load package

```
library(GillespieSSA)
```

Define parameters

parms <- c(b = 2, d = 1, K = 1000) # Parameters tf <- 10 # Final time simName <- "Logistic growth"

Define initial state vector

x0 <- c(N = 500)

Define state-change matrix

nu <- matrix(c(+1, -1),ncol = 2)

Define propensity functions

a <- c("b*N", "(d+(b-d)*N/K)*N")

Run simulations with the Direct method

set.seed(1) out <- ssa( x0 = x0, a = a, nu = nu, parms = parms, tf = tf, method = ssa.d(), simName = simName, verbose = FALSE, consoleInterval = 1 ) ssa.plot(out, show.title = TRUE, show.legend = FALSE)

Run simulations with the Explict tau-leap method

set.seed(1) out <- ssa( x0 = x0, a = a, nu = nu, parms = parms, tf = tf, method = ssa.etl(tau = .03), simName = simName, verbose = FALSE, consoleInterval = 1 ) ssa.plot(out, show.title = TRUE, show.legend = FALSE)

Run simulations with the Binomial tau-leap method

set.seed(1) out <- ssa( x0 = x0, a = a, nu = nu, parms = parms, tf = tf, method = ssa.btl(f = 5), simName = simName, verbose = FALSE, consoleInterval = 1 ) ssa.plot(out, show.title = TRUE, show.legend = FALSE)

Run simulations with the Optimized tau-leap method

set.seed(1) out <- ssa( x0 = x0, a = a, nu = nu, parms = parms, tf = tf, method = ssa.otl(), simName = simName, verbose = FALSE, consoleInterval = 1 ) ssa.plot(out, show.title = TRUE, show.legend = FALSE)

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