Wolbachia are intracellular, maternally inherited bacteria that are widespread among arthropods and commonly induce a reproductive incompatibility between infected male and uninfected female hosts known as unidirectional cytoplasmic incompatibility (CI). If infected and uninfected populations occur parapatrically, CI acts as a post-zygotic isolation barrier. We investigate the stability of such infection polymorphisms in a mathematical model with two populations linked by migration. We determine critical migration rates below which infected and uninfected populations can coexist. Analytical solutions of the critical migration rate are presented for mainland-island models. These serve as lower estimations for a more general model with two-way migration. The critical migration rate is positive if either Wolbachia causes a fecundity reduction in infected female hosts or its transmission is incomplete, and is highest for intermediate levels of CI. We discuss our results with respect to local adaptations of the Wolbachia host, speciation, and pest control.
The intracellular bacterium Wolbachia is maternally inherited via egg cytoplasm and has been found to infect two animal phyla, arthropods and nematodes (reviewed in O'Neill et al., 1997; Stouthamer et al., 1999; Stevens et al., 2001). In insects, estimations of Wolbachia infection incidence range from 16% to 76% infected species (Werren et al., 1995; Jeyaprakash & Hoy, 2000). As striking as the wide distribution of Wolbachia is its ability to manipulate the host reproductive system to its own advantage. Wolbachia is known to induce feminization of genetic male hosts, parthenogenesis, male-killing and – most commonly – cytoplasmic incompatibility (CI). From the bacteria's point of view, these manipulations are advantageous because they enhance the frequency of infected females in host populations (Werren, 1997). Unidirectional CI is a mating incompatibility that leads to a reduced number of offspring in matings between uninfected female hosts and infected male hosts (for a review, see Bourtzis et al., 2003). It is called unidirectional because the reciprocal cross (infected female hosts × uninfected male hosts) results in the same number of offspring as crosses where mating partners are both uninfected or both infected.
Basic mathematical models of CI show that Wolbachia spreads in a single panmictic host population if Wolbachia transmission is perfect and the infection does not affect host fecundity (Caspari & Watson, 1959; Fine, 1978). However, if either the transmission is imperfect or the Wolbachia infection reduces host fecundity, there exists a threshold infection frequency for the spread of Wolbachia (Fine, 1978; Hoffmann et al., 1990). Crucial to understanding these effects is that selection on CI is frequency dependent. Frequency dependence also explains the stability of bidirectional CI (where matings between partners bearing infections with two different Wolbachia strains result in fewer offspring) between parapatric host populations (Telschow et al., 2005b) and the spatial spread of Wolbachia in host populations because of unidirectional CI (Turelli & Hoffmann, 1991; Schofield, 2002).
A growing number of experimental studies examine the spatial distribution of Wolbachia. Although some studies clearly show a spatial spread of CI-inducing Wolbachia (Turelli & Hoffmann, 1991; Riegler & Stauffer, 2002; Hiroki et al., 2005), other studies suggest a stable pattern with some populations infected but others not (Shoemaker et al., 1999; Riegler & Stauffer, 2002; Keller et al., 2004; Vala et al., 2004; J. Jaenike, K.A. Dyer, C. Cornish & M.S. Minhas, unpublished data). These studies raise the question under which conditions infected and uninfected host populations can persist in face of migration. This question is especially important because in this case, unidirectional CI acts as a post-zygotic isolation barrier between the populations which might be a factor in host speciation as experimental and theoretical work suggest (Shoemaker et al., 1999; J. Jaenike, K.A. Dyer, C. Cornish, M.S. Minhas, unpublished data; A. Telschow, M. Flor, P. Hammerstein, J.H. Werren, unpublished data).
In the present study, we extend the single population model of Fine (1978) and analyse the Wolbachia dynamics in two host populations that are linked by migration (see Fig. 1). In the focus of our interest are the conditions under which infected and uninfected host populations can stably coexist because only then does unidirectional CI cause post-zygotic isolation between the populations. For the purposes of this article, we will use the term infection polymorphism for such a coexistence of infected and uninfected populations. This should not be confused with a polymorphism resulting from infections with different Wolbachia strains. We follow Telschow et al. (2005b) and analyse the stability problem in terms of a critical migration rate which is defined as the highest migration rate below which the stable coexistence of infected and uninfected populations is possible. We will demonstrate analytically that the critical migration rate is positive (and therefore post-zygotic isolation possible) if either Wolbachia causes a fecundity reduction in infected female hosts (section Model with fecundity reductions) or the transmission rate of Wolbachia is imperfect (section Model with imperfect Wolbachia Transmission).
Given the existence of an invasion threshold frequency in a panmictic host population (Fine, 1978), it is straightforward to argue that an infection may spread between populations if coupling via migration is strong enough. However, system dynamics and equilibrium states are changed by migration in a nontrivial fashion. Invasion thresholds derived for an isolated panmictic population cannot be directly applied to a population in the face of migration, and even less to two populations with migration in both directions. In addition, previous theoretical work rather suggests that CI-inducing Wolbachia inevitably spread through spatially structured host populations (Turelli & Hoffmann, 1991; Wade & Stevens, 1994; Schofield, 2002). Thus, based on parameters commonly used to describe Wolbachia dynamics, the critical migration rates derived in the present work allow to determine parameter regions of Wolbachia spread, of Wolbachia extinction, and of stable infection polymorphism. Our results allow to make general conclusions and are therefore a solid base for discussions about infection patterns of Wolbachia, its use in pest control and the potential role of unidirectional CI in speciation.
Model parameters and population structure
Following Fine (1978), we use three parameters to describe the Wolbachia dynamics: transmission rate, t; fecundity reduction, f; and CI level, lCI. The host organisms reproduce in discrete, nonoverlapping generations. Hosts may be either infected with Wolbachia or not. Wolbachia is assumed to be transmitted strictly maternally via egg cytoplasm. However, not all offspring of infected female hosts may inherit the infection. We define the Wolbachia transmission rate, t, as the fraction of infected eggs among all eggs of an infected female host. Further, infected female hosts may suffer a fecundity reduction [note that in Fine's (1978) model, both sexes’ fecundity was assumed to be affected by Wolbachia], i.e. the number of eggs is reduced by a certain fraction, f. Cytoplasmic incompatibility is described by the CI level, lCI, which is defined as the proportion of zygotes that die if the egg was uninfected but fertilized by sperm from an infected male host. All three parameters may range from 0 to 1. Experimental studies have shown that the transmission rate is usually very high, at 95–100% (see Hoffmann & Turelli, 1997, for a review), whereas fecundity reductions are low or absent, e.g. 10–20% in Drosophila simulans (Hoffmann et al., 1990). CI levels are much more variable; for example in the Drosophila species complex, they range from low levels of < 10% up to nearly complete incompatibility (Bourtzis et al., 1996; Merçot & Charlat, 2004). Also note that CI levels have been found to be dynamic. For example in D. melanogaster, they decline with the age of male hosts (Reynolds et al., 2003).
We will analyse the Wolbachia dynamics in three different population structures shown in Fig. 1: (a) a mainland–island model with an uninfected mainland, (b) a mainland–island model with an infected mainland and (c) a model with two populations and two-way migration. In all three cases, migration is described by the migration rate, m; i.e. each generation, a fraction m of the target population is replaced by migrants from the source population. In the model with two-way migration, we allow migration rates to be different in both directions. We define mi as the fraction of population i that is replaced by migrants. In our model, the migration rate may take values between zero and one.
Model with fecundity reductions
In the first part of our analysis, we assume that the Wolbachia infection reduces the fecundity of infected female hosts (f > 0) whereas Wolbachia transmission is perfect (t = 0). We start this section with a short summary of the dynamics of Wolbachia within a single panmictic host population. These results are a baseline for the analysis of the mainland–island models and the model with two-way migration. As we are interested in the stability of infection polymorphism, we always include starting conditions with one population infected but the other not.
Dynamics in a single host population
We follow Fine (1978) and describe the dynamics of unidirectional CI in a panmictic host population by a difference equation. Let x and x′ denote the Wolbachia frequencies in consecutive generations. Considering both unidirectional CI and the effects of Wolbachia on female fecundity, we can formulate the infection dynamics in the following way:
The fixpoints of eqn 1 can be calculated by simple algebraic operations: = 0, x = f/lCI and = 1. If f = 0 then = = 0 is unstable. However, if f > 0 then fixpoint is stable for any level of CI and represents an equilibrium where Wolbachia is absent from the population. The stability of the other two fixpoints depends on the specific values of the parameters. If lCI > f then is stable and characterizes the situation where Wolbachia has gone to fixation in the population. Furthermore, the fixpoint is unstable in this case and can be interpreted as a threshold frequency which determines the equilibrium state the system converges to. If lCI ≤ f then is unstable. In this case, for any starting frequency smaller than one, the system converges to , and Wolbachia goes to extinction. An important insight from this analysis is that a Wolbachia infection can stably persist in a panmictic population only if lCI > f (cf. Fine, 1978; Hoffmann et al., 1990). We will refer to this result repeatedly below.
Building upon the dynamics within a single panmictic host population, we discuss the effects of migration on the dynamics of Wolbachia. In this section, we allow migration to occur only in one direction, from a mainland to an island population. We will consider two different scenarios. First, we analyse the case of an uninfected mainland. Secondly, we will assume that the Wolbachia infection is at fixation in the mainland population.
We analyse the dynamics of Wolbachia in an island population receiving migration from an uninfected mainland. Let x and x′ denote the Wolbachia frequencies on the island in consecutive generations. Employing function G(f, lCI, x), defined in eqn 1, we can state the following difference equation describing the Wolbachia dynamics on the island:
In order to determine the fixpoints x* of this system, we solve eqn 2 for x* = x′ = x. This yields:
A stability analysis shows that, if f ≥ 0 and m > 0 then is stable for all levels of CI. Wolbachia can persist on the island if and only if (a) are real numbers, and if additionally (b) 0 < < 1. Otherwise, is the only stable equilibrium frequency, and Wolbachia goes to extinction on the island for arbitrary starting conditions. Condition (a) is fulfilled if and only if the square root in eqn 4 is positive or zero, condition (b) if condition (a) is satisfied and additionally lCI > f (compare section Dynamics in a single host population). If both conditions are met, it holds that 0 < < 1; and is stable and describes the situation where Wolbachia can persist on the island despite permanent migration of uninfected individuals from the mainland. Moreover, fixpoint is unstable in that case and denotes a threshold frequency (unstable equilibrium frequency). If the frequency of Wolbachia on the island is above this threshold at the beginning then the system converges towards ; but if it is below , it converges towards .
The analysis shows that a stable coexistence between an uninfected mainland and an infected island is possible if and only if conditions (a) and (b) are fulfilled. We follow Telschow et al. (2005b) and describe the stability in terms of a critical migration rate. For the model with an uninfected mainland, the critical migration rate is defined as the highest migration rate below which Wolbachia can stably persist on the island. The existence of the critical migration rate follows from the strictly monotone decrease of the square root in eqn 4 if interpreted as a function of m, and the fact that this function has a root. Figure 2a illustrates the critical migration rate by showing the equilibrium frequencies as a function of the migration rate. For low migration rates, three equilibrium frequencies exist. With increasing migration, the equilibrium frequency decreases whereas the threshold frequency increases. The distance between the two frequencies becomes smaller until they coincide at the critical migration rate. For migration rates above this critical value, the infection frequency converges towards for arbitrary starting frequencies.
The critical migration rate can be calculated analytically. We first note that the critical migration rate is zero if lCI ≤ f. Under these circumstances, the infection cannot stably persist in the island population even without migration of uninfected hosts, as shown in section Dynamics in a single host population. However, for lCI > f, positive critical migration rates occur. As Telschow et al. (2005b) have pointed out, the critical migration rate, mcrit, is the migration rate for which = . From eqn 4, it follows that mcrit must fulfil the equation:
The critical migration rate therefore is:
for lCI > f and zero for lCI ≤ f.
The critical migration rate is a function of the fecundity reduction and the CI level. Figure 3 shows how the critical migration rate depends on these two parameters. Graph (a) illustrates that the critical migration rate is zero for lCI < f. For higher levels of CI, however, it increases strictly monotonously. This is because, for increasing levels of CI, an increasing number of uninfected migrants is needed to supplant the infection on the island. Graph (b) on the other hand shows that the critical migration rate decreases strictly monotonously with increasing fecundity reduction if the CI level is fixed. In this case, less migration is needed to supersede the infection that is now linked to growing fitness costs. Note that the highest possible critical migration rate is mcrit = 0.25. It is achieved for lCI = 1 and f = 0.
Next, we analyse the situation where an island receives migration from a mainland with a stable Wolbachia infection at fixation. Again, we use function G(f, lCI, x), defined in eqn 1, to state a difference equation describing the Wolbachia dynamics on the island:
Note that these dynamics imply that lCI > f because otherwise the infection could not persist on the mainland, and the additive term ‘+ m’ would not be appropriate. Eqn 7 has three fixpoints:
If f ≥ 0 and m > 0 then is stable for all CI levels, lCI > f, and describes the state of Wolbachia fixation on the island. The spread of Wolbachia can be prevented if and only if are real numbers. Then, is stable whereas is unstable. characterizes a state of low infection frequency because of recurrent migration of infected individuals, and again marks a threshold frequency. How these fixpoints depend on the migration rate is shown in Fig. 2b.
According to our analysis, a stable infection polymorphism – i.e. an infected mainland and a (mainly) uninfected island – is possible if and only if the island infection frequency is at . Again, the stability can be described by a critical migration rate, mcrit. Here, the critical migration rate is defined as the highest migration rate below which the island population can maintain a state of low Wolbachia frequency. Both and are real numbers if m < mcrit. For migration rates above this critical value, the infection converges towards independent of its starting frequency. The critical migration rate is the migration rate where = and computes to:
if lCI > f. If lCI ≤ f, the critical migration rate is zero.
In Fig. 4, the critical migration rate is plotted as a function of both CI level and fecundity reduction. As can be seen in Fig. 4a, the infection polymorphism is stable if lCI ≥ f, and if migration is below the critical migration rate. For higher migration rates, the Wolbachia infection spreads to the island and becomes fixed in both populations, thus destroying the infection polymorphism. For fixed fecundity reduction, the potential of Wolbachia to spread becomes greater with increasing CI levels, resulting in decreasing critical migration rates. Because of the same reason, increasing fecundity reduction leads to higher critical migration rates for constant CI levels (Fig. 4b). The highest critical migration rates are achieved if CI levels equal the fecundity reductions. Then, the effect of CI is merely strong enough to keep Wolbachia within the mainland population, and – especially if fecundity reductions are large – high migration rates are necessary to permit the spread onto the island.
Model with two-way migration
The model can be generalized to incorporate migration between two populations in both directions (see Fig. 1c). After reproduction has occurred, a fraction m of each population is replaced by migrants from the other one. Let the populations be labelled A and B from left to right as presented in Fig. 1c. Then, denoting with xA and xB the infection frequencies in populations A and B, respectively, the system dynamics become a set of two coupled difference equations:
where G(f, lCI, x) models reproduction within each population according to eqn 1.
In contrast to the mainland–island models, we were not able to solve this system analytically. Therefore, we conducted numerical simulations to determine equilibrium states for the two-way migration model. We started all of our simulations with population A being completely uninfected and population B consisting exclusively of infected hosts (cf. Fig. 1c). We numerically iterated the dynamics for 106 generations or until an equilibrium had been reached. We considered the system to be in equilibrium if Wolbachia frequency differences between two consecutive generations were smaller than 10−11 and still declining.
We screened the parameter space of the two-way migration model in order to determine under which conditions a stable infection polymorphism is possible. In general, the simulations confirmed that Wolbachia cannot stably persist if lCI < f. This is true for arbitrary migration rates. In addition, we found that a stable coexistence of an infected and an uninfected host population is possible if (a) lCI ≥ f and if (b) migration rates stay within certain parameter regions.
Figure 5a illustrates these findings for symmetrical migration, m = mA = mB, and fixed fecundity reduction at f = 0.1. In the parameter plane spanned by migration rate and CI level, three regions can be distinguished: (a) a region where a stable infection polymorphism is possible, (b) a region where the Wolbachia infection spreads to fixation in both populations and (c) a region where the infection is lost in both populations. In the latter two cases, infection polymorphism and thus reproductive isolation between the populations is destroyed. Note that the stability of the infection polymorphism is highest for intermediate levels of CI.
For asymmetrical migration, Fig. 5c depicts equilibrium states of the system in the parameter plane spanned by the two migration rates, mA and mB (fecundity reduction and CI level are fixed at f = 0.1 and lCI = 0.4 respectively). Because lCI > f, a region exists where the uninfected population A and the infected population B can stably coexist. However, outside of this region, Wolbachia either spreads or becomes extinct in both populations.
In general, the regions of stable infection polymorphism for the model with two-way migration could not be determined analytically. However, good approximations of these regions can be derived using the results of the previous section. In the mainland–island models, the cytotype that is fixed in the mainland population (i.e. either Wolbachia or ‘no infection’) is favoured in comparison with the model with two-way migration. Thus, if migration occurs in both directions, the spread of Wolbachia or ‘no infection’ occurs at higher migration rates than in both mainland–island models; the minimum of the critical migration rates for the mainland–island models can therefore serve as a lower estimation for the boundary of the region of stable infection polymorphism. Using eqn 6 and eqn 10, it holds that for symmetrical migration, m = mA = mB, with given fecundity reduction and level of CI (where lCI ≥ f), a stable infection polymorphism is possible if:
This is illustrated in Fig. 5b. The approach can be generalized for the case of asymmetrical migration: if population A is uninfected and population B infected in the beginning, and if lCI ≥ f, then the infection polymorphism is stable if:
For example, if we choose the parameters f = 0.1 and lCI = 0.4 then the critical migration rates for the mainland–island models are mcrit ≈ 6.25% for an uninfected mainland and mcrit ≈ 0.66% for an infected mainland. Thus, from inequalities (14) it follows that the infection polymorphism is stable for any pair (mA, mB) with mA ≤ 0.66% and mB ≤ 6.25%. In Fig. 5c, this region is illustrated by dashed lines, and the open circle marks the pair of migration rates (0.66 %, 6.25%). As can be seen in the graph, the whole range of stable infection polymorphism is slightly larger but can be approximated in the described way using inequalities (14).
Model with imperfect Wolbachia transmission
In this section, we assume that Wolbachia transmission is imperfect (t < 1) and that the infection does not affect host fecundity (f = 0). Again, we begin with a short summary of the dynamics in a single host population, then discuss the mainland–island models and finally the model with two-way migration (cf. Fig. 1). Because the results of this section are very much akin to those of the previous section with fecundity reduction, we will describe them only briefly.
Dynamics in a single host population
For the model with imperfect Wolbachia transmission, the infection dynamics within an isolated host population can be written as a difference equation (Fine, 1978) and read:
Equation 15 has the fixpoints = 0 and If t = 1 then = = 0 is unstable, and = 1 is stable. Wolbachia goes to fixation for any positive level of CI in that case. But if t < 1 then is stable for any CI level and characterizes the state of Wolbachia extinction. Furthermore, if t < 1 then Wolbachia can persist stably in the population if and only if (a) are real numbers and (b) 0 ≤ < 1. Condition (a) is fulfilled if and only if lCI≥4t(1−t), condition (b) if and only if condition (a) is fulfilled and t > 0.5. Note that conditions (a) and (b) imply that 0 < ≤ 1. Then, is stable and characterizes a state where Wolbachia has reached a stable equilibrium frequency. Further, is unstable and marks a frequency threshold determining whether the system converges towards (if starting from below ) or (if starting from above ).
As in the previous model with fecundity reductions, we discuss the effects of migration on the infection dynamics for the case of imperfect Wolbachia transmission. In this section, we consider two scenarios with one-way migration from a mainland to an island population. In the first scenario, the mainland is uninfected whereas it is infected in the second.
Using function H(lCI, t, x), defined in eqn 15, we can formulate the infection dynamics in an island population that is affected by migration from an uninfected mainland:
If t < 1 and m > 0 then (the state of Wolbachia extinction) is stable for any level of CI. A stable persistence of Wolbachia on the island is possible if and only if (a) are real numbers and (b) 0 ≤ < 1. These conditions imply that . Condition (b) is fulfilled if and only if condition (a) is satisfied and if additionally t > 0.5. If both conditions are satisfied, a standard stability analysis shows that is stable whereas is unstable. In that case, represents a state of stable persistence of Wolbachia on the island, and a threshold frequency separating the ranges of attraction of the two stable fixpoints, and . However, if either of the two conditions is not fulfilled, is the only stable equilibrium frequency. In this case, Wolbachia goes to extinction independent of its starting frequency. Note that if the Wolbachia frequency on the island is at , the infection polymorphism (an uninfected mainland and a mainly infected island) is stable.
We again apply the concept of the critical migration rate to analyse the stability of the infection polymorphism. For condition (a) to be fulfilled, lCI>4t(1−t) must hold, and the migration rate must be below a critical value, the critical migration rate. The critical migration rate is zero for lCI≤4t(1−t) or t < 0.5, but it is positive otherwise. Following Telschow et al. (2005b), we can calculate it by solving equation = for m. Thus, the critical migration rate computes to:
for lCI>4t(1−t) and t > 0.5. If migration occurs at a rate larger than mcrit then is the only equilibrium frequency, and Wolbachia goes to extinction independent of its starting frequency. The critical migration rate is a function of CI level and transmission rate. At a fixed rate of Wolbachia transmission, the critical migration rate is a linear function that grows monotonously with increasing CI level (see Fig. 6a).
In this section, we discuss the influence of migration of infected hosts on the Wolbachia dynamics in an island population. As Wolbachia transmission is imperfect, the infection on the mainland is not assumed to be fixed but to be at the stable equilibrium frequency xmain(lCI,t)=, as derived in section Dynamics in a single host population for an isolated host population. Then, the dynamics on the island can be formulated as follows:
where function H(lCI, t, x) describes reproduction within the island population according to eqn 15. Note that these dynamics imply that lCI≥4t(1−t) and t > 0.5 (otherwise the infection could not persist on the mainland, and the additive term ‘+ mxmain(lCI,t)’ would be inappropriate). Solving eqn 20 for x* = x′ = x yields three fixpoints. The first equals the Wolbachia frequency on the mainland and does not depend on the migration rate, the other two can be calculated by polynomial division:
where and s(lCI,m,t) =1 + m − (1 − m) r(lCI,t). If m > 0 and t < 1 then a standard stability analysis shows that is stable for all levels of CI, representing a state of stable persistence of Wolbachia on the island. Prevention of a Wolbachia spread onto the island is possible if and only if are real numbers. Then, is stable, and is unstable. If the infection on the island is at the low frequency , infection polymorphism of the mainland and island populations is stable. again is a threshold frequency between the two stable equilibria. If are complex numbers, Wolbachia inevitably spreads in the island population, and the infection polymorphism is lost.
Again, infection polymorphism stability can be expressed in terms of a critical migration rate: the square root in eqn 22 is positive or zero (and thus are real-valued) if migration is below or equal to the critical migration rate respectively. The critical migration rate is zero for lCI ≤ 4t(1−t) or t < 0.5. However, it is positive otherwise and can be calculated by solving = for m:
for lCI>4t(1−t) and t > 0.5. The critical migration rate is a function of the CI level and the rate of Wolbachia transmission. For fixed transmission rate, mcrit monotonously decreases with increasing CI level (see Fig. 6b).
where xA and xB represent the infection frequencies in populations A and B respectively; and where H(lCI, t, x) models reproduction within each population according to eqn 15. We were not able to calculate fixpoints of this system analytically and therefore analysed the two-way migration model by numerically iterating the dynamical equations (for details on how the simulations were conducted, refer to section Model with two-way migration in Model with fecundity reductions).
The simulations show that the critical migration rates for the model with imperfect Wolbachia transmission resemble those found for the fecundity reduction model. If lCI<4t (1−t) or t < 0.5, the Wolbachia infection gets lost in both populations even without migration (cf. section Dynamics in a single host population). However, for lCI>4t(1−t) and t > 0.5, stable coexistence of an infected and an uninfected population is possible if migration rates stay within certain regions. For symmetrical migration rates (m = mA = mB), the parameter plane spanned by migration rate and CI level is shown in Fig. 6c (for transmission rate fixed at t = 0.95). Wolbachia goes to extinction if lCI<4t(1−t)=0.19. However, for larger CI levels the infection polymorphism is stable if the migration rate stays within the region shaded in dark grey. Outside this region, Wolbachia goes to extinction in both populations for low levels of CI, and spreads from the infected to the uninfected population for high CI levels.
As in the model with fecundity reduction, the region of stable infection polymorphism can be approximated by using the analytical solutions of the critical migration rates for the mainland–island models, i.e. eqn 19 and eqn 23. Thus, for symmetrical migration, it holds that an uninfected and an infected host population can stably coexist if:
provided that lCI>4t(1−t) (see Fig. 6c). Equivalent to the model with fecundity reduction, the region where a stable infection polymorphism is possible, can be approximately described by two conditions for the general case of asymmetrical migration when Wolbachia transmission is imperfect:
where again lCI>4t (1−t), and where population A is uninfected and population B infected in the beginning. For example, if in the mainland–island models lCI = 0.4 and t = 0.95, then mcrit ≈ 5.82% for an uninfected mainland, and mcrit ≈ 0.20% for an infected mainland. Thus, if two parapatric host populations with polymorphic infection status are linked by two-way migration, the infection polymorphism is stable as long as migration does not exceed 0.20% from the infected to the uninfected population whereas it can be as high as 5.82% in the opposite direction.
In the present study, we investigated the dynamics of Wolbachia-induced unidirectional CI in a spatial model with two host populations and migration between them. Our main result is that infected and uninfected host populations can stably coexist if migration is below a critical migration rate and if infection is costly or imperfectly transmitted. Under these circumstances, unidirectional CI acts as a post-zygotic isolation mechanism between the populations. We determined the critical migration rates analytically for mainland–island models and showed that these solutions are lower estimations for the general model with two-way migration. In general, critical migration rates are positive if either Wolbachia causes fecundity reductions in infected female hosts or Wolbachia transmission is imperfect. Note that the combination of these two barriers against the spread of Wolbachia (f > 0 and t < 1) will generally result in increased critical migration rates. In the case of symmetric two-way migration, infection polymorphism is most stable for intermediate levels of CI.
Such mosaically structured infection patterns might strongly influence the evolution of both Wolbachia and the host. This is because of unidirectional CI causing an asymmetric gene flow reduction between infected and uninfected populations (Telschow et al., 2002a,b, A. Telschow, M. Flor, P. Hammerstein, J.H. Werren, unpublished data). Because of this gene flow distortion, infected populations are converted into population genetic sinks. Local adaptations are therefore favoured in uninfected host populations compared with infected ones. Further, host adaptation to Wolbachia is impeded. This might have huge impact on the coevolution of host and symbiont. For male-killing inducing Wolbachia, it was shown theoretically that asymmetric gene flow can prevent adaptation in a host population infected by a male killer (Telschow et al., 2006).
Our results support the view that unidirectional CI could be a factor in Wolbachia host speciation. This is because unidirectional CI causes post-zygotic isolation in hybrid zones between infected and uninfected host populations. Post-zygotic isolation, however, is widely believed to play a crucial role in speciation because it presumably selects for female mating preference (see Coyne & Orr, 2004, for a review). Thus, in uninfected host populations that receive migration from infected populations, unidirectional CI might select for premating isolation by reinforcement of female mating preferences if the infection polymorphism is stable, and thus facilitate speciation (Shoemaker et al., 1999; J. Jaenike, K.A. Dyer, C. Cornish, M.S. Minhas, unpublished data; A. Telschow, M. Flor, P. Hammerstein, J.H. Werren, unpublished data).
From a theoretical point of view, our model approach is a natural extension of single population models. Fine (1978) has shown that in order for CI to spread in a single host population, Wolbachia has to be at a frequency above a certain threshold. The critical migration rate introduced in this study is the two-population analogon of Fine's threshold frequency. Wolbachia spreads from one population to another only if migration is above a threshold, the critical migration rate. Previous theoretical studies on spatial CI dynamics have not described such a threshold (Hoffmann & Turelli, 1997; Schofield, 2002). The reason is that these models are based on partial differential equations with migration characterized by a diffusion term. These approaches, however, do not incorporate enough population structure to observe the effects described in this study.
Symbiotic bacteria have been proposed as means to introduce useful genes in insect pest populations (Beard et al., 1993; Sinkins et al., 1997). For example in insect disease vectors, such useful genes may render the insect refractory to infection with a pathogenic agent, or reduce the competence to transmit these agents. With the self-spreading power of unidirectional CI, Wolbachia could provide the vehicle for the introduction of foreign genes in targeted arthropods. Our results imply that such measures can only then be effectively applied if migration between host populations generally occurs with rates above the critical migration rate. For biological pest control, high levels of CI are desirable because propagation of useful genes within host populations will be fastest under these conditions, but also because critical migration rates are low for CI levels close to one, facilitating the spread between populations. Thus, knowing the spatial structure of pest populations may prove essential for a successful application of this mechanism.
Previously, it was shown that Wolbachia-induced bidirectional CI is stable in face of two-way migration at high rates of up to 15% per generation (Telschow et al., 2005b). The critical migration rates observed for unidirectional CI are roughly one order of magnitude lower. However, even for unidirectional CI, high critical migration rates might be observed if additional genetical or ecological factors are included in the model. In the case of bidirectional CI, it was shown that local adaptations in the host can significantly increase the critical migration rates (Telschow et al., 2005a). This is because linkage disequilibria build up that stabilize both infection differences and local adaptations. We expect the same stabilizing effect to be effective for unidirectional CI. Other reasonable scenarios resulting in higher critical migration rates are (a) local selection against Wolbachia because of naturally occurring antibiotics and (b) local infections with a Wolbachia strain that causes a sex ratio distortion rather than CI (Engelstädter et al., 2004).
In summary, our results suggest that a stable coexistence between infected and uninfected parapatric host populations is possible under biologically reasonable conditions. We analysed the role of migration in detail and showed that, if migration is below a critical value, then unidirectional CI acts as a post-zygotic isolation mechanism. These results might have important implications for host evolution including speciation, for the coevolution of Wolbachia and its hosts, and for utilization of CI-inducing Wolbachia in biological pest control.
We thank Jan Engelstädter and two anonymous reviewers for helpful comments on the mansucript. This article was partly supported by NSF EF-0328363, the Deutsche Forschungsgemeinschaft (SFB 618), the grant of Biodiversity Research of 21st Century COE (A14), and the Japanese Society for the Promotion of Science (JSPS).