Sources and sinks: a stochastic model of evolution in heterogeneous environments

Abstract

We study evolution driven by spatial heterogeneity in a stochastic model of source-sink ecologies. A sink is a habitat where mortality exceeds reproduction so that a local population persists only due to immigration from a source. Immigrants can, however, adapt to conditions in the sink by mutation. To characterize the adaptation rate, we derive expressions for the first arrival time of adapted mutants. The joint effects of migration, mutation, birth, and death result in two distinct parameter regimes. These results may pertain to the rapid evolution of drug-resistant pathogens and insects.

FIG. 1

Source-sink model. We consider 2 patches and 2 genotypes. The wild-type can reproduce only in patch 1 while the mutant can grow in both patches. (a) Organisms mutate and migrate (indicated by arrows) and die at rate δ. (b) A wild-type population in patch 1 will in time give rise to a mutant in patch 2. Arrows indicate the two competing pathways.

FIG. 2

Mean first arrival time (MFAT) vs. parameters, for (a) path ↱ and (b) path ⬏. In each case the MFAT depends on only two lumped parameters: c_i and κ_i. In both plots clearly two regimes can be discerned as a function of κ_i.

FIG. 3

Mean first arrival time T as a function of parameters. Unless specified, parameters are: r = 1, δ = 10⁻¹, ν = 10⁻³, μ_b = 10⁻⁴, μ_b = 10⁻⁷, K = 10⁵. (a) T as a function of the carrying capacity K, for various mutation rates μ_f. Data points are averages over 10⁴ simulations; lines are corresponding theoretical predictions. (b) T versus migration rate ν, for various death rates δ. (c) As predicted by the theory all data points from Figs (a) and (b) collapse on a single curve after rescaling. The plot shows two regimes, for κ_↱ << 1 and κ_↱ >> 1, corresponding to mutation- or migration-limited dynamics.

FIG. 4

T vs. reproduction rate of wild-type in the sink, r_2w. As long as r_2w ≈ δ + ν, T is insensitive to r_2w because path ⬏ is dominant. When r_2w ≈ δ + ν, T_↱ ≈ T_⬏ and the process speeds up about two-fold. If r_2w ≈ δ + ν patch 2 is no longer a sink. Also shown is the mean time before n_2m = 500. Clearly the growth of the mutant colony slows down dramatically as r_2w approaches r. Parameters used: r = 1, δ = 10⁻¹, ν = 10⁻³, K = 10⁵, μ_f = 10⁻⁷, μ_b = 10⁻⁴.