---
title: "Lotka predator-prey model (Gillespie, 1977; Kot, 2001)"
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---

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```{r, setseed, echo=FALSE}
set.seed(1)
knitr::opts_chunk$set(fig.width = 8, fig.height = 6)
```

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
```{r}
library(GillespieSSA)
```

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

Define initial state vector
```{r}
x0 <- c(Y1=1000, Y2=1000)
```

Define state-change matrix
```{r}
nu <- matrix(c(+1, -1, 0, 0, 1, -1), nrow = 2, byrow = TRUE)
```

Define propensity functions
```{r}
a  <- c("c1*Y1", "c2*Y1*Y2","c3*Y2")
```

Run simulations with the Direct method
```{r direct}
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
```{r etl}
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
```{r btl}
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
```{r otl}
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)
```
