---
title: "Pearl-Verhulst Logistic growth model (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)
```

The logistic growth model is given by `dN/dt = rN(1-N/K)` where
`N` is the number (density) of indviduals at time `t`, 
`K` is the carrying capacity of the population, 
`r` is the intrinsic growth rate of the population. 
We assume `r=b-d` where 
`b` is the per capita p.c. birth rate and 
`d` is the p.c. death rate. 

This model consists of two reaction channels,
```
 N ---b--->  N + N
 N ---d'---> 0
```
where `d'=d+(b-d)N/K`. The propensity functions are `a_1=bN` and `a_2=d'N`.

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

Define parameters
```{r}
parms <- c(b = 2, d = 1, K = 1000)      # Parameters
tf <- 10                                # Final time
simName <- "Logistic growth" 
```

Define initial state vector
```{r}
x0 <- c(N = 500)
```

Define state-change matrix
```{r}
nu <- matrix(c(+1, -1),ncol = 2)
```

Define propensity functions
```{r}
a  <- c("b*N", "(d+(b-d)*N/K)*N")
```

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 = .03),
  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 = 5),
  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(),
  simName = simName,
  verbose = FALSE,
  consoleInterval = 1
) 
ssa.plot(out, show.title = TRUE, show.legend = FALSE)
```