Global Migration Can Lead to Stronger Spatial Selection than Local Migration

Abstract

The outcome of evolutionary processes depends on population structure. It is well known that mobility plays an important role in affecting evolutionary dynamics in group structured populations. But it is largely unknown whether global or local migration leads to stronger spatial selection and would therefore favor to a larger extent the evolution of cooperation. To address this issue, we quantify the impacts of these two migration patterns on the evolutionary competition of two strategies in a finite island model. Global migration means that individuals can migrate from any one island to any other island. Local migration means that individuals can only migrate between islands that are nearest neighbors; we study a simple geometry where islands are arranged on a one-dimensional, regular cycle. We derive general results for weak selection and large population size. Our key parameters are: the number of islands, the migration rate and the mutation rate. Surprisingly, our comparative analysis reveals that global migration can lead to stronger spatial selection than local migration for a wide range of parameter conditions. Our work provides useful insights into understanding how different mobility patterns affect evolutionary processes.

Appendix

In this appendix, we present details for studying evolutionary competition of two strategies in finite islands with local migration. The case of global migration can be analyzed in a similar fashion.

Specifically, let us consider two binary strategies A and B with the 2×2 payoff matrix, which gives the payoff for the column player:

(11)

Denote by x _A (x _B ) the fraction of A (B) individuals, and we have x _A +x _B =1. Denote by $x_{A}^{i}$ ($x_{B}^{i}$) the fraction of A (B) individuals in the site i, and we have $\sum_{i = 1}^{M} x_{A}^{i} = x_{A}$ and $\sum_{i = 1}^{M} x_{B}^{i} = x_{B}$. The interaction pattern of strategies is characterized by [I _AA ,I _AB ;I _BA ,I _BB ], where I _XY denotes the total number of interactions between X and Y. By symmetry, we have I _AB =I _BA . The payoff of an A-individual is P _A =(aI _AA +bI _AB )/(Nx _A ) and its fitness is given by f _A =exp(βP _A ), where β denotes the intensity of selection.

Perturbation method has proved to be particularly useful for studying evolutionary dynamics in the limit of weak selection. Following previous practice [53, 56, 81], we can obtain the equilibrium fraction of A individuals (up to the first order of the selection strength β):

(12)

where F _A and F _B are the total payoff of A and B, respectively, and the subscript “0” indicates that 〈⋅〉 is averaged from all possible population configurations at neutrality with β=0.

Natural selection favors A over B if 〈x _A 〉>1/2. This condition is equivalent to

(13)

which can further boil down to

(14)

At neutrality, the two types A and B act like different colors of individuals, and thus index permutations do not bring in any changes. We have 〈x _B I _AA 〉₀=〈x _A I _BB 〉₀ and 〈x _B I _AB 〉₀=〈x _A I _BA 〉₀. Therefore, we get the following σ-dominance condition for A to be favored [56]:

(15)

where the coefficient σ summarizes the spatial effect, and its formula can be given by [55]

(16)

For the simplified prisoner’s dilemma, cooperation is favored by selection if the benefit-to-cost ratio satisfies:

(17)

Substituting $I_{AA} = N^{2} \sum_{i = 1}^{M} x_{A}^{i} x_{A}^{i}$, $I_{AB} = N^{2} \sum_{i = 1}^{M} x_{A}^{i} x_{B}^{i}$ into the above inequality, we can obtain the critical $(\frac{b}{c} )^{*}$:

(18)

We note that for notational simplicity, we drop the sums and cancel the common factor N ² in the above equation.

Using the symmetry condition under neutrality, we can further simplify the above condition. To be concrete, $\langle x_{B} x_{A}^{i} x_{A}^{i} \rangle_{0} + \langle x_{B} x_{B}^{i} x_{A}^{i} \rangle_{0} = \langle x_{B} x_{*}^{i} x_{A}^{i} \rangle_{0} = \langle x_{B} x_{*}^{i} x_{*}^{i} \rangle_{0} - \langle x_{B} x_{B}^{i} x_{*}^{i} \rangle _{0}$, where $x_{*}^{i}$ denotes the fraction of individuals in the site i. The first term $\langle x_{B} x_{*}^{i} x_{*}^{i}\rangle_{0}$ can be interpreted as the probability that for three randomly chosen individuals, the first individual is a B strategist, and the second and the third are in the same site. The second term $\langle x_{B} x_{B}^{i} x_{*}^{i} \rangle_{0}$ represents the probability that for three randomly chosen individuals, the first and the second individuals have the same B strategy while the second and the third belong to the same site. Using the fact that both strategies A and B are equally present in the population, $\langle x_{B} x_{*}^{i} x_{*}^{i} \rangle_{0} - \langle x_{B} x_{B}^{i} x_{*}^{i} \rangle_{0}$ can be equivalently written as $\frac{1}{2} \operatorname {Prob} \{G_{i} = G_{j} \} - \frac{1}{2} \operatorname {Prob} \{S_{i} = S_{j}, G_{j} = G_{k} \}$. In a similar way, $\langle x_{B} x_{A}^{i} x_{A}^{i} \rangle_{0} - \langle x_{B} x_{B}^{i} x_{A}^{i} \rangle_{0} = \langle x_{A}^{i} x_{A}^{i} \rangle_{0} - \langle x_{A} x_{A}^{i} x_{A}^{i} \rangle_{0} - ( \langle x_{B} x_{B}^{i} x_{*}^{i}\rangle_{0} - x_{B} x_{B}^{i} x_{B}^{i}\rangle_{0} ) = \langle x_{A}^{i} x_{A}^{i} \rangle_{0} - \langle x_{B} x_{B}^{i} x_{*}^{i}\rangle_{0} = \frac{1}{2} \operatorname {Prob} \{S_{i} = S_{j}, G_{i} = G_{j} \} - \frac{1}{2} \operatorname {Prob} \{S_{i} = S_{j}, G_{j} = G_{k} \}$. Therefore we get the critical b/c ratio in the following form [53]:

(19)

where these pair/triplet correlations are taken at neutral evolution. The key for these calculations is to calculate $\operatorname {Prob} \{G_{i} = G_{j} \}$, the probability that any two randomly individuals are from the same site.

To calculate these quantities using coalescent theory [115], it is useful to consider the rescaled mutation and migration rates, μ=Nu and ν=Nv.

Let us label the M sites from 0,…,M−1. Taking into accounting the fact that these sites are located at a regular ring with periodic boundary conditions, we can use an integer l∈(−∞,∞), computed modulo M, to denote an individual i’s site number G _i as G _i =l mod M. In this way, we can regard the migration of an individual between neighboring sites as an unbiased random walk on a one-dimensional plane with integer coordinates l∈(−∞,∞).

After their coalescence, the signed displacement at time τ, d=G _i (τ)−G _j (τ), between any randomly chosen two individuals i and j, follows the probability distribution

(20)

where I _|d|(ντ) is the modified Bessel function of its first kind. It is easy to see that individuals i and j are in the same site if and only if d mod M=0. Thus, we can calculate the pair correlation $\operatorname {Prob} \{G_{i} = G_{j} \}$ as follows:

(21)

$\operatorname {Prob} \{S_{i} = S_{j}, G_{i} = G_{j} \}$ can be calculated as follows.

(22)

Calculating the remaining quantity $\operatorname {Prob} \{S_{i} = S_{j}, G_{j} = G_{k} \}$ requires a little bit more work, but we are able to obtain the explicit expression (however, it is too tedious to be included here). Substituting these quantities into Eq. (19), we obtain the explicit formula for the critical b/c ratio.

For completeness, we show the explicit expression of the structural coefficient σ _l =c ₁/c ₂ for the local migration case, where c ₁ and c ₂ are given by

(23)

(24)

For the global migration case, the formula for σ _g can be given by

(25)

For low mutation and infinitely many sites, the above formulae for σ _g and σ _l can be greatly simplified

(26)

(27)

It is easy to check that σ _l <σ _g holds for any ν>0 in above equations. Therefore, for low mutation and infinitely many sites, global migration always leads to stronger spatial effect than local migration.