density-dependence.Rmd

---
title: "Density dependence"
output: html_document
editor_options: 
  chunk_output_type: console
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

## Packages

```{r}
library(purrr)
library(deSolve)
library(tibble)
```


## Functions

```{r}
seq2 <- function(...) seq(..., le = 512)
plot2 <- function(..., col = 4) plot(..., type = "l", lwd = 2, col = col)
lines2 <- function(..., col = 4) lines(..., lwd = 2, col = col)
ode2 <- function(...) as_tibble(as.data.frame(deSolve::ode(...)))
```


## Models

Model 1:

$$
\frac{dN}{dt} = \frac{1}{(1 + N)^\alpha} b N
$$
Model 2:

$$
\frac{dN}{dt} = \left(1 - \frac{N}{K}\right) b N
$$


```{r}
xs <- seq2(0, 10)
plot(NA, xlim = range(xs), ylim = 0:1, xlab = "density", ylab = "per capita rate")
walk(c(.1, .5, 1, 2, 3), ~ lines2(xs, 1 / (1 + xs)^.x))
title("model 1")
```

```{r}
xs <- seq2(0, 10)
plot(NA, xlim = range(xs), ylim = 0:1, xlab = "density", ylab = "per capita rate")
walk(c(1, 3, 5, 8, 10), ~ lines2(xs, (1 - xs / .x)))
title("model 2")
```


```{r}
model <- function(N0, b, alpha, times) {
  ode2(c(N = N0),
       times,
       function(time, state, pars) {
         with(as.list(c(state, pars)), {
           dN <- b * N / (1 + N) ^ alpha
           list(dN)
         })
       },
       c(b     = b,
         alpha = alpha))
}
```

```{r}
with(model(1, 1, 1, seq2(0, 500)), plot2(time, N, ylim = c(0, 1000)))
```