In the previous instalment, we introduced the idea of pair approximation, by which we try to understand a system by tracking the joint probability distributions for pairs of its pieces. Now, we’ll look at this machinery in more detail by focusing on a specific example. The ecosystem which we shall study will contain one species living on a regular lattice, and the individual organisms of that species can move about, give birth and die. That is, our pair dynamics will include three processes, each occurring stochastically with its own characteristic rate: movement or migration,birth and death. We follow van Baalen (in Dieckmann et al. (2000), chapter 19).
(Petri net pictures will go here.)
We write $z$ for the “coordination number” of the lattice. Birth:
$R 0 \rightarrow R R,$
with rate $b/z$.
Death:
$R a \rightarrow 0 a,$
with rate $d/z$.
Movement or migration:
$R 0 \rightarrow 0 R,$
with rate $m/z$.
$\begin{array}{rcl}
\frac{d p_{R 0}}{d t} & = & -p_{R 0}[b/z + d + (z - 1)q_{0|R0}m/z + (z - 1)q_{R|0R}(b + m)/z] \\
& & + p_{00} (z - 1) q_{R|00} (b + m)/z \\
& & + p_{R R} [d + (z - 1) q_{0|R R} m/z].
\end{array}$
If we ignore spatial structure altogether, we can say that
$q_{0|R} = p_0,$
which by normalization of probability means
$q_{0|R} = 1 - p_R.$
So,
$\frac{d p_R}{d t} = (b(1 - p_R) - d) p_R.$
This should look familiar: it’s a logistic equation for population growth, with growth rate $b - d$ and equilibrium population $1 - d/b$.
It’s worth pausing a moment here and using this result to touch on a more general concern. Often, a logistic-growth model is presented with the growth rate and the equilibrium population size as its parameters. When we see the model in that form, we naturally start thinking of those parameters as independently variable quantities. We imagine that a mutation or a change in the environmental conditions could change one without affecting the other. However, if the growth rate and the equilibrium population size are both functions of other parameters taken together, then the changes which are biologically reasonable to consider will likely affect both of them. To understand which quantities we should treat as independent, we need to spend time looking at how the numbers which apply to population-scale phenomena arise from the smaller-scale physiological and ecological goings-on (Fox 2011).
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