This page has been superceded by my thesis.
Continued from Part 1.
Example 1: Birth, Death, Movement
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 $m/z$.
$\begin{array}{rcl}
\frac{d p_{R 0}}{d t} & = & p_{R 0}[b/z + d + (z  1)q_{0R0}m/z + (z  1)q_{R0R}(b + m)/z] \\
& & + p_{00} (z  1) q_{R00} (b + m)/z \\
& & + p_{R R} [d + (z  1) q_{0R R} m/z].
\end{array}$
$\frac{d p_{0 0}}{d t} = p_{00} 2(z  1) q_{R00} (b + m)/z + p_{R 0} 2 [d + (z  1) q_{0R 0} m/z].$
$\frac{d p_{R R}}{d t} = p_{R 0} 2[b/z + (z  1)q_{R0 R} (b + m)/z]  p_{R R} 2[d + (z  1) q_{0R R} m/z].$
$p_i = \sum_j p_{i j}.$
$\frac{d p_R}{d t} = (b q_{0R}  d) p_R.$
If we ignore spatial structure altogether, we can say that
$q_{0R} = p_0,$
which by normalization of probability means
$q_{0R} = 1  p_R.$
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 logisticgrowth 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 populationscale phenomena arise from the smallerscale physiological and ecological goingson (Fox 2011).
Continued in Part 3.
References
 H. Matsuda, N. Ogita, A. Sasaki, and K. Sato (1992), “Statistical mechanics of population”, Progress of Theoretical Physics 88, 6: 1035–49 (web).
 U. Dieckmann, R. Law, and J. A. J. Metz, eds., The Geometry of Ecological Interactions: Simplifying Spatial Complexity. Cambridge University Press, 2000.
 T. Gross, C. J. Dommar D’Lima and B. Blasius (2006), “Epidemic dynamics on an adaptive network”, Physical Review Letters 96, 20: 208701 (web). arXiv:qbio/0512037.
 A.L. Do and T. Gross (2009), “Contact processes and moment closure on adaptive networks”, in T. Gross and H. Sayama, eds., Adaptive Networks: Theory, Models and Applications. Springer.
 B. Allen (2010), Studies in the Mathematics of Evolution and Biodiversity. PhD thesis, Boston University (web).
 J. A. Damore and J. Gore (2012), “Understanding microbial cooperation”. Journal of Theoretical Biology 299: 31–41, DOI:10.1016/j.jtbi.2011.03.008 (pdf).
 B. Simon, J. A. Fletcher and M. Doebeli (2012), “Hamilton’s rule in multilevel selection models” Journal of Theoretical Biology 299: 55–63 PMID:21820447.

 B. C. Stacey, A. Gros and Y. BarYam (2014), “EcoEvolutionary Feedback in Host–Pathogen Spatial Dynamics”, arXiv:1110.3845 [nlin.AO].
 J. D. Van Dyken, T. A. Linksvayer and M. J. Wade (2011), “Kin Selection–Mutation Balance: A Model for the Origin, Maintenance, and Consequences of Social Cheating” The American Naturalist 177, 3: 288–300. JSTOR:10.1086/658365 (pdf).