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

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

Load package

    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)