---
title: "Rosenzweig-MacArthur Predator-Prey model"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Rosenzweig-MacArthur Predator-Prey model}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
bibliography: ../inst/REFERENCES.bib
---

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```{r, setseed, echo=FALSE}
set.seed(1)
knitr::opts_chunk$set(fig.width = 6, fig.height = 4)
if("package:GillespieSSA" %in% search()) detach("package:GillespieSSA", unload=TRUE) 
```

Rosenzweig-MacArthur predator-prey model [@PinedaKrch2007].

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

Define reactions
```{r}
reactions <- list(
  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
```{r exact}
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
```{r etl}
set.seed(1)
out <- ssa(
  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
```{r btl}
set.seed(1)
out <- ssa(
  initial_state = initial_state,
  reactions = reactions,
  params = params,
  final_time = final_time,
  method = ssa_btl(),
  sim_name = sim_name
) 
plot_ssa(out)
```

## References
