---
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 = 6, fig.height = 4)
if("package:GillespieSSA" %in% search()) detach("package:GillespieSSA", unload=TRUE) 
```

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`.

Define parameters
```{r}
library(GillespieSSA2)
sim_name <- "Pearl-Verhulst Logistic Growth model"
params <- c(b = 2, d = 1, K = 1000)
final_time <- 10
initial_state <- c(N = 500)
```

Define reactions
```{r}
reactions <- list(
  reaction("b * N", c(N = +1)),
  reaction("(d + (b - d) * N / K) * N", c(N = -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 = .03),
  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 = 5),
  sim_name = sim_name
) 
plot_ssa(out)
```
