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

```
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
```

Define parameters

```
library(GillespieSSA2)
<- "Rosenzweig-MacArthur Predator-Prey model"
sim_name <- c(
params b = 2,
d = 1,
K = 1000,
alpha = 0.005,
w = 0.0025,
c = 2,
g = 2
)<- 10
final_time <- c(N = 500, P = 500) initial_state
```

Define reactions

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

Run simulations with the Exact method

```
set.seed(1)
<- ssa(
out 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)
<- ssa(
out initial_state = initial_state,
reactions = reactions,
params = params,
final_time = final_time,
method = ssa_etl(tau = .01),
sim_name = sim_name
) plot_ssa(out)
```

Run simulations with the Binomial tau-leap method

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

Pineda-Krch, Mario, Hendrik J. Blok, Ulf Dieckmann, and Michael Doebeli. 2007. “A Tale of Two Cycles – Distinguishing Quasi-Cycles and Limit Cycles in Finite Predator–Prey Populations.” *Oikos* 116 (1): 53–64. https://doi.org/https://doi.org/10.1111/j.2006.0030-1299.14940.x.