# Lotka Predator-Prey model

This version of the Lotka predator-prey model (Gillespie 1977; Kot 2001) 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

Define parameters

    library(GillespieSSA2)
    sim_name <- "Lotka Predator-Prey model"
    params <- c(c1 = 10, c2 = .01, c3 = 10)
    final_time <- 2
    initial_state <- c(Y1 = 1000, Y2 = 1000)

Define reactions

    reactions <- list(
      reaction("c1 * Y1", c(Y1 = +1)),
      reaction("c2 * Y1 * Y2", c(Y1 = -1, Y2 = +1)),
      reaction("c3 * Y2", c(Y2 = -1))
    )

Run simulations with the Exact method

    set.seed(1)
    out <- ssa(
      initial_state = initial_state,
      reactions = reactions,
      params = params,
      final_time = final_time,
      method = ssa_exact(),
      sim_name = sim_name
    )
    plot_ssa(out)

Run simulations with the Explict tau-leap method

    set.seed(1)
    out <- ssa(
      initial_state = initial_state,
      reactions = reactions,
      params = params,
      final_time = final_time,
      method = ssa_etl(tau = .002),
      sim_name = sim_name
    )
    plot_ssa(out)

Run simulations with the Binomial tau-leap method

    set.seed(1)
    out <- ssa(
      initial_state = initial_state,
      reactions = reactions,
      params = params,
      final_time = final_time,
      method = ssa_btl(mean_firings = 100),
      sim_name = sim_name
    )
    plot_ssa(out)

Gillespie, Daniel T. 1977. “Exact Stochastic Simulation of Coupled Chemical Reactions.” The Journal of Physical Chemistry 81 (25): 2340–61. https://doi.org/10.1021/j100540a008.
Kot, Mark. 2001. Elements of Mathematical Ecology. Cambridge University Press. https://doi.org/10.1017/cbo9780511608520.