Limits to growth 1

In mathematical/practical investigations it is often useful to create a standardized equation through suitable rescaling.The most obvious rescaling here is to scale the \(n_t\) dependent variable relative to \(n_{+\infty}\). Such a rescaling could affect the parameters in the related logistic equation.

Let us define \(n_{st}=n_t/n_{+\infty}\). Substituting \(n_t=n_{st}n_{+\infty}\) in the logistic equation \(n_{s(t+1)}n_{+\infty}=(k_0-k'n_{st}n_{+\infty})n_{st}n_{+\infty}\implies n_{s(t+1)}=(k_0-k'n_{st}n_{+\infty})n_{st}\).

We can read off the rescaled parameters: \(k_{s0}=k_0,k_s'=k'n_{+\infty}\). But if we consider the definition \(n_{+\infty}=(k_0-1)/k'\), \(k_s'=k_0-1\). The scaled logistic now just has one parameter: \(n_{s(t+1)}=(k_0-(k_0-1)n_{st})n_{st}\). The rescaling has gobbled up \(k'\). The new parameters give \(n_{s+\infty}=1\) by design.

The next rescaling is a lot more ticklish. Up to now we have taken the time steps to be 1. The nuts behaviour we noted on the previous page is intimately connected with the discrete time used. When we take a continuum model for the time, the behaviour is much better controlled — and less interesting.

We might say: “But time IS continuous. Why bother with a discrete time at all?” Personally I have a wide range of responses to such an attitude, not all of them fit for publication. But how do we know time is continuous? I can imagine practical models where discrete time provides a better description of reality: for example, there are yearly effects on the way the world economy works in terms of farming, distribution, retail, etc. One thing that used to drive me mad was looking at economic reports that compared quarter year performance to the previous quarter rather than the same quarter in the previous year. Everyone knows (or should) that there is increased retail activity in the month of December in historically Christian countries (and in others that have been influenced by colonial incursions and neocolonial domination). This increase is prepared for upstream in the supply chain (design, tooling, production and so on) at different parts of the year.

Here we will be assuming that the changes in \(n\) become smaller as the time step reduces. I understand there are stochastic processes where this assumption is not true. I also suspect the point is moot in quantum systems relative to the Heisenberg uncertainty principle. These are points I might explore in more depth at some point.

As mentioned elsewhere on this site, the Greek upper-case letter \(\Delta\) (“delta”) is used to represent changes. The lower-case version \(\delta\) is also sometimes used according to taste and with the added implication that a limiting process is being proposed. The actual limit then becomes a Latin letter “d”.

Let us look at how this works with our scaled equation \(n_{(t+1)}=(k_0-(k_0-1)n_{t})n_{t}\). We hope it is alright to drop the “s” from the subscript as unnecessary clutter. The change at \(t\) is: \(\Delta n_t = n_{t+1} - n_t = (k_0-1)(1-n_t)n_t\). The change in \(t\) itself is \(\Delta t =1\), so one could write with a view to time scaling: \[\Delta n_t=(k_0-1)(1-n_t)n_t \Delta t\].

For those familiar with differential equations, this becomes in the limit \(\delta t \rightarrow 0\): \[\dv{n}{t}=\kappa (1-n)n\] where I have replaced \(k_0 -1\) with \(\kappa\).

The behaviour of the solution is detemined by the value at one point in time: \[n(t)=\frac{1}{1+e^{-\kappa (t-t_0)}}\] with \(t_0\) being a parameter fixed by the value of \(n\) at the given time, e.g. \(t=0\).

We can further standardize by taking \(\kappa=1\) and scaling the \(t\) variable suitably, \(n_\kappa(t)=n_1(\kappa t)\): \[n_1(t)=\frac{1}{1+e^{-(t-t_0)}}\] (with a different \(t_0\) value).

The \(n_1\) curve is the theoretical limit as \(\Delta t \rightarrow 0\) described above. The \(n\) curve is the approximation that has the discrete logistic form: \[n_{t+\Delta t}=n_t + \Delta t (1-n_t)n_t\] — in essence the step size \(\Delta t\) has taken the place of \(\kappa\) and the \(t\)-scale has been redefined. Reducing \(\Delta t\) to zero brings the curves closer together.