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

<!-- github markdown built using 
rmarkdown::render("vignettes/lotka_predator_prey.Rmd", output_format = "github_document")
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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) 
```

This version of the Lotka predator-prey model [@Gillespie1977; @Kot2001] 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
```{r}
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
```{r}
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
```{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 = .002),
  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(mean_firings = 100),
  sim_name = sim_name
) 
plot_ssa(out)
```
