Limits to growth 0
Introduction
Mainly our thoughts here have been directed to constant growth and decay processes, but it is also interesting to consider limiting factors in a simple model. Here we will look at what is called the “logistic” equation, which implements a very simple method for describing growth limits.
Let’s look first at a standard discrete growth equation: \(n_{t+1}=k n_t\). If \(k\) is constant, this has the simple solution: \(n_t=k^tn_0\). If \(k>1\), the number \(n_t\) grows “exponentially” with time \(t\). The \(k<1\) behaviour is an exponential decay to zero. Finally, when \(k=1\) \(n_t\) is constant.
In real life, or “irl” as the twitterati would put it, things don't grow at the same rate forever. External factors come into play: food supplies run out, new technology is developed (in a human context), and so on. There are also internal factors that control biological processes such as cell division and so on.
A simple model for such limitation is to allow \(k\) to depend on the existing population \(n_t\). [One could also just make \(k\) time dependent, but that would lead to rather arbitrary results of little interest.] We expect \(k\) to decrease towards 1 as \(n\) increases. Perhaps the simplest way to implement a model for this would be to have \(k=k_0 - k' n_t\) with \(k_0,k'\) constants.
Discrete logistics
| \(k_0\): | \(k'\): | \(n_0\): |
If you generate the graph with the default parameters, duplicating the head image, you should get a graph showing initial growth and eventual stability at a value of more than 150. The stability occurs when the growth factor \(k\) is 1. This happens when \(n_{+\infty} = (k_0-1)/k'\). With \(k_0=1.5,k'=0.03\), this is \(n_{+\infty} = 500/3 \approx 167\).
Initially, we will explore this behaviour in the continuum limit (essentially, \(k_0\rightarrow1\)), but eventually we will be more interested in extreme discrete situations such as \(k_0=4\) — try it out — wtf, right? Our predicted \(n_{+\infty}=1000\) is only a very rough average of apparently “chaotic” swings. The value of \(k_0=5\) swings \(n_t\) “out of the ballpark”.