# Pearl-Verhulst Logistic growth model (Kot, 2001) In GillespieSSA: Gillespie's Stochastic Simulation Algorithm (SSA)

```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`.

```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)
```

## Try the GillespieSSA package in your browser

Any scripts or data that you put into this service are public.

GillespieSSA documentation built on July 27, 2019, 1:02 a.m.