Righteous modeling: the competence of classical population genetics

Abstract

In a recent article, “Wayward Modeling: Population Genetics and Natural Selection,” Bruce Glymour claims that population genetics is burdened by serious predictive and explanatory inadequacies and that the theory itself is to blame. Because Glymour overlooks a variety of formal modeling techniques in population genetics, his arguments do not quite undermine a major scientific theory. However, his arguments are extremely valuable as they provide definitive proof that those who would deploy classical population genetics over natural systems must do so with careful attention to interactions between individual population members and environmental causes. Glymour’s arguments have deep implications for causation in classical population genetics.

Appendix

Glymour considers a hypothetical population whose dynamics classical population genetics putatively cannot predict or explain. Here is his description of the system:

Let K ₁ – K _n be a list of potential prey items, ordered in decreasing profitability. According to optimal foraging theory, potential prey items ought elicit attack from a predator either always or never (the so-called zero–one rule). Whether or not items of kind K _k ought to elicit attack depends on the energetic demands of the predator, the profitability of K _k , and the profitability and frequencies of prey items of kinds K ₁–K _k-1. The kinds upon which a predator in fact preys identify a choice rule under which it is operating: ‘Prey on items of type K ₁–K _k-1 if opportunity arises, but never on items of kinds K _k –K _n ’. If the frequencies of items of kind K ₁–K _k-1 decrease, then the choice rule ought to change—items of lower profitability ought to be added to the diet. The rule for adding or removing items from the list of those preyed upon is, in our terms, a meta-choice rule; when the choice rules adopted by individual predators over the course of their lives change with time according to a meta-choice rule, there will be interactive causes of reproductive success. The frequency of K ₂ in the current generation is an interactive cause of reproductive success in that generation for any type that employs the meta-rule: the frequency of K ₂ has an effect on reproductive success if, but only if, the frequency of K ₁ is sufficiently low; hence the frequency of K ₁ in the current generation is also an interactive cause of reproductive success in the current generation. (Glymour 2006, 379)

What follows is a Wright-Fisher model that is my best guess at an instance of what Glymour had in mind: it is a single-locus biallelic Wright-Fisher model featuring three prey types. It could easily be generalized; any of those last specifying assumptions could be discharged. Despite the possibility of incongruence between the population considered below and Glymour’s hypothetical population, the model will serve to bring out some of the points made in the body of the paper.

$$ \begin{aligned} p^{\prime } & = \frac{1}{{\bar{w}}}\left\{ {p_{t}^{2} \left[ {K_{{1_{t} }} \left( {S_{{D_{{K_{{1_{t} }} }} }} \cdot P_{{D_{{K_{1} }} }} } \right) + K_{{2_{t} }} \left( {S_{{D_{{K_{{2_{t} }} }} }} \cdot P_{{D_{{K_{2} }} }} } \right) + K_{{3_{t} }} \left( {S_{{D_{{K_{{3_{t} }} }} }} \cdot P_{{D_{{K_{3} }} }} } \right)} \right]} \right. \\ & \left. { + p_{t} q_{t} \left[ {K_{{1_{t} }} \left( {S_{{H_{{K_{{1_{t} }} }} }} \cdot P_{{H_{{K_{1} }} }} } \right) + K_{{2_{t} }} \left( {S_{{H_{{K_{{2_{t} }} }} }} \cdot P_{{H_{{K_{2} }} }} } \right) + K_{{3_{t} }} \left( {S_{{H_{{K_{{3_{t} }} }} }} \cdot P_{{H_{{K_{3} }} }} } \right)} \right]} \right\} \\ \bar{w} & = p_{t}^{2} \left[ {K_{{1_{t} }} \left( {S_{{D_{{K_{1} }} }} \cdot P_{{D_{{K_{1} }} }} } \right) + K_{{2_{t} }} \left( {S_{{D_{{K_{2} }} }} \cdot P_{{D_{{K_{2} }} }} } \right) + K_{{3_{t} }} \left( {S_{{D_{{K_{3} }} }} \cdot P_{{D_{{K_{3} }} }} } \right)} \right] \\ & + 2p_{t} q_{t} \left[ {K_{{1_{t} }} \left( {S_{{H_{{K_{1} }} }} \cdot P_{{H_{{K_{1} }} }} } \right) + K_{{2_{t} }} \left( {S_{{H_{{K_{2} }} }} \cdot P_{{H_{{K_{2} }} }} } \right) + K_{{3_{t} }} \left( {S_{{H_{{K_{3} }} }} \cdot P_{{H_{{K_{3} }} }} } \right)} \right] \\ & + q_{t}^{2} \left[ {K_{{1_{t} }} \left( {S_{{R_{{K_{1} }} }} \cdot P_{{R_{{K_{1} }} }} } \right) + K_{{2_{t} }} \left( {S_{{R_{{K_{2} }} }} \cdot P_{{R_{{K_{2} }} }} } \right) + K_{{3_{t} }} \left( {S_{{R_{{K_{3} }} }} \cdot P_{{R_{{K_{3} }} }} } \right)} \right] \\ S_{{D_{{K_{{2_{t} }} }} }} & = \left\{ {\begin{array}{*{20}c} {C_{{K_{2} }} \left( {K_{{1_{t} }} - T_{{D_{{K_{2} }} }} } \right)} & {{\text{if}}\,K_{{1_{t} }} \le T_{{D_{{K_{2} }} }} } \\ 0 & {{\text{if}}\,K_{{1_{t} }} > T_{{D_{{K_{2} }} }} } \\ \end{array} } \right\} \\ S_{{D_{{K_{{3_{t} }} }} }} & = \left\{ {\begin{array}{*{20}c} {C_{{K_{3} }} \left( {K_{{1_{t} }} + K_{{2_{t} }} - T_{{D_{{K_{3} }} }} } \right)} & {{\text{if}}\,K_{{1_{t} }} + K_{{2_{t} }} \le T_{{D_{{K_{2} }} }} } \\ 0 & {{\text{if}}\,K_{{1_{t} }} + K_{{2_{t} }} \le T_{{D_{{K_{3} }} }} } \\ \end{array} } \right\} \\ S_{{H_{{K_{{2_{t} }} }} }} & = \left\{ {\begin{array}{*{20}c} {C_{{K_{2} }} \left( {K_{{1_{t} }} - T_{{H_{{K_{2} }} }} } \right)} & {{\text{if}}\,K_{{1_{t} }} \le T_{{H_{{K_{2} }} }} } \\ 0 & {{\text{if}}\,K_{{1_{t} }} > T_{{H_{{K_{2} }} }} } \\ \end{array} } \right\} \\ S_{{H_{{K_{{3_{t} }} }} }} & = \left\{ {\begin{array}{*{20}c} {C_{{K_{3} }} \left( {K_{{1_{t} }} + K_{{2_{t} }} - T_{{H_{{K_{3} }} }} } \right)} & {{\text{if}}\,K_{{1_{t} }} + K_{{2_{t} }} \le T_{{H_{{K_{2} }} }} } \\ 0 & {{\text{if}}\,K_{{1_{t} }} + K_{{2_{t} }} \le T_{{H_{{K_{3} }} }} } \\ \end{array} } \right\} \\ S_{{R_{{K_{{2_{t} }} }} }} & = \left\{ {\begin{array}{*{20}c} {C_{{K_{2} }} \left( {K_{{1_{t} }} - T_{{R_{{K_{2} }} }} } \right)} & {{\text{if}}\,K_{{1_{t} }} \le T_{{R_{{K_{2} }} }} } \\ 0 & {{\text{if}}\,K_{{1_{t} }} > T_{{R_{{K_{2} }} }} } \\ \end{array} } \right\} \\ S_{{R_{{K_{{3_{t} }} }} }} & = \left\{ {\begin{array}{*{20}c} {C_{{K_{3} }} \left( {K_{{1_{t} }} + K_{{2_{t} }} - T_{{R_{{K_{3} }} }} } \right)} & {{\text{if}}\,K_{{1_{t} }} + K_{{2_{t} }} \le T_{{R_{{K_{2} }} }} } \\ 0 & {{\text{if}}\,K_{{1_{t} }} + K_{{2_{t} }} \le T_{{R_{{K_{3} }} }} } \\ \end{array} } \right\} \\ K_{{1_{t} }} & = b_{1} K_{{1_{t - 1} }} \left( {1 - p_{t - 1}^{2} \cdot S_{{D_{{K_{{1_{t - 1} }} }} }} - 2pq \cdot S_{{H_{{K_{{1_{t - 1} }} }} }} - q^{2} \cdot S_{{H_{{K_{{1_{t - 1} }} }} }} } \right) \\ K_{{2_{t} }} & = b_{2} K_{{2_{t - 1} }} \left( {1 - p_{t - 1}^{2} \cdot S_{{D_{{K_{{2_{t - 1} }} }} }} - 2pq \cdot S_{{H_{{K_{{2_{t - 1} }} }} }} - q^{2} \cdot S_{{H_{{K_{{2_{t - 1} }} }} }} } \right) \\ K_{{3_{t} }} & = b_{3} K_{{3_{t - 1} }} \left( {1 - p_{t - 1}^{2} \cdot S_{{D_{{K_{{3_{t - 1} }} }} }} - 2pq \cdot S_{{H_{{K_{{3_{t - 1} }} }} }} - q^{2} \cdot S_{{H_{{K_{{3_{t - 1} }} }} }} } \right) \\ \end{aligned} $$

where p′ = next generation frequency of the A allele; p = this generation frequency of the A allele; q = this generation frequency of the a allele; D, H, and R subscripts reflect dominant, heterozygote, and recessive zygotes; K parameters reflect the frequency of prey types 1, 2, and 3; S parameters represent the rates at which zygotes strike prey; P parameters represent the profitability of different prey types for different zygotes; T parameters are thresholds for striking less profitable prey; C parameters are constants; b parameters reflect the natural rate at which prey populations grow; t is generation number.

The above system of equations is fully recursive and can be used to infer the dynamics of the population, though not, so far as I can tell, by analytic means. Provided some of the variables other than frequency terms are set at different values, the population will exhibit the dynamics of a system under selection. So, for instance, if the T parameters for the different types of zygotes are different, such that they are willing at different times to pursue different prey, the resulting dynamics of the system will be governed by the above equations. The prey types themselves change in frequency according to how often they are struck by the various types of zygotes combined with a natural birth rate. This leads to the phenomenon that Glymour emphasizes in his description: the environment changes from generation to generation because the prey change in frequency and the behavior of the population members changes accordingly, since their willingness to strike different types of prey is contingent on the frequencies of those types of prey. If the most attractive type of prey is sufficiently common, they strike only it, and unless the two most attractive types of prey are sufficiently rare, the least attractive is never struck.

In a variety of ways, the model is simple and idealized, and it should be re-iterated that it may not match exactly what Glymour meant to posit. However, the model serves well enough to justify the general point being pursued above: by means of fitness functions classical population genetics can be used to model the dynamics of the sorts of natural populations Glymour describes in his argument for that theory’s incompetence.