This page is a blog article in progress, written by Blake Stacey. To discuss as it is being written, go to the Azimuth Forum.
Moment closures are a way of forgetting information about a system in a controlled fashion, in the hope that an incomplete, fairly heavily “coarse-grained” picture of the system will still be useful in figuring out what will happen to it. Sometimes, this is a justifiable hope, but in other cases, we are right to wonder whether all the algebra it generates actually leads us to any insights. Here, we’ll be concerned with a particular application of this technology: studying the vulnerability of an ecosystem to invasion. We shall find expressions for invasion fitness, the expected relative growth rate of an initially-rare species or variety.
Consider a lattice, each site of which can occupied by an individual of “resident” type ($R$), occupied by a mutant ($M$), or empty ($0$). The difference between the mutant-type and resident-type individuals is encoded in the choice of transition rules representing death, birth and migration. We can get an aggregate measure of the situation by finding the probability that a randomly chosen site will be in state $a$, where $a$ can take values in the set $\{R, M, 0\}$. A finer degree of distinction is provided by the conditional probabilities $q_{a|b}$, where, for example, $q_{R|M}$ denotes the probability that a randomly chosen neighbor site to a randomly chosen mutant is of resident type. Note that if a mutant is injected into a native resident population and its offspring form a geographical cluster, $q_{M|M}$ can be much larger than $p_M$: few individuals are mutants overall, but the probability of a mutant life-form interacting with another mutant is high.
The pair dynamics of the system involves the time evolution of the probabilities $p_{a b}$, that is, the probability that a randomly selected lattice edge will have $a$ on one end and $b$ on the other. The differential equation for $d p_{R M}/d t$, for example, will have terms reflecting the processes which can form and destroy $R M$ pairs: $R M\rightarrow R R$ is one possibility, and $R M\rightarrow M M$ is another. Death, which comes for organisms and leaves empty spaces behind, introduces processes like $R M \rightarrow R 0$, $R M \rightarrow 0 M$ and $R M \rightarrow 0 0$. Reproduction can lead to formerly empty spaces becoming occupied: $R 0 \rightarrow R R$ and $M 0 \rightarrow M M$. In language more like we’ve used in the Azimuth discussions of stochastic Petri net theory, we’ve moved beyond just creating and annihilating residents and mutants, and now we’re dynamically changing the number of “resident–resident” and “resident–mutant” pairs. So, if we put our minds to it, we could make a Petri net diagram for these processes, but the labels on the boxes might start looking a little funny. We’ll see in more detail how these diagrams might look next time.
Each term in our differential equations will have a transition rate dependent upon a conditional probability of the form $q_{a|b c}$, denoting the probability that a $b$ of a $b c$ pair will have a neighbor of type $a$. The differential equations for the pair probabilities $p_{a b}$ thus depend on triplet probabilities $p_{a b c}$, which depend upon quadruplet probabilities and so forth. To make progress, we truncate this hierarchy, brutally cutting off higher-order correlations by declaring that
This imposition, a pair approximation, destroys information about spatial structure and thereby introduces bias which in an ideal world ought to be accounted for. In theoretical ecology, this maneuver dates back at least to Matsuda et al. (1992), though it has antecedents in statistical physics, going back to the kinetic theory work of Bogoliubov, Born, Green, Kirkwood and Yvon, for whom the “BBGKY hierarchy” is named.
Invasion fitness is judged in the following manner. We start with a lattice devoid of mutants ($p_{M a} = 0$) and find the equilibrium densities $p_{R R}^*$ and $p_{R 0}^*$ by setting
The exact form of $p_{R R}^*$ and $p_{R0}^*$ will depend upon interaction details which we won’t worry about just yet. We then inject a mutant strain into this situation; as the mutants are initially rare, we can say they do not affect the large-scale dynamics of the resident population. Summarizing the pair probabilities $p_{M a}$ with the shorthand $\underline{p}$, we write the differential equation in matrix form
where the matrix $T(q_{a|b c})$ encapsulates the details of our chosen dynamics. The pair approximation, in which we discard correlations of third and higher order, lets us simplify this to
When people started doing simulations of lattice models like these, they found that the conditional probabilities $q_{a|M}$ equilibrate. That is to say, even if the global density of mutants $p_M$ changes, the local statistical structure of a mutant cluster remains constant. This is the key statement which allows us to linearize the dynamics and write the behavior of $\underline{p}$ in terms of eigenvectors and eigenvalues:
The dominant eigenvalue $\lambda$ of $T$ is the “invasion exponent” which characterizes whether an invasion will fail ($\lambda \lt 0$) or succeed ($\lambda \gt 0$). The eigenvector $\underline{e}_A$ associated with $\lambda$ describes the vehicle of selection for the mutants’ particular genetic variation, by summarizing the structure of their geographical cluster.
The failure of the mean-field theory and PA model implies that the long-range correlation is essentially important for the pattern formation.
The extent to which pair-dynamics models are satisfactory depends on the goal of the modeler. As we have seen, these models do not capture all of the phenomena that can be observed in simulations of fully spatial probabilistic cellular automata. Basically, the approximation fails whenever spatial structures arise that are difficult to “describe” using pairs alone. More technically, the method fails whenever significant higher-order correlations arise – that is, whenever the frequency of parituclar triplets (or triangles, squares, or all sorts of star-like configurations) starts to diverge from what one would expect on the basis of pair densities. Thus, pair-dynamics models satisfactorily describe probabilistic cellular automata in which only “small-scale” patterns arise. Larger, “meso-scale” patterns such as spirals are difficult to capture using this method.
—in Dieckmann et al. (2000), chapter 19.
It’s also pretty easy for the algebra involved in a pair-approximation calculation to blow up far beyond the point of being useful. For example, Dobrinevski et al. (2014) study a four-species system, where the pair approximation turns out to require 256 coupled differential equations. The only way to tackle that problem is to give it back to the computer and solve those equations numerically—and when they do that, it doesn’t even work all that well!
Continued in Part 2.