Section 5.3 Extreme Values of Models
Subsection 5.3.1 Models and Asymptotic Analysis
I can use our tools of limits and asymptotic analysis to analyze models. Limits at finite values tell me how models behave near their undefined points; in particular, whether they diverge to infinity or remain bounded. This leads to an understanding of the limitations of a model and how it behaves at its own extreme situations.
Limits at infinity and asymptotic analysis tells me about the growth and long term behaviour of a model. I can compare population growth models asymptotically to get an idea of which is fundamentally a faster growth model. The analysis of algorithms in computing science is also done almost entirely with asymptotic analysis; the asymptotic order of an algorithm is a good measure of how fast it can operate.
The notion of stability is frequently the focus of the study of limits and models. The word ‘stability’ has various technical definitions in various pieces of applied mathematics, but it always relates to the limits of the model and the behaviour near those limits.
Considering popluation models, here are four categories of long term asymptotic behaviour (though many other behaviours are possible).
- The function can grow without bound, as in the exponential growth function \(p(t) = p_0 e^{at}\text{.}\)
- The function can decay to zero, as in the exponential decay function \(p(t) = p_0 e^{-at}\text{.}\)
- The function can approach a steady state, as in the logistic growth function.\begin{equation*} p(t) = \frac{p_0 K e^{at}}{K + p_0 (e^{at}-1)} \end{equation*}
- The function can oscillate without ever reaching a steady state, possibly with chaotic behaviour. A non-chaotic oscillating function is this periodic version of logistic growth.\begin{equation*} p(t) = \left(\frac{p_0 K e^{at}}{K + p_0 (e^{at}-1)} \right) \left( 1 + \frac{1}{5} \sin (bt) \right) \end{equation*}
Asymptotic analysis can also give information about the extreme values in a model. Consider the ideal gas law (where \(P\) is pressure, \(V\) is volume, \(n\) is the amount of gas, \(T\) is temperature and \(R\) is a constant). I will assume that \(P\) and \(V\) are the variables, and \(n\text{,}\) \(R\) and \(T\) are constant.
\begin{equation*}
PV = nRT
\end{equation*}
I can ask what happens to pressure at low volumes, expressed as a limit as \(V \rightarrow 0\text{.}\)
\begin{equation*}
\lim_{V \rightarrow 0} P = \lim_{V \rightarrow 0} \frac{nRT}{V}
= \infty
\end{equation*}
This tells me that low volumes result in high pressures. I could equivalently ask what happens at very high volumes.
\begin{equation*}
\lim_{V \rightarrow \infty} P = \lim_{V \rightarrow \infty}
\frac{nRT}{V} = 0
\end{equation*}
Unsurprisingly, very large volume result in very low pressures.
The behaviour at extreme values depends on the particulars of the models. A more complicated gas law is the Vander Waal’s gas law (where \(a\) and \(b\) are new positive constants).
\begin{equation*}
\left( P + \frac{n^2 a}{V^2} \right) \left( V - nb \right) =
nRT
\end{equation*}
What happens for these gasses at high pressures? It’s difficult to solve directly for volume, but I can analyze the equation in its current form. If the term involving pressure grows very large, the term only involving volume must grow to zero, so that the product remains the same. That happens when \(V\) is very close to the value \(nb\text{.}\)
\begin{equation*}
\lim_{P \rightarrow \infty} V = nb
\end{equation*}
Therefore, in this model, extreme pressures happen near the fixed, but non-zero, volume \(V = nb\text{.}\)
