# Lotka predator-prey model (Gillespie, 1977; Kot, 2001)

This version of the Lotka predator-prey model is given by

    dY1/dt = c1*Y1 - c2*Y1*Y2
    dY2/dt = c2*Y1*Y2 - c3*Y2

consisting of the three reaction channels,

          Y1 --c1--> Y1 + Y1
     Y1 + Y2 --c2--> Y2 + Y2
          Y1 --c3--> 0

Load package

    library(GillespieSSA)

Define parameters

    parms <- c(c1 = 10, c2 = .01, c3 = 10)
    tf <- 2                                        # Final time
    simName <- "Lotka predator-prey model"         # Name

Define initial state vector

    x0 <- c(Y1=1000, Y2=1000)

Define state-change matrix

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

Define propensity functions

    a  <- c("c1*Y1", "c2*Y1*Y2","c3*Y2")

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 = .002),
      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 = 100),
      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(epsilon = .1),
      simName = simName,
      verbose = FALSE,
      consoleInterval = 1
    )
    ssa.plot(out, show.title = TRUE, show.legend = FALSE)