# Rosenzweig-MacArthur predator-prey model (Pineda-Krch et al., 2007) In GillespieSSA: Gillespie's Stochastic Simulation Algorithm (SSA)

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

Rosenzweig-MacArthur predator-prey model (Pineda-Krch et al., 2007, Pineda-Krch, 2008)

```dN/dt = r(1-N/K - alpha/(1+wN))NP
dP/dt = c*alpha/(1+wN))NP
```

This model has five reactions with the following per capita rates,

```prey birth:     b
prey death:     d+(b-d)N/K
predation:      alpha/(1+wN)
predator birth: c*alpha/(1+wN)N
predator death: g
```

Propensity functions:

```a1 = b * N
a2 = (d+(b-d)N/K) * N
a3 = alpha/(1+wN) * N * P
a4 = c*alpha/(1+wN) * N * P
a5 = g * P
```

```library(GillespieSSA)
```

Define parameters

```parms <- c(b=2, d=1, K=1000, alpha=0.005,
w=0.0025, c=2, g=2)
tf <- 10                                               # Final time
simName <- "Rosenzweig-MacArthur predator-prey model"  # Name
```

Define initial state vector

```x0  <- c(N=500, P=500)
```

Define state-change matrix

```nu  <- matrix(c(+1, -1, -1,  0,  0,
0,  0,  0, +1, -1),
nrow=2,byrow=TRUE)
```

Define propensity functions

```a <- c(
"b*N",
"(d+(b-d)*N/K)*N",
"alpha/(1+w*N)*N*P",
"c*alpha/(1+w*N)*N*P",
"g*P"
)
```

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 = .01),
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(),
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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GillespieSSA documentation built on March 18, 2022, 7:55 p.m.