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Royama.md

Royama’s log-linear AR2 model phase plane

11/02/2021

Here is some R code to draw Royama’s phase plane for the log-linear AR2 model (Royama, T. 1992. Analytical Population Dynamics), as used in this paper.

plot(1, 1, col= NULL, xlim= c(-2, 2), ylim= c(-1, 1), 
    xlab= expression(paste("1 + ", Omega[1])), 
    ylab= expression(Omega[2]), pch= 21, bg= NULL, cex.lab= 1.2)

segments(c(-2, 0, -2, 0), 
    c(-1, 1, -1, -1), 
    c(0, 2, 2, 0), 
    c(1, -1, -1, 0), 
    col= grey(0.4))

curve(-x^2 / 4, -2, 2, col= grey(0.4), add= T)

cycleperiod<- c(3, 5:100)

w= 2 * pi / cycleperiod

for(i in 1){
    curve(-0.25 * x^2 * (tan(w[i])^2 + 1), -sqrt(1 / (0.25 * (tan(w[i])^2 + 1))), 0,
    add= T, col= grey(0.4))
}

for(i in 2:97){
    curve(-0.25 * x^2 * (tan(w[i])^2 + 1), 0, sqrt(1 / (0.25 * (tan(w[i])^2 + 1))),
    add= T, col= grey(0.4))
}

text(x= c(-1.34, -0.65, 0.05, 0.47, 0.73, 0.91, 1.04), 
    y= c(-0.516, -0.516, -0.516, -0.516, -0.516, -0.516, -0.516), 
    labels= 2:8, col= grey(0.5), cex= 1.2)

The triangle represents the region of the parameter space that produces stationary dynamics, either stable (top part) or cyclical (below the parabola, with numeric labels indicating some of the shorter cycle lengths).

Royama.rmd

---
title: "Royama's log-linear AR2 model phase plane"
date: "`r format(Sys.time(), '%d/%m/%Y')`"
editor_options:
  chunk_output_type: console
output: github_document
---


Here is some R code to draw Royama's phase plane for the log-linear AR2 model (Royama, T. 1992. Analytical Population Dynamics), as used in [this paper](https://scholar.google.co.uk/scholar?hl=en&as_sdt=0%2C5&q=Europe-Wide+Dampening+of+Population+Cycles+in+Keystone+Herbivores&btnG=).



```{r, fig.height= 7.5, fig.width= 8}
plot(1, 1, col= NULL, xlim= c(-2, 2), ylim= c(-1, 1), 
    xlab= expression(paste("1 + ", Omega[1])), 
    ylab= expression(Omega[2]), pch= 21, bg= NULL, cex.lab= 1.2)

segments(c(-2, 0, -2, 0), 
    c(-1, 1, -1, -1), 
    c(0, 2, 2, 0), 
    c(1, -1, -1, 0), 
    col= grey(0.4))

curve(-x^2 / 4, -2, 2, col= grey(0.4), add= T)

cycleperiod<- c(3, 5:100)

w= 2 * pi / cycleperiod

for(i in 1){
    curve(-0.25 * x^2 * (tan(w[i])^2 + 1), -sqrt(1 / (0.25 * (tan(w[i])^2 + 1))), 0,
    add= T, col= grey(0.4))
}

for(i in 2:97){
    curve(-0.25 * x^2 * (tan(w[i])^2 + 1), 0, sqrt(1 / (0.25 * (tan(w[i])^2 + 1))),
    add= T, col= grey(0.4))
}

text(x= c(-1.34, -0.65, 0.05, 0.47, 0.73, 0.91, 1.04), 
    y= c(-0.516, -0.516, -0.516, -0.516, -0.516, -0.516, -0.516), 
    labels= 2:8, col= grey(0.5), cex= 1.2)

```

> The triangle represents the region of the parameter space that produces stationary dynamics, either stable (top part) or cyclical (below the parabola, with numeric labels indicating some of the shorter cycle lengths).

unnamed-chunk-1-1.png