Stationary solutions for metapopulation Moran models with mutation and selection

We construct an individual-based metapopulation model of population genetics featuring migration, mutation, selection, and genetic drift. In the case of a single “island,” the model reduces to the Moran model. Using the diffusion approximation and time-scale separation arguments, an effective one-variable description of the model is developed. The effective description bears similarities to the well-mixed Moran model with effective parameters that depend on the network structure and island sizes, and it is amenable to analysis. Predictions from the reduced theory match the results from stochastic simulations across a range of parameters. The nature of the fast-variable elimination technique we adopt is further studied by applying it to a linear system, where it provides a precise description of the slow dynamics in the limit of large time-scale separation.

Figure 1

This figure shows the fractions of type $X$ individuals on each island $i,n_i/{\beta }_{i}N$, for a neutral six-island metapopulation Moran model. Over short times (upper panel) the proportions of individuals on each island become almost equal. After this the trajectories of each island are coupled, so that over long times (lower panel) the system behaves approximately as a well-mixed model with effective parameters.

Figure 2

Trajectories of a neutral system with three islands. It can be seen that after a short time, the stochastic trajectories denoted by orange and red points come to lie in the region of the one-dimensional center manifold, the thick blue line $x_1=x_2=x_3$. The stochastic trajectories follow an approximately deterministic trajectory to the center manifold, indicated by the black arrows.

Figure 3

Deterministic trajectories (gray) plotted for the two-island Moran model with mutation. The slow subspace, Eq. (14), is plotted as a blue dashed line. A histogram of trajectories of the original IBM is overlaid in orange. Having relaxed to the slow subspace, they can be seen to be confined to its vicinity.

Figure 4

Stationary pdf of a metapopulation Moran model with mutation. The system has four islands of equal size $N=300$ connected by a symmetric migration matrix with diagonal entries $m_{ii}=0.9$. The mutation rates are ${\kappa }_{1i}={\kappa }_{2i}=7\times {10}^{-4}$ for each island. The filled orange histogram is obtained from stochastic simulations of the IBM, while the solid black line is obtained from theory [Eq. (24)]. The dashed line is the theoretical prediction of a well-mixed model with the same average mutation rates and a population size $\mathcal{D}N$.

Figure 5

Similar plots to Fig. 4 but for a two-island system with asymmetric migration, differing islands sizes, and varying mutation rates. While for brevity most parameters are not stated here, we note that the mutation rates differ by many magnitudes over the islands (see Appendix pp3). Numerical investigations suggest this is necessary in order to enhance the bistability of the system relative to the well-mixed unstructured analog (with the same total population size and mean mutation rates), the pdf of which is plotted here as a dashed black line.

Figure 6

The stationary pdf on the slow subspace for a range of systems with various parameters that are omitted here for brevity but which can be found in Appendix pp3. Once again, the solid black line is obtained from theory, the orange histogram from simulations of the original IBM, and the dashed line from a well-mixed model with the same total system size and average mutation rates (weighted by island size).

Figure 7

The stationary pdf in the slow variable $z$ for systems in which both selection and mutation are present. The precise parameters can once again be found in Appendix pp3. Once more, the orange histogram is obtained from simulations of the original IBM, while the solid black line is obtained from reduced theory. The theory once again matches simulations extremely well, especially in comparison with the predictions of a well-mixed model with the same total system size and average mutation rates weighted by island size (black dashed line).

Figure 8

Stationary pdfs for two-island systems exhibiting migration-selection balance. In both cases, mutation is relatively small (see Appendix pp3). In these plots, along with the results from simulations (orange histograms), the predictions from the approximation Eq. (32) (solid black lines), and the predictions from a well-mixed system without the metapopulation structure (black dashed lines), the prediction of Eq. (32) at first order in $s$ has also been given (blue dotted line).