lambda_equation_guide.Rmd

---
#title: "equation guide"
#author: "quinn_asena"
#date: "28 March 2019"
#output: html_document
---

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

##### __Equation Guide__

* $N$ is the population at a given time-step (t)  
* $\lambda$ is the finite rate of increase
    * $\lambda \gtrapprox 1.02$ is positive growth (at $N = 10$ and $K = 1000$)
    * $\lambda \lessapprox 1.01$ is negative growth (at $N = 10$ and $K = 1000$)
    * $\lambda = 1.03$ is stable (at $N = 42$ and $K = 1442$)
* $K$ is the carrying capacity


##### __Limit Cycles and Chaos__
At $N = 10$ and $K = 1000$

* $\lambda < 3$ population reaches $K$ with dampening oscillations
* $3 < \lambda < 3.4$: two point stable limit cycle
* $3.4 < \lambda < 3.6$: limit cycle increase geometrically (2, 4, 6, 8...)
* $\lambda > 3.6$: aperiodic (irregular) patterns known as chaos.