The Azimuth Project
Invasion fitness in moment-closure treatments (Rev #2)

This page is a blog article in progress, written by Blake Stacey.

Consider a lattice, each site of which can be empty (00), occupied by a selfish individual (SS) or occupied by an altruist (AA). The difference between selfishness and altruism 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 aa, where aa can take values in the set {0,A,S}\{0, A, S\}. A finer degree of distinction is provided by the conditional probabilities q a|bq_{a|b}, where, for example, q S|Aq_{S|A} denotes the probability that a randomly chosen neighbor site to a randomly chosen altruist is selfish. Note that if a selfish mutant is injected into a population of altruists and its selfish offspring form a geographical cluster, q S|Sq_{S|S} can be much larger than p Sp_S: few individuals are selfish overall, but the probability of a selfish life-form interacting with one of the same behavior type is high.

The pair dynamics of the system involves the time evolution of the probabilities p abp_{a b}, that is, the probability that a randomly selected lattice edge will have aa on one end and bb on the other. The differential equation for dp S0/dtd p_{S 0}/d t, for example, will have terms reflecting the processes which can form and destroy S0S0 pairs: 00S000\rightarrow S 0, S000S 0\rightarrow 00, S0SSS0\rightarrow S S, SSS0S S\rightarrow S 0, S0SAS 0\rightarrow S A and SAS0S A\rightarrow S 0, let us say. In language more like we’ve used in the Azimuth discussions of stochastic Petri net theory, we’ve moved beyond just creating and annihilating altruists and selfishists, and now we’re dynamically changing the number of “altruist–altruist” and “altruist–selfish” pairs. So, we probably could make a Petri net diagram for these processes, but the labels on the boxes might start looking a little funny.

Each term in our differential equations will have a transition rate dependent upon a conditional probability of the form q a|bcq_{a|b c}, denoting the probability that a bb of a bcb c pair will have a neighbor of type aa. The differential equations for the pair probabilities p abp_{a b} thus depend on triplet probabilities p abcp_{a b c}, which depend upon quadruplet probabilities and so forth. To make progress, we truncate this hierarchy in a manner reminiscent to cutting off the BBGKY hierarchy in statistical mechanics: we impose the pair approximation that

q a|bcq a|b. q_{a|b c} \approx q_{a|b}.

This approximation destroys information about spatial structure and thereby introduces bias which in an ideal world ought to be accounted for.

(It just occurred to me that if a Petri net specifies a symmetric monoidal category, then successive truncations of the moment-dynamics hierarchy yield mappings between categories. Going from a pair approximation to a mean-field approximation, for example, transforms a Petri net whose circles are labelled with pair states to one labelled by site states. Category theory might be able to say something interesting here. Anything which can tame the horrible spew of equations which arises in these problems would be great to have.)

Invasion fitness is judged in the following manner. We start with a lattice devoid of altruists (p Aa=0p_{A a} = 0) and find the equilibrium densities p SS *p_{S S}^* and p S0 *p_{S 0}^* by setting

dp S0dt=dp SSdt=0. \frac{d p_{S0}}{d t} = \frac{d p_{S S}}{d t} = 0.

The exact form of p SS *p_{S S}^* and p S0 *p_{S0}^* will depend upon interaction details which do not concern us here. We then inject an altruistic strain into this situation; as the altruists are initially rare, we can say they do not affect the large-scale dynamics of the selfish population. Summarizing the pair probabilities p Aap_{A a} with the shorthand p̲\underline{p}, we write the differential equation in matrix form

dp̲dt=M(q a|bc)p̲, \frac{d\underline{p}}{d t} = M(q_{a|b c})\underline{p},

where the matrix M(q a|bc)M(q_{a|b c}) encapsulates the details of our chosen behavior modes. The pair approximation, in which we discard correlations of third and higher order, lets us simplify this to

dp̲dt=M(q a|b)p̲. \frac{d\underline{p}}{d t} = M(q_{a|b})\underline{p}.

A result supported by numerical investigation is that the conditional probabilities q a|Aq_{a|A} equilibrate. That is to say, even if the global density of altruists p Ap_A changes, the local statistical structure of an altruistic cluster remains constant. This is the key statement which allows us to linearize the dynamics and write the behavior of p̲\underline{p} in terms of eigenvectors and eigenvalues:

p̲(t)=Ce̲ Aexp(λt). \underline{p}(t) = C\underline{e}_A \exp(\lambda t).

The dominant eigenvalue λ\lambda of MM is the “invasion exponent” which characterizes whether an invasion will fail (λ<0\lambda \lt 0) or succeed (λ>0\lambda \gt 0). The eigenvector e̲ A\underline{e}_A associated with λ\lambda describes the vehicle of selection for the altruist’s particular genetic variation, by summarizing the structure of the altruists’ cluster.

All of this, of course, is only as good as our linearization!

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.
  • B. Allen, Studies in the Mathematics of Evolution and Biodiversity. PhD thesis, Boston University, 2010. (web)
  • J. A. Damore and J. Gore (2011). “Understanding microbial cooperation”. Journal of Theoretical Biology DOI:10.1016/j.jtbi.2011.03.008 (pdf).
  • B. C. Stacey, A. Gros and Y. Bar-Yam (2011), “Beyond the mean field in host-pathogen spatial ecology”, arXiv:1110.3845 [nlin.AO].

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