Consumer resource interactions and the evolution of migration

Abstract

Theoretical studies have demonstrated that selection will favor increased migration when fitnesses vary both temporally and spatially, but it is far from clear how pervasive those theoretical conditions are in nature. While consumer-resource interactions are omnipresent in nature and can generate spatial and temporal variation, it is unknown even in theory whether these dynamics favor the evolution of migration. We develop a mathematical model to address whether and how migration evolves when variability in fitness is determined at least in part by consumer-resource coevolutionary interactions. Our analyses show that such interactions can drive the evolution of migration in the resource, consumer, or both species and thus supplies a general explanation for the pervasiveness of migration. Over short time scales, we show the direction of change in migration rate is determined primarily by the state of local adaptation of the species involved: rates increase when a species is locally maladapted and decrease when locally adapted. Our results reveal that long-term evolutionary trends in migration rates can differ dramatically depending on the strength or weakness of interspecific interactions and suggest an explanation for the evolutionary divergence of migration rates among interacting species.

INTRODUCTION

Populations at equilibrium with local selection should be adapted to their local environments (Kawecki and Ebert 2004). This should, in turn, favor philopatry over migration to different environments since offspring are likely to have higher fitness in their parental environment. In other words, local adaptation selects against migration (Balkau and Feldman 1973; Teague 1977). Yet, the vast majority of species exhibit some level of migration, mediated by behavior or morphology (Ronce 2007). Evolutionary biologists have not discovered a general solution to this puzzle.

Environmental variability was thought to be a potentially potent mechanism that can favor higher migration rates as a bet hedging strategy (Gadgil 1971). However, subsequent theoretical studies showed that spatial environmental variation alone selects against migration (Balkau and Feldman 1973; Teague 1977). Those studies modeled a species inhabiting a heterogonous environment wherein high fitness in one patch was combined with low fitness in the other. This trade-off in which migration “would lead more offspring to reach lower quality rather than higher quality habitats” (Leturque and Rousset 2002) sets up selection against migration.

Yet other studies show that increased migration rates are selectively favored by particular forms of spatial variability when combined with temporal variability in fitness (Gillespie 1981; McPeek and Holt 1992; Holt and McPeek 1996; Blanquart and Gandon 2011). Gillespie (1981) demonstrated that selection will favor increased migration when fitnesses vary both temporally and spatially. McPeek and Holt (1992) came to the same conclusion when analyzing models in which carrying capacities have spatiotemporal variation. Even spatially asynchronous chaotic population dynamics can favor some amount of migration (Holt and McPeek 1996). These studies confirm that direct selection for migration requires spatiotemporal variation in fitness, but also that the particular details of variation that favor migration must be finely tuned. While these studies show conditions in theory exist that favor migration, it is far from clear how pervasive those conditions are in nature.

It seems unlikely that the specific patterns of spatiotemporal variation required for migration to evolve could be produced ubiquitously by abiotic environmental conditions. However, consumer-resource dynamics routinely produce capricious temporal variation in fitness for interacting species, potentially leading to continuously changing selection (Maynard Smith 1978). Numerous consumer-resource coevolutionary models confirm the cyclical temporal variability in fitness, notably leading to selection for sex and recombination under the Red Queen hypothesis (Nee 1989; Peters and Lively 1999; Otto and Nuismer 2004; Agrawal and Otto 2006; Gandon and Otto 2007; Salathé et al. 2008b; Salathé et al. 2009). Furthermore, cyclical temporal variation operating out of phase in different communities can generate spatial variation in fitness and local adaptation (Dybdahl and Storfer 2003; Salathé et al. 2008a; Lively 2010). Beyond theoretical predictions, these dynamics have been observed in natural host-parasite systems (Dybdahl and Lively 1998; Decaestecker et al. 2007; Jokela et al. 2009).

While consumer-resource interactions are virtually omnipresent in nature, it is not known even in theory whether such interactions can generate the spatial and temporal dynamics of selection necessary to favor the evolution of migration. A serious cause for doubt is that any appreciable migration might quickly synchronize coevolutionary dynamics over space nullifying any initial advantage to migration. Here, we develop theory specifically to address whether and how migration evolves when variability in fitness is determined at least in part by antagonistic coevolutionary interactions and thereby assess whether such interactions might provide a general explanation for the common occurrence of migration in species.

MODEL

We develop our theory using a basic population genetics model of a consumer-resource species pair, which we call host and parasite for convenience. The model assumes both species are haploid, have discrete non-overlapping generations, and that the species co-occur in, and can move between two distinct patches (A and B). The proportion of the host population in patch A is designated by c_H with the remaining 1 − c_H found in patch B. Similarly, the fraction of parasite population in patch A is designated by c_P.

We assume species encounter one another at random within a patch and that each has a biallelic locus (with alleles denoted “0” and “1”) that determines fitness outcomes when they interact. The frequency of the interaction allele “1” in species X in patch k (k = A, B) is noted by L_X,k, where here and below we let “H” and “P” denote host and parasite respectively for the species subscript X. We assume these fitness outcomes are governed by a matching alleles model (Frank 1992) such that an encounter between a host and parasite with matching genotypes decreases host fitness (by β) and increases parasite fitness (by α); mismatched genotypic encounters are assigned a fitness of 1 for both species.

The rate an individual migrates is determined genetically by a second biallelic locus that may be linked to the fitness locus. Individuals with the “0” or “ancestral” migration allele at this locus move to the other patch at rate m_X whereas those with the ”1” or “mutant” allele migrate at rate m_X + δ_X. We let M_X,k denote the frequency of the mutant migration allele for species X in patch k.

We show below how a migration mutation can evolve either in response to direct selection or as a correlated response to selection on the fitness interaction locus. The indirect response depends on the disequilibrium between the two loci. We let D_X,k designate the disequilibrium coefficient for species X in patch k. Any disequilibrium will be reduced at the species-specific rate of recombination between the two loci r_x.

We use an alternative notation when we want to emphasize the genetic structure of a species across both patches. Specifically, we use an over-bar to indicate the weighted mean across both patches

$${\overline{L}}_X=c_X{L}_{X,A}+\left(1-c_X\right)L_{X,B},$$ (1a)

$${\overline{M}}_X=C_X{M}_{X,A}+\left(1-c_X\right)M_{X,B},$$ (1b)

$${\overline{D}}_X=c_X{D}_{X,A}+\left(1-c_X\right)D_{X,B},$$ (1c)

and a hat to designate the difference between patches

$${\widehat{L}}_X=L_{X,A}-L_{X,B},$$ (2a)

$${\widehat{M}}_X=M_{X,A}-M_{X,B},$$ (2b)

$${\widehat{D}}_X=D_{X,A}-D_{X,B}.$$ (2c)

The dynamics of the patch-specific variables are mathematically equivalent to the dynamics of these global variables. Finally, note that the overall mean rate of migration in species X is

$$\overline{m}=m_X+{\overline{M}}_X{\delta }_X.$$ (3)

If δ_X > 0, eq. (3) implies that increases or decreases in the mean frequency of the migration mutation correspond directly to increases or decreases, respectively, in the mean migration rate of that species.

In Appendix A, we derive a deterministic system of recursions for the allele frequencies and disequilibria in terms of frequencies of the four possible two-locus haplotypes (11, 10, 01, and 00) for each species in each patch (eqs. A1–A14). Table 1 summarizes all notation, parameters, and variables. The system describes the joint evolution of migration and adaptation within each species. The recursions assume that the life cycles of both host and parasite are fully synchronized and ordered as selection → migration → random mating → recombination → reproduction. They also assume that population sizes of each species are regulated locally after selection and then regulated globally following migration. Therefore, the fraction of the population in each patch is a dynamic variable that can fluctuate in response to differences in local migration rates but not local selection.

Table 1.

Summary of notation, parameters, and variables

Notation	Definition
X (=P,H)	Species, either parasite or host
i (= 0,1)	Migration allele
j (= 0,1)	Interaction allele
k (= A,B)	Patch
ȲX	Weighted mean of variable Y across both patches in species X
ŶX	Difference of variable Y between patches in species X
Parameter	Definition
α, β	Fitness interaction coefficients (parasite or host respectively)
mX	Ancestral migration rate in species X
δX	Mutant migration rate modifier in species X
rX	Recombination rate between the two loci
Variables	Definition
LX,k	Frequency of the interaction allele “1” in species X in patch k
MX,k	Frequency of the mutant migration allele in species X in patch k
DX,k	Linkage disequilibrium coefficient for species X in patch k
Xijk	Frequency of two-locus haplotypes in species X with migration allele i and interaction allele j in patch k
cX	Proportion of species X in patch A

The choice to assume that post-migration densities are regulated globally is a modeling strategy that allows us to isolate the influence of species interactions on the evolution of migration. Indeed, we prove in Appendix B that post-migration global density regulation has no effect on the mean migration rate over patches. This contrasts with models of the evolution of migration that assume population abundances are regulated separately within each patch after migration (Hamilton and May 1977; McPeek and Holt 1992; Leturque and Rousset 2002; Billiard and Lenormand 2005). As these studies show, migration can evolve in response to the kin competition generated by local density regulation. Thus, our assumption that densities are regulated globally after migration in effect eliminates kin competition and allows us to attribute any changes in migration to the particular influence of interspecific interactions.

ANALYSES

Numerical iterations

The recursion equations presented in Appendix A, while complex, are relatively straightforward to explore numerically by forward iteration. For a given set of parameters and initial conditions, we iterated equations (A1–A14) using MATLAB (R2010a, The Mathworks) for 10,000 generations. To represent the space of initial frequencies for the interaction alleles, we randomly sampled 1000 sets of four allele frequencies, one frequency for each species in each patch. (Since the recursions are deterministic, our sampling of initial allele frequencies is the only source of randomness in our analyses.) We used the same 1000 sets as initial conditions for each combination of parameter values tested. Each species’ mutant migration allele was initialized at frequency 0.05 in linkage equilibrium with the interaction locus in both patches. With this low initial frequency, we in effect examine how selection would act on a new mutation in the population. After 10,000 generations, we calculated the cumulative change in frequency of the mutant migration allele.

Our computations found larger overall changes in migration with lower recombination rates (fig. 1) or with stronger interactions (fig. 2). With tight linkage (r_X = 10⁻⁴), we found large magnitude changes at the migration locus (fig. 1a) compared to loci that recombine relatively more freely (r_X = 10⁻¹, fig. 1d). When the interaction coefficients were small (α = β = 10⁻⁴), very little change occurred at migration loci (fig. 2a) compared to when interactions had a large effect on fitness, (fig. 2d; α = β = 0.1).

Figure 1.

The effect of recombination on the change in the migration mutant frequency in the host and parasite assuming weak selection and migration. Shown is the total change in frequency of the migration mutant (“1”) allele from initial frequency 0.05 calculated by iterating eqs (1–14) for 10,000 generations for 1000 sets of initial allele frequencies at the interaction locus and initial linkage equilibrium. Each simulation was color coded based on the average local maladaptation across both patches over the course of the entire 10,000 generations (red: host; blue: parasite). Recombination rate (r_X) was a) 10⁻⁴ b) 10⁻³ c) 10⁻² d) 10⁻¹. Parameter values for all simulations: ancestral migration rate (m_X) was 10⁻⁴ and interaction strength coefficients (α and β) were 10⁻⁴, migration mutant effect (δ_x) was 10⁻⁵ (10% of the migration rate).

Figure 2.

The effect of the strength of the interaction on the change in the migration mutant frequency in the host and parasite. Details and parameter values are the same as figure 1 except that α, β, and m_X varied a) 10⁻⁴ b) 10⁻³ c) 10⁻² d) 10⁻¹. For all simulations the migration mutant effect (δ_x) was fixed at 10% of the migration rate (m_x) and r_X = 0.01.

Previous results suggest that the state of local adaptation should modulate when migration is favored to increase or decrease (Billiard and Lenormand 2005; Blanquart and Gandon 2011). Higher migration rates tend to be favored given local maladaptation whereas the reverse is true for local adaptation. For matching alleles models the spatial covariance of host and parasite interaction allele frequencies, cov(L_H,L_P), measures host local malaptation (Gandon and Nuismer 2009). Because of the symmetrical nature of our model, host local maladaptation is equivalent to parasite local adaptation. The host is locally maladapted and the parasite is locally adapted in a given generation if cov(L_H,L_P) > 0. When this occurs, host and parasite interaction allele frequencies vary in parallel across patches, which is detrimental to the host because the parasites tend to match their hosts, leading to more infection and host maladaptation. The opposite applies when the covariance between allele frequencies is negative. In our model, this measure of host local maladaptation is computed as cov(L_H,L_P) = L̂_HL̂_P/4, which weights patches equally.

We determined the mean state of local maladaptation for a particular case by averaging cov(L_H,L_P) calculated at each generation over all 10,000 generations. Our results show that when the host was on average locally maladapted, the frequency of its mutant migration allele increases whereas it decreases when the host tends to be locally adapted (compare red vs. blue points in figs. 1 and 2). The same pattern was shown in the parasite. The magnitude of change in the migration rate corresponds well to the magnitude of local maladaptation. Generally, migration increases in one species and decreases in the other because when one species is locally maladapted, then the other must be locally adapted.

We examined the effects of baseline migration rate (m_X) and interaction strengths (α and β) on the evolution of migration by varying these parameters over three orders of magnitude from 10⁻³ to 10⁻¹. Our results (figs. 3 and 4) reveal two general patterns. First, in cases where interactions are weak (fig. 3: first column) or migration is strong (fig.3, top row), the species that is locally maladapted on average tends to evolve a higher migration rate, while the other species evolves a lower migration rate. Second, in cases where the strength of interactions (α and β) are an order of magnitude more than the baseline migration rate (m_X) (fig. 3: panels f, h, and i), both species evolve higher migration rates over the long run and the mean state of local adaptation is not associated with long-term changes in migration. These strong interaction simulations lead to a co-escalation of migration rates in host and parasite. Figure 4 shows an example typical of the joint evolutionary trajectories of migration rates in both species. From the figure, one can see that the migration rate eventually rises in each species via an attenuating ratcheting process. Indeed, over the short term the migration rate increases in one species and decreases in other but over an entire cycle, the increases dominate the decreases. Consistent with the other results, short-term increases or reductions in migration corresponded to periods of local maladaptation and adaptation, respectively (red vs. blue points in fig. 4). Nevertheless, over the long-term migration rates increase in both species.

Figure 3.

A comparison of the change in the migration mutant frequency in the host and parasite. The figure shows the total change in frequency of the migration mutant (“1”) allele from starting frequency 0.05 calculated by iterating eqs (1–14) for 10,000 generations for 1000 sets of initial allele frequencies at the interaction locus. Each simulation was color coded based on which species was on average local maladapted across both patches over the course of the entire 10,000 generations (red: host; blue: parasite). Ancestral migration rate (m_X) is indicated on the right side for each row and interaction strength coefficients (α and β) is indicated across the top for each column. Parameter values for all simulations: migration mutant effect (δ_x) was fixed at 10% of the migration rate and recombination rate (r_X) was fixed at 0.01.

Figure 4.

A sample simulation of the change in the migration mutant frequency in the host and parasite assuming strong interactions. The simulation starts in the lower left where the change in frequency is 0 for each species. Plotted is the cumulative change in frequency of the migration mutant after each generation for 5,000 generations. The migration mutant was introduced into the population an initial frequency of 0.05 in linkage equilibrium with the interaction locus. At each generation, we calculated which species was locally maladapted across both patches (red: host; blue: parasite). Interaction alleles were initiated at frequencies L_HA = 0.75, L_HB = 0.25, L_PA = 0.55, and L_PB = 0.45. Ancestral migration rate (m_X) was 10⁻⁴ and interaction coefficients (α and β) were 10⁻². The migration mutant effect (δ_x) was 10⁻⁵ (10% of the migration rate) and recombination rate (r_X) was fixed at 0.01.

A small parameters approximation

The complexity of the co-evolutionary recursions makes it difficult to identify specific processes driving the evolution of migration. To address this, we derived an approximate but more transparent set of recursion equations assuming weak selection, weak mutational effects on migration, and equal patch sizes. This is similar to approaches used by M’Gonigle et al. (2009) to study the evolution of mutation rate in a host-parasite interaction and by Blanquart and Gandon (2011) in their analysis of the evolution of migration with spatiotemporal variation.

Our “small parameters” approximation assumes the migration parameters m_H, m_P, δ_H, and δ_P, and the interaction coefficients, α and β, are all of order ε, where 0 < ε ≪ 1. In addition, we assume equal patch sizes in both species (c_H = c_P = ½). It is enlightening to write the approximation in terms of the strengths of local selection for each species. We let

$$s_{H,k}=-\beta \left(L_{P,k}-1/2\right)$$ (4a)

be the strength of selection in patch k against matching in the host and let

$$s_{P,k}=\alpha \left(L_{H,k}-1/2\right)$$ (4b)

be the corresponding strength of selection for matching in the parasite. These definitions imply, for example, that where the “1” interaction allele in the parasite is in the majority (i.e., where L_P,k > ½) the matching allele is selected against in the host (s_H,k < 0 because β > 0). In contrast, if the “1” allele is predominant in the host (L_H,k > ½), the matching allele is favored in the parasite (s_P,k > 0). Using these assumptions and definitions, and ignoring terms of order ε² and smaller, we derived approximate recursion equations for the between-generation change in the crosspatch mean frequency of the migration rate mutant in each species

$$\mathrm{\Delta }{\overline{M}}_H=2\left[cov(s_H,D_H)+{\overline{s}}_H{\overline{D}}_H\right]$$ (5a)

and

$$\mathrm{\Delta }{\overline{M}}_P=2\left[cov(s_P,D_P)+{\overline{s}}_P{\overline{D}}_P\right]$$ (5b)

where

$${\overline{s}}_H=-\beta \left({\overline{L}}_P-1/2\right)$$ (6a)

and

$${\overline{s}}_P=\alpha \left({\overline{L}}_H-1/2\right)$$ (6b)

describe, respectively, the mean strength of selection for mismatching in the host and for matching in the parasite over both patches and cov(s_X, D_X) = (s_X,A − s_X,B)D̂_X/4 is the covariance between local selection and local linkage disequilibrium in species X.

Equations (5) are identical in form to an expression derived by Blanquart and Gandon (2011) for the evolution of migration in a single species experiencing spatiotemporal fitness variation in the absence of direct migration costs (see the first term in their eq. 8a). The resemblance is striking given considerable differences between their model, which assumes an infinite number of patches with extrinsically determined variation in local fitness, and ours, which assumes two patches and internally driven fluctuations in local fitness. This suggests that equations like (5) describe a general process by which migration rates evolve in response to spatiotemporal variation in fitnesses. This “indirect selection” on migration can then be combined with direct selection on migration via migration costs, as was done by Blanquart and Gandon (2011).

Equations (5) also reveal an important difference between our model and Blanquart and Gandon’s. Their model assumes that patches experience unsynchronized fluctuations in selection such that the average selection over all patches is identically zero at all times. This means, in effect, that s̄_X = 0 identically and so indirect selection on migration is completely determined by cov(s_X, D_X). In our model, by contrast, s̄_X will generally differ from zero in any generation. This suggests that overall adaptation, in addition to spatial variation, could contribute to the evolution of migration in our model. Given that the sign and magnitude of s̄_X are expected to change over time, however, it is not immediately clear what long term evolutionary trends this second term should drive.

As equations (5) make apparent, it is necessary to consider the dynamics of linkage disequilibrium to gain further insight into the processes that drive the evolution of migration. We show in Appendix C that, given the weak parameters assumptions and reasonably strong recombination between the migration and local adaptation loci, the sizes of linkage disequilibria quickly decay to order ε. Since the selection coefficients are also of order ε, it is evident from (eq. 5) that the concomitant changes in migration rates would be of order ε². In Appendix C, we show that the linkage disequilibria would continue to evolve an order of magnitude faster unless they assume specific “quasi-linkage equilibrium” (QLE) forms in terms of the allele frequencies and parameters (eqs. C3). At QLE, linkage disequilibria evolve at a rate of order ε². However, migration allele frequencies would also evolve at the same slow pace. This precludes the standard application of “separation of time scales” approaches (e.g., Barton and Turelli 1991). Still the evolutionary dynamics of migration rates (eq. 5) at a QLE provide an interesting analytic counterpart to our numerical iterations described in the section just above. Those QLE dynamics are (substituting eqs. C3 into eqs. 5)

$$\mathrm{\Delta }{\overline{M}}_H^{\text{QLE}}=4\frac{1-r_H}{r_H}\beta \left[cov\left(L_H,L_P\right)V_H^M{\delta }_H-cov\left(L_H,M_H\right)\left({\overline{L}}_P-\frac{1}{2}\right)\left(2m_H+{\delta }_H\right)\right],$$ (7a)

and

$$\mathrm{\Delta }{\overline{M}}_P^{\text{QLE}}=-4\frac{1-r_P}{r_P}\alpha \left[cov\left(L_H,L_P\right)V_P^M{\delta }_P-cov\left(L_P,M_P\right)\left({\overline{L}}_H-\frac{1}{2}\right)\left(2m_P+{\delta }_P\right)\right],$$ (7b)

where $V_X^M={\overline{M}}_X\left(1-{\overline{M}}_X\right)+{\widehat{M}}_X^2/4$ is a measure of genetic diversity at the migration locus. Consistent with our numerical results (figs. 1–3), equations (7) show that migration rates evolve more rapidly with tighter linkage (smaller r_X) and with stronger fitness interactions (larger values of α and β).

The QLE approximations (eqs. 7) highlight two causes of evolution at the migration locus. These can be understood in reference to the two terms inside the square brackets of the equations. Note that the first term in both equations includes the factor cov(L_H, L_P) which, as discussed above, is a measure of local maladaptation for the host and of local adaptation for the parasite. If the host is locally maladapted (i.e., cov(L_H, L_P) > 0), this term will drive its migration rate higher and the associated term for the locally adapted parasite will favor less migration. The reverse holds if the host is locally adapted (cov(L_H, L_P) < 0). The opposing impacts of this first term on host and parasite migration rates mirror to a remarkable degree our numerical results that assume weak migration and selection (figs 1, 2 and 3g). This suggests that local adaptation is the dominant influence on the evolution of migration in these cases.

The second terms inside the square brackets of equations (7) describe how global directional selection on the fitness locus imposed by the parasite on the host, or vice versa, is converted into changes at the migration locus via the between-patch component of total linkage disequilibrium, cov(L_x,M_X) = L̂_XM̂_X/4. Thus, migration can increase (or decrease) in one or both species as a “correlated response” to global selection, even when neither species is locally maladapted, i.e., when cov(L_H, L_P) = 0. For example, migration would increase in the host via this mechanism were the migration locus in positive linkage disequilibrium with the fitness locus, i.e. cov(L_H, M_H) > 0, and were L̄_P < ½, in which case the “1” allele is favored in the host since then hosts carrying that allele are less likely to encounter matching parasites and thus suffer less harm overall than hosts with “0” alleles. It is not difficult to imagine scenarios that would favor higher migration rates in both species, lower rates in both species, or an increase in one and decrease in the other. This component of evolution could work in concert with—or in complete opposition to—the component determined by local maladaptation/adaptation. We suspect that the correlated responses to global selection in the host and parasite may be responsible for the numerical results with strong selection (fig. 3: panels f, h, and i) in which both the host and parasite evolved higher migration rates regardless of one being locally maladapted and the other being locally adapted on average over time.

Encouraged by this idea, we returned to our numerical iterations and assessed the impact of correlated responses on the net evolution of migration. The correlated response (second term in eqs. 5) is the product of the mean strength of selection in the host or parasite over both patches (s̄_X) and the mean linkage disequilibrium (D̄_X). For each species X we computed in each generation the correlated response of migration, ${\overline{s}}_H{\overline{D}}_H=-\beta \left({\overline{L}}_P-\frac{1}{2}\right){\overline{D}}_H$ for the host and ${\overline{s}}_P{\overline{D}}_P=\alpha \left({\overline{L}}_H-\frac{1}{2}\right){\overline{D}}_P$ for the parasite; summing across the 10000 generations gave the net correlated response. This was repeated for each combination of parameter values and initial condition. We calculated mean linkage disequilibrium (D̄_X) by taking the weighted mean of the patch specific disequilibria, D_x,k, defined by equations (A19–A20). The results are summarized in figure 5.

Figure 5.

Contributions from correlated responses for the evolution of migration. Using the same set of simulations described in figure 3, for each species X in each patch k we computed the correlated response of migration to local adaptation (second term in eq. 5) using the patch specific disequilibria. In each species, at each generation, we computed the mean value across both patches and summed these values for the 10,000 generations for each of the 1000 initial conditions for a given parameter combination. Bars indicate the mean and 75% confidence intervals of the simulations. Bars labeled H and P correspond to host and parasite, respectively. Color-coding has been used to indicate simulations where the host (black) or parasite (white) was locally maladapted. Note the different ordinate scales for each panel.

The overall magnitude of the net correlated response was found to be small in most cases and is almost always positive (or at least non-negative) (fig. 5). Mathematically, this means that the between-patch disequilibrium and mean strength of selection tend to have the same sign. If we focus only on the strong selection cases, the correlated response was always relatively large and positive regardless of the state of local adaptation (fig. 5: panels f, h, and i; compare the ordinate scales of those panels to the others). Under strong selection, the correlated response may explain the long-term escalation of migration rates in both the host and parasite (fig. 3: panels f, h, and i). In effect, the correlated response more than compensates for the decline in migration rate favored during periods when a species is locally adapted (e.g. fig. 4).

DISCUSSION

Our analyses show that consumer-resource interactions can drive the evolution of migration in one or both species. Over shorter time scales, we found that the direction of change in migration rate is determined primarily by the state of local adaptation of the species involved: rates increase when a species is locally maladapted and decrease when it is locally adapted. We also discovered that long-term evolutionary trends in migration rates depend markedly on the strength of interspecific interactions. To wit, when the species have relatively weak reciprocal effects on each other’s individual fitness, one species evolves a higher migration rate while the rate of its partner declines. In contrast, migration rates eventually increase in both species when their interactions have strong fitness consequences.

Our results clarify the immediate impacts of local adaptation on selection of migration rates. At a given time, local adaptation depends on how a species is distributed over the landscape (Kawecki and Ebert 2004). This is especially pertinent to interacting species since their patterns of local adaptation are determined interdependently by the distributions of partner species. This means, for example, that a host with little population structure can nevertheless experience considerable fitness differences across space if an associated parasite has substantial spatial population structure. A host genotype with high fitness in the presence of local parasites might have substantially lower fitness in the presence of allopatric parasites. Only those host individuals that then do not migrate avoid this fitness cost. Consequently, a sessile host genotype would leave more descendants than a vagile one. Local adaptation of the host population thus favors less migration. This same logic applies to species that inhabit spatially heterogeneous environments, which previous studies have shown selects against migration (Balkau and Feldman 1973; Teague 1977). Those models assume a fixed trade-off in fitness across environments. In our model, fitness trade-offs tend to shift continually over time because of on-going consumer-resource coevolution and lead to alternating periods of local adaptation and maladaptation.

The converse of the argument above implies that local maladaptation favors migration. In our host-parasite model, local maladaptation can be quantified by the spatial covariance of matching allele frequencies in each species at the locus that mediates interspecific fitness effects. A positive covariance means more frequent matching of hosts and parasites which signals host local maladaptation. In this case, the host fitness is higher in the presence of allopatric than sympatric parasites and only those individuals that migrate can reap this fitness benefit. Migrating genotypes, having a higher fitness in allopatric demes, would produce more descendants. In this way, local maladaptation favors increased migration.

Consumer-resource interactions often exhibit complex temporal dynamics with frequency dependent fluctuations. We know from previous theoretical work that consumer-resource coevolution leads to alternating periods of local adaptation and maladaptation (Nee 1989; Jokela et al. 1999; Lively 1999; Nuismer 2006). The intuition above suggests that higher migration rates will be favored some times and lower rates will be favored at others. But this leaves open the larger question of how migration rates in the consumer and resource species will evolve over the long term.

Our results show that, when the species interactions are weak, long-term evolutionary trends in migration mirror short-term evolutionary responses. We found that migration rates climb over the long run in species that tend to be locally maladapted and to fall over time in species that are locally adapted on average. These long-term trends can be interpreted as the direct sum of short-term decreases and increases in migration induced, respectively, by the alternating periods of local adaptation and local maladaptation that characterize consumer-resource interactions. In our matching alleles model, whenever a species is locally adapted its partner species must be locally maladapted and whether the consumer or resource species is more often locally maladapted is determined by ancestral conditions. So while we always expect one species to evolve a higher migration rate and the other a lower rate when interspecific interactions are weak, the identities of the specific species that evolve to migrate more and less depend on their history.

Pomiankowski and Bridle (2004) pointed out, in the context of the host-parasite interactions and the evolution of sex, that the assumption of a weak interspecific interactions may overlook cases of potential biological importance. For example, strong interactions (e.g. Red Queen dynamics) can give evolutionary outcomes that are qualitatively different from those produced by weak interactions (see review by Salathé et al. 2008a). In accord with this, our results revealed a strikingly different long-term pattern in the evolution of migration rates for strong consumer-resource interactions than what we observed for weakly interacting species. Specifically, we found that interspecific interactions with strong fitness effects ultimately cause migration rates in both species to escalate despite the occurrence of short-term increases and decreases in migration consistent with the intuition outlined above. As a result, a species that tends to be locally adapted eventually evolves more migration, opposite to what is expected given weak interactions.

Our analyses suggest that the distinct evolutionary dynamic when interactions are strong is driven by correlated responses at the migration locus to direct selection at the interaction locus. The comparatively strong frequency-dependent selection causes large fluctuations in the frequencies of the fitness-determining interaction alleles and, in concert with migration, generates substantial linkage disequilibrium because recombination fails to break down linkage disequilibrium fast enough to counter selection. Migration rates increase via this correlated response because the sign of linkage disequilibrium is often the same sign as the mean strength of selection on the interaction locus. When this occurs, selection favors two-locus genotypes that carry the migration rate mutation more often than were the two loci statistically independent. Combined with ongoing direct selection on migration stemming from local adaptation and maladaptation, this consistently positive correlated response buffers declines and magnifies increases in migration yielding a net increase in migration after each coevolutionary cycle (fig. 4).

We were unable to derive an analytic cause for the tendency of selection and disequilibrium to have the same sign when interactions are strong, however, Blanquart and Gandon (2011) observed a similar pattern in their study of migration evolution in a periodically changing environment. Indeed they found that when selection changes rapidly, linkage disequilibrium tends to lag behind local selection such that they tend to share the same sign. Since strong interactions in our model generate rapid fluctuations in fitness for both species, the same process may be responsible for the joint increases in migration rates found in our numerical results. This possibility calls for further investigation.

When interactions are weak, the same correlated responses that favor more migration when interactions are strong are completely overwhelmed by the costs and benefits of migration from patterns of local adaptation. Weak interactions decrease the rate of frequency-dependent fluctuations in selection. With strong recombination compared with selection, linkage disequilibrium between the two loci is broken down more quickly. As a consequence, the signs of selection and linkage disequilibrium are opposite of one another (Blanquart and Gandon 2011). Directional selection at the interaction locus leads to local adaptation, and like static heterogonous environments, favors sessile individuals (Balkau and Feldman 1973).

Our study focused on the fate of isolated migration mutations introduced at a low initial frequency. This means the maximum increase in migration at the population level in our model is capped by the effect size of the mutant allele on the migration rate. A more complete analysis might consider the sequential substitution of multiple mutations or more complex patterns of standing variation (such as diploid polygenic inheritance) for migration. Such phenomena are beyond the scope of this study, but our results suggest that even were migration rates in both species to increase, at some point migration would match interaction strength (fig. 3, top row) the result of which would be continued increases in migration in one species but lower migration in the other.

While our main concern has been on how local selection affects the evolution of migration, migration itself can affect the ability of a species to track a fluctuating environment by enriching local genetic variation in fitness. Alternative mechanisms such as mutation and recombination can also create genetic variation. Both mechanisms have been investigated previously in the context of host-parasite coevolutionary interactions generating selection for modifiers that increase these forces (Gandon and Otto 2007; M’Gonigle et al. 2009).

Directional selection imposed by host-parasite interactions decreases variation, but frequency dependent cycles can occur that maintain polymorphism. M’Gonigle et al. (2009) found that when cycles are generated by host-parasite coevolution, selection favors modifiers for higher mutation rates. These modifiers had the direct effect of increasing the genetic variation of the population and allowing selection to act more efficiently. Consistent with our results, M’Gonigle et al. (2009) found that stronger recombination weakened the effects of host-parasite interactions on selection for a mutation modifier. Interestingly, they found that drift due to small population sizes broadened the parameter space consistent with increased mutation rates. In these cases, stochastic fluctuations in allele frequencies supplement the deterministic host-parasite cycles, generating continued selection for mutation modifiers. Additional studies have shown these effects can create conditions favoring higher migration rates (Billiard and Lenormand 2005). Our analyses did not consider stochastic evolutionary processes such as random genetic drift but in additional simulations of our model assuming finite population sizes (results not shown), we observed outcomes consistent with these previous results.

Another interesting parallel between our study and M’Gonigle et al. (2009) is their finding that host-parasite cycles disappear with sufficiently high mutation rates. In their model, the dynamic feedback between host-parasite cycling and continued selection for increased mutation rate eventually dampens coevolutionary cycles, which eliminates the selective advantage of modifiers for increased mutation. We see an analogous process ensue in our model as spatial variation in fitness is wiped out by high amounts of migration. In our case, migration is favored only when there is spatial genetic variation in the host and parasite. Increased migration homogenizes populations across space and thereby reduces the potential for subsequent increases in migration.

Blanquart and Gandon (2011) recently analyzed the evolutionary forces generating selection on migration in a spatio-temporally varying environment. Their model describes a single species meta-population with an infinite number of demes. They assumed selection within each deme fluctuates over time with a fixed period, favoring and then disfavoring genotypes, and is spatially asynchronous among demes at all times such that no genotype is favored meta-population wide. Because it considers spatially variable oscillating fitnesses, their model provides an initial understanding of how host-parasite coevolution could favor increased migration when fitness variation is driven directly by interspecific interactions. As noted above, our strong selection results are congruent with their finding that more rapid fluctuations in fitness favor higher migration rates via correlated evolutionary responses enabled by significant amounts of linkage disequilibrium with the same sign as selection. Blanquart and Gandon (2011) also derived a key analytical result that reveals how local maladaptation generates selection for migration. Despite the considerable differences between their model and ours, we demonstrated that the identical process of migration evolution applies to both species in consumer-resource interactions.

Previous theoretical studies have examined the connection between migration and local adaptation in coevolutionary interactions (Gandon et al. 1996; Gandon and Michalakis 2002; Nuismer 2006). While they did not investigate the evolution of migration rates per se, the studies did find that fixed differences in migration rates between interacting species resulted in predictable patterns of local adaptation in antagonistic interactions (Gandon et al. 1996; Gandon and Michalakis 2002). These analyses implicated not just the magnitude of migration, but, more specifically, the relative rates of migration among the interacting species. When absolute rates of migration are low (on the order of 0.1%), then the species with the greater rate of migration will tend to be locally adapted (Gandon et al. 1996; Gandon and Michalakis 2002). The intuition for this is that migration replenishes local genetic variation, thereby increasing the efficiency of selection.

The prediction that the interacting species with the higher migration rate will be locally adapted has been examined empirically many times (reviewed by Greischar and Koskella 2007). However, this theoretical result assumes that migration rates are fixed in both species. If migration rates can evolve, then our model suggests that this outcome may be uncommon. In particular, we showed that weakly interacting species evolve different migration rates and that a species that tends to be locally adapted should have a lower migration rate than the species that was locally maladapted. The latter result is contrary to expectations from the previous analyses that assume fixed migration rate differences.

The model we examined here does not include a number of factors that could impede or impel the evolution of migration, such as direct migration costs, kin selection, random genetic drift, and conditional migration. In doing so, our analysis proves that consumer-resource interactions per se can drive the evolution of migration, perhaps in combination with these other factors. It is inconceivable that any wild species exists completely free from being consumed, harmed, or exploited by other species. Our demonstration that consumer-resource interactions favor the evolution of migration thus supplies a general explanation for the pervasiveness of migration, and also provides accounts for the evolutionary divergence of migration rates among interacting species.

APPENDIX A: Derivation of Recursions

This appendix derives recursion equations that describe the evolution of migration in two interacting species under the assumptions described in the main text. Let P_ijk denote the frequency of the two-locus parasite haplotypes with migration allele i (= 0,1) and interaction allele j (= 0,1) in patch k (= A,B). Similarly, we let H_ijk be the frequency of the host haplotype with migration allele i and interaction allele j in patch k.

Given an encounter with a host with a matching interaction locus allele, a parasite’s fitness is incremented by α > 0. We assume hosts and parasites encounter one another at random within a patch so that the average fitness of a parasite with the two-locus haplotype ij in patch k is

$$w_{\mathit{ijk}}^P=1+\alpha \left(H_{0jk}+H_{1jk}\right).$$ (A1)

The expression in parentheses is the post-reproduction frequency of the hosts in the same patch that match the parasite interaction allele. The corresponding expression for host fitness is

$$w_{\mathit{ijk}}^H=1-\beta \left(P_{0jk}+P_{1jk}\right)$$ (A2)

where β > 0 is the decrement in host fitness given an encounter with a parasite with matching interaction allele.

After selection the within patch haplotype frequencies of the parasites and host species are, respectively,

$$P_{\mathit{ijk}}^{\prime }=P_{\mathit{ijk}}\frac{w_{\mathit{ijk}}^P}{{\overline{w}}_k^P}$$ (A3)

$$H_{\mathit{ijk}}^{\prime }=H_{\mathit{ijk}}\frac{w_{\mathit{ijk}}^H}{{\overline{w}}_k^H}$$ (A4)

where

$${\overline{w}}_k^P=\sum _{i=0}^1\sum _{j=0}^1{P}_{\mathit{ijk}}{w}_{\mathit{ijk}}^P$$ (A5)

is the parasite mean fitness in patch k and

$${\overline{w}}_k^H=\sum _{i=0}^1\sum _{j=0}^1{H}_{\mathit{ijk}}{w}_{\mathit{ijk}}^H$$ (A6)

is the corresponding host mean fitness.

We assume migration follows selection and that selection does not alter the relative fractions of a species found in each patch. However, we do allow those fractions to be altered by migration, with the new proportions being maintained during subsequent global density regulation. Assuming global density regulation within each species after migration assures there is no kin selection on migration so that any evolution of migration must trace to effects of the interspecific interaction locus (Appendix B).

Let $o_{kk}^P$ denote the average fraction of parasites in patch k that stay and let $o_{ik}^P$ be the fraction in patch i that migrate to patch k (i ≠ k). These parasite forward migration rates are

$$o_{kk}^P=\left(1-m_P\right)\sum _{j=0}^1{P}_{0jk}^{\prime }+\left[1-\left(m_P+{\delta }_P\right)\right]\sum _{j=0}^1{P}_{1jk}^{\prime },$$ (A7a)

$$o_{ik}^P=m_P\sum _{j=0}^1{P}_{0ji}^{\prime }+\left(m_P+{\delta }_P\right)\sum _{j=0}^1{P}_{1ji}^{\prime }.$$ (A7b)

Let $c_k^P$ be the fraction of parasites in patch k before migration. Note that $c_{\mathrm{A}}^p+C_{\mathrm{B}}^P=1$. After migration the fraction of parasites in patch k is

$$c_k^{\prime P}=c_k^P{o}_{kk}^P+c_i^P{o}_{ik}^P.$$ (A8)

We assume parasite densities are regulated globally after migration such that the fractions $c_k^{\prime P}$ are not changed. The analogous expressions for the host are

$$o_{kk}^H=\left(1-m_H\right)\sum _{j=0}^1{H}_{0jk}^{\prime }+\left[1-\left(m_H+{\delta }_H\right)\right]\sum _{j=0}^1{H}_{1jk}^{\prime },$$ (A9a)

$$o_{ik}^H=m_H\sum _{j=0}^1{H}_{0ji}^{\prime }+\left(m_H+{\delta }_H\right)\sum _{j=0}^1{H}_{1ji}^{\prime },$$ (A9b)

and

$$c_k^{\prime H}=c_k^H{o}_{kk}^H+c_i^H{o}_{ik}^H.$$ (A10)

Finally, following migration and global density regulation, the haplotype frequencies are (j = 0,1)

$$P_{0jk}^{″}=\frac{c_k^P\left(1-m_P\right)P_{0jk}^{\prime }+c_i^P{m}_P{P}_{0ji}^{\prime }}{c_k^{\prime P}}$$ (A11a)

$$P_{1jk}^{″}=\frac{c_k^P\left[1-\left(m_P+{\delta }_P\right)\right]P_{1jk}^{\prime }+c_i^P\left(m_P+{\delta }_P\right)P_{1ji}^{\prime }}{c_k^{\prime P}}$$ (A11b)

for the parasite and

$$H_{0jk}^{″}=\frac{c_k^H\left(1-m_H\right)H_{0jk}^{\prime }+c_i^H{m}_H{H}_{0ji}^{\prime }}{c_k^{\prime H}},$$ (A12a)

$$H_{1jk}^{″}=\frac{c_k^H\left[1-\left(m_H+{\delta }_H\right)\right]H_{1jk}^{\prime }+c_i^H\left(m_H+{\delta }_H\right)H_{1ji}^{\prime }}{c_k^{\prime H}},$$ (A12b)

for the host.

After random mating, recombination, and reproduction the parasite haplotype frequencies are

$$P_{\mathit{ijk}}^{‴}=\{\begin{array}{c}{P}_{\mathit{ijk}}^{″}-r_P{D}_k^{″P},i=j\\ P_{\mathit{ijk}}^{‴}+r_P{D}_k^{″P},i\ne j\end{array}$$ (A13)

where $D_k^{″P}=P_{00k}^{″}{P}_{11k}^{″}-P_{01k}^{″}{P}_{10k}^{″}$ is the coefficient of linkage disequilibrium between the interaction locus and the migration locus for the parasite population in patch k, and r_P is the rate of recombination between the interaction and migration loci in the parasite. After random mating, recombination, and reproduction, the host haplotype frequencies are

$$H_{\mathit{ijk}}^{‴}=\{\begin{array}{c}{H}_{\mathit{ijk}}^{″}-r_H{D}_k^{″H},i=j\\ H_{\mathit{ijk}}^{″}+r_H{D}_k^{″H},i\ne j\end{array}$$ (A14)

where $D_k^{″H}=H_{00k}^{″}{H}_{11k}^{″}-H_{01k}^{″}{H}_{10k}^{″}$ is the host’s linkage disequilibrium coefficient in patch k, and r_H is the recombination rate in the host.

Recursions (A1)–(A14) can be written, equivalently, in terms of allele frequencies at each locus and disequilibria. The frequency of the mutant migration allele (“1” migration allele) in patch k is

$$M_{H,k}=\sum _{j=0}^1{H}_{1jk}$$ (A15)

in the host and

$$M_{P,k}=\sum _{j=0}^1{P}_{1jk}$$ (A16)

in the parasite. The frequency of the “1” interaction allele in patch k is

$$L_{H,k}=\sum _{i=0}^1{H}_{i1k}$$ (A17)

in the host and

$$L_{P,k}=\sum _{i=0}^1{P}_{i1k}$$ (A18)

in the parasite. Finally, the host and parasite disequilibria in patch k are, respectively

$$D_{H,k}=H_{00k}{H}_{11k}-H_{01k}{H}_{10k}$$ (A19)

and

$$D_{P,k}=P_{00k}{P}_{11k}-P_{01k}{P}_{10k}.$$ (A20)

APPENDIX B: Migration rates do not evolve with global density regulation

We show in this appendix that overall migration rates are unaffected when post-migration densities are regulated globally. To this end, we consider the recursion equations in Appendix A but assume there is no genetic variation at the interaction locus, in which case both species—and their migration rates—evolve completely independently of one another. To simplify notation, let be the q_k frequency of the “1” migration allele in one of the species in patch k and let c_k be the fraction of that species found in that patch such that c_A + c_B = 1. After migration and global density regulation, the frequency of the mutant allele in patch k is

$$q_k^{\prime }=\frac{c_k{q}_k\left(1-m-\delta \right)+c_i{q}_i\left(m+\delta \right)}{c_k^{\prime }},$$ (B1)

where

$$c_k^{\prime }=c_k\left(1-m-q_k\delta \right)+c_i\left(m+q_i\delta \right)$$ (B2)

is the new fraction of the population in patch i and the patch indicators k and i take values A and B, with i ≠ k. Equations (B1) and (B2) together describe the within-species evolution at the migration locus in the absence of interspecific interactions.

Now, consider the evolution of the mean migration rate, m̄ = m +(c_Aq_A + c_Bq_B)δ. Using (B1), it follows that

$$\begin{array}{l}{\overline{m}}^{\prime }=m+\left[c_{\mathrm{A}}^{\prime }{q}_{\mathrm{A}}^{\prime }+c_B^{\prime }{q}_B^{\prime }\right]\delta \\ =m+\left[c_{\mathrm{A}}{q}_{\mathrm{A}}\left(1-m-\delta \right)+c_{\mathrm{B}}{q}_{\mathrm{B}}\left(m+\delta \right)+c_{\mathrm{B}}{q}_{\mathrm{B}}\left(1-m-\delta \right)+c_{\mathrm{A}}{q}_{\mathrm{A}}\left(m+\delta \right)\right]\delta \\ =m+\left[\left(c_{\mathrm{A}}{q}_{\mathrm{A}}+c_{\mathrm{B}}{q}_{\mathrm{B}}\right)\left(1-m-\delta \right)+\left(c_{\mathrm{A}}{q}_{\mathrm{A}}+c_{\mathrm{B}}{q}_{\mathrm{B}}\right)\left(m+\delta \right)\right]\delta \\ =m+\left[\left(c_{\mathrm{A}}{q}_{\mathrm{A}}+c_{\mathrm{B}}{q}_{\mathrm{B}}\right)\left(1-m-\delta +m+\delta \right)\right]\delta \\ =m+\left(c_{\mathrm{A}}{q}_{\mathrm{A}}+c_{\mathrm{B}}{q}_{\mathrm{B}}\right)\delta \\ =\overline{m}\end{array}$$

Thus, the overall mean rate of migration does not change when population abundances are regulated globally after migration. This result is consistent with a previous model using slightly different genetics (McPeek and Holt 1992). By comparison, if populations are regulated locally after migration, then the overall migration rate can evolve via kin selection (Hamilton and May 1977; Leturque and Rousset 2002). Assuming global density regulation after migration in effect neutralizes kin selection on migration and thus allows us to isolate the impacts of interspecific interactions on the evolution of migration in our analyses.

APPENDIX C: Approximate linkage disequilibrium with small parameters

We use the small parameters assumptions to approximate local linkage disequilibria. The approximations assume that migration parameters, m_H, m_P,δ_H, and δ_P, and the interaction coefficients, α and β, are all of order ε (ε ≪1) and that patches are equal sized, c_H = c_P = ½. Substituting these values into the recursions (A1–A14) using change of variables formulas (A15–A20), and expanding as series around ε = 0, shows that

$$\mathrm{\Delta }{\overline{D}}_X=-r_X{\overline{D}}_X+\mathrm{O}(\epsilon ),$$ (C1)

in the host (X = H) and parasite (X = P). If recombination rates in both species are not too small (ε ≪ r_X ≤ 0.5), equation (C1) shows that disequilibria will decay rapidly, i.e., after a short period of time the disequilibria will become small---of order ε. After this, further changes in the disequilibria are of order ε

$$\mathrm{\Delta }{\overline{D}}_X=-r_X{\overline{D}}_X+\left(1-r_X\right){\widehat{L}}_X{\widehat{M}}_X\left(2m_X+{\delta }_X\right)/2+\mathrm{O}\left({\epsilon }^2\right),$$ (C2a)

$$\mathrm{\Delta }{\widehat{D}}_X=-r_X{\widehat{D}}_X-2\left(1-r_X\right){\delta }_X{\widehat{L}}_X\lfloor {\overline{M}}_X\left(1-{\overline{M}}_X\right)+{\widehat{M}}_X^2/4\rfloor +\mathrm{O}\left({\epsilon }^2\right).$$ (C2b)

From (C2), one can see that both D̄_X and D̂_X will evolve even more slowly, at rate ε², if they assume the particular forms

$${\overline{D}}_X^{\text{QLE}}=\frac{1-r_X}{r_X}{\widehat{L}}_X{\widehat{M}}_X\left(2m_X+{\delta }_X\right)/2,$$ (C3a)

$${\widehat{D}}_X^{\text{QLE}}=-2\frac{1-r_X}{r_X}{\delta }_X{\widehat{L}}_X\left[{\overline{M}}_X\left(1-{\overline{M}}_X\right)+{\widehat{M}}_X^2/4\right].$$ (C3b)