Individual-based model description

The landscape structure comprised 20 × 20 patches on a square lattice. Spatial heterogeneity was introduced by assigning patch-specific mean carrying capacities, K̄_i sampled from a normal distribution with landscape mean, K̄, and a spatially uncorrelated variance, . A lower bound for K̄_i in the landscape, further denoted as K_min, was specified to compare the effect of minimum patch carrying capacity on demographic parameters and reflects a minimum patch quality before individuals will settle. Hence, patch specific carrying capacities were distributed according to K̄_i = max{N(K̄,  ), K_min}. Temporal variation in patch carrying capacities was implemented at each yearly time step, to which the K_min lower bound was again applied, by sampling from the distribution K_i(t) = max{N(K̄_i,  ), K_min}.

We considered an asexual species consisting of dispersing and sedentary individuals as well as a sexual species where the dispersal phenotype is determined by a two-allele mechanism, with the sedentary allele being dominant (equivalent to the genetic determination of macropterous and brachypterous individuals in our empirical system, see Empirical model system: Density of dispersing and sedentary individuals in relation to population size in real landscapes). Only dispersing individuals were able to colonize other patches in the landscape and sedentary individuals always stay in their natal patch.

We implemented three different dispersal methods. First, dispersers dispersed globally and all patches were equally likely to receive immigrants. In the second case, dispersal distance was restricted and dispersers were only able to colonize neighboring patches. Third, dispersal was conditional on the patch density,

such that underpopulated patches were more likely to receive immigrants. Here, a dispersing individual chose a patch at random in the landscape with probability

For the asexual species, dispersal was followed by reproduction and population regulation. For the sexual species, the series of events was modeled according to the life history of the study species Pterostichus vernalis (see Empirical model system: Density of dispersing and sedentary individuals in relation to population size in real landscapes), i.e., individuals first mate with each other in their natal patch, disperse afterward and reproduce once settled in a particular patch.

Regulation of population densities followed the Beverton-Holt model where each of N_i(t) adults within a patch produces λ offspring, which we allowed to be different for sedentary and dispersing individuals, that compete with each other leading to patch specific population sizes equal to

with parameter a_i(t) = [(λ − 1)/K_i(t)].

After population regulation, the number of dispersing and sedentary individuals was recorded each generation, and these individuals made up the next generation that respectively underwent dispersal, reproduction and population regulation.

The cost of dispersal was implemented as the probability of mortality during the dispersal phase, m. Whereas under sexual conditions brachypterous individuals and alleles can be introduced to local populations through eggs produced by immigrant females that were mated in patches with brachypterous males, that was not possible in the asexual model. Therefore, to allow the introduction of sedentary individuals after stochastic extinction in a patch in the model, we implemented a small reciprocal probability of mutation between sedentary and dispersing genotypes in the asexual model only. Annotated computer code used to perform the simulations can be found in Supplement 1.

In a first series of our simulation experiments, we investigated how the dispersal method affects the dynamics and absolute numbers of both dispersal phenotypes within local populations. This was performed under the following population and landscape parameters: K̄ = 50, σ_sp = 50, σ_temp = 5 and a minimal carrying capacity at which we assumed a local population can persist or settle, i.e., K_min, equal to 20 individuals. To obtain the average frequencies of both dispersal phenotypes per patch, further denoted as and for sedentary individuals and dispersers respectively, we first visually inspected the dynamics and selected the last 500 generations for which no monotonic increase or decrease in the proportion of dispersing individuals was present. This was further confirmed by the absence of a significant relationship between the total proportion of dispersing individuals in the landscape and generation number. and were then obtained by averaging N_Si(t) and N_Di(t) for these last 500 generations. In addition, we inspected the distribution of both phenotypes within a single generation in order to compare the simulated data with the empirical data, which also originate from a single generation sampling. For these single generation data, we investigated the extent to which the variance in frequency of dispersers in each patch differs from a random Poisson expectation by calculating the Pearson X² overdispersion parameter (φ). Next, we investigated how variation in σ_sp, σ_temp, K_min, m, and a dispersal-related fecundity trade-off affects the equilibrium numbers of both dispersal phenotypes in each local population.

Empirical model system: Density of dispersing and sedentary individuals in relation to population size in real landscapes

Individual-based simulation

Considering an asexual species, global dispersal, and the absence of dispersal mortality, the average number of individuals in each patch with the sedentary phenotype, , correlated almost perfectly with the mean carrying capacity of the patch (Fig. 1a). Only for the smallest patches in the landscape (K_i ≤ 40 in Fig. 1a), the number of sedentary individuals per patch decreased abruptly, indicating that the sedentary phenotype does not persist in these smaller patches. The number of individuals with the dispersing phenotype in each patch, , appeared independent of the local patch capacity and was approximately equal among all patches in the landscape. The average number of , 23.97, closely approximated the predicted value of K_min = 20. These results were robust to dispersal method, including dispersal restricted to neighboring patches (Fig. 1b) and even global dispersal with immigration proportional to N_i(t)/K_i(t) (Fig. 1c), for which was equal to 23.90 and 23.46, respectively.

For the sexual species, a similar pattern was observed with an even tighter relationship between patch capacity and the number of sedentary individuals per patch and a homogeneous distribution of dispersing individuals across patches (Fig. 1d). As for the asexual species, the results were highly consistent for the different implemented dispersal methods (Fig. 1d–f). However, compared to the asexual case, the number of dispersers in each patch was considerably lower than the minimum patch capacity, with being equal to 8.25, 8.24, and 8.70 under global, neighboring, and density-dependent immigration, respectively. Within a single generation the number of dispersers within a patch was more variable, and larger than expected from a random Poisson distribution for all dispersal methods (φ_NDi = 3.12, φ_NDi = 3.25, and φ_NDi = 2.70 under global, neighboring, and density-dependent immigration, respectively; Fig. 1g–i).

To further investigate the dependency of to K_min, simulations were first performed with an asexual species with global dispersal, but we varied the minimum patch capacity in the landscape from K_min = 0, which represents the presence of deterministic sink patches in the landscape, to K_min = 60. Within each local population, the average number of dispersers in each patch corresponded almost perfectly with the lowest patch capacity present in the landscape (Fig. 2a, solid circles). This pattern remained unaffected for simulations with different values of temporal variation (e.g., σ_temp = 20; Fig. 2a, solid triangles). Also, changing spatial variation did not affect the strong dependency of to K_min, but resulted in a slightly higher number of dispersers in each patch, in particular for lower values of K_min (e.g., σ_sp = 50; Fig. 2a, open circles). These simulations further demonstrated that individuals with the dispersing phenotype persist when sinks are present in the landscape.

Although this pattern remained unchanged when a low dispersal mortality was assumed (m = 0.01; Fig. 2b), strong deviations from this global pattern were observed under higher levels of dispersal mortality. Here, the dispersal strategy was no longer favored due to the high dispersal costs, but this effect only occurred when the minimal patch capacity in the landscape was large.

Considering a sexual species and global dispersal, the strong relationship between minimum patch capacity and number of dispersers per patch persisted, albeit that this number was substantially higher than expected when sinks are present in the landscape (i.e., K_min = 0) and lower than expected for higher values of K_min (Fig. 2a, open triangles). All else being equal, incorporating a dispersal related fecundity trade-off, which was implemented by changing λ of the dispersing phenotype while keeping λ = 3 for the sedentary phenotype did not appear to affect the number of sedentary individuals per patch ( = 8.32, 8.72, and 8.62 for λ_disp = 2.5, 2.0, and 1.5, respectively).

Empirical model system: Density of dispersing and sedentary individuals in relation to population size in real landscapes

A first prediction that emerges from this theory is that variation in the frequency of macropterous individuals among the different patches within a given landscape is low and primarily due to stochastic variation. For brachypterous individuals, in contrast, it is expected that the variation in number among patches is more variable and relates to the size of the local populations.

In accordance with these predictions, the variance of the number of captured macropterous individuals among trap sets within a landscape was only about twice as high as expected under a random distribution (φ_mac = 2.17; bootstrap P < 0.0001), while the variance in the number of captured brachypterous individuals was substantially larger and more than nine times higher than expected under a Poisson distribution (φ_brac = 9.43; bootstrap P < 0.0001).

Differences in the total number of captured individuals per trap set was significantly and positively related to the amount of suitable habitat both when assessed across all sampled sites (slope = 0.0038, SE = 0.0012, F_1, 157 = 6.67, P = 0.008) as well as within landscapes (slope = 0.0016, SE = 0.0008, F_1, 148 = 5.28, P = 0.02). Although both the number of brachypterous and macropterous individuals captured per trap set was significantly correlated with the total number of individuals captured in each trap set (P < 0.01), the strength of this correlation was much stronger and almost perfectly linear for the brachypterous individuals (r = 0.99) compared to the macropterous individuals (r = 0.27; Fig. 3a and b). As a consequence, the proportion of macropterous individuals in each local population was negatively correlated with local population size (logit(Pr_mac) vs. N_tot: slope = −0.051; SE = 0.0122; F_1,94 = 18.00, P < 0.0001; Fig. 3c).

Moreover, the number of captured individuals per trap set averaged across the landscape did not differ between landscapes for macropterous individuals (F_9,91 = 0.76, P = 0.7), but was highly different for brachypterous individuals (F_9,91 = 7.84, P < 0.0001; Fig. 4). Consequently, the proportion of macropterous individuals was strongly negatively correlated with the total number of individuals captured in the landscape (r_S = −0.8, P = 0.005).

Distribution of fixed dispersal phenotypes: individual-based simulations

Results of our numerical simulations are congruent with these observations, and moreover showed that this result holds under a wide variety of dispersal types. Hence, irrespective of whether dispersal is global, restricted to neighboring patches, or dependent on local population size, a similar outcome was observed. Moreover, these simulations demonstrated that the frequency of dispersing individuals in each patch is strongly determined by the carrying capacities of the smallest habitat patches in the landscape. Indeed, if all individuals in the landscape emigrate and dispersal is random over all patches, the number of individuals colonizing the smallest patches in the landscape will on average be substantially larger than their carrying capacities. Under this condition, the dispersal strategy will be selected against until the number of dispersing individuals in each patch approaches the carrying capacity of the smallest patches and random dispersal will in this case not lead to overpopulation in inferior patches.

However, results of our simulations also show some subtle, though important deviations from these classical IFD predictions. First, the number of dispersers in each patch is slightly but consistently higher than the specified minimum patch capacity (Fig. 2a). Consequently, and contrary to deterministic IFD predictions, the dispersal phenotype does not go extinct when sinks are present in the landscape. These results are consistent with those of Schreiber (2012), who recently showed that high levels of spatially uncorrelated environmental stochasticity still favor the selection of deterministic sink patches resulting in overmatching of low quality patches. Within the context of a coalition of a low and high unconditional dispersal strategy, this is equivalent to the persistence of high dispersers in the landscape and their occupation of sink patches. Second, for the asexual model, the number of sedentary individuals does not decrease linearly with decreasing patch capacity for patches that approach K_min (e.g., patches with K_i ≤ 40 in Fig. 1). This is likely due to the fact that for patches with low carrying capacities, competition between sedentary and dispersing individuals may cause stochastic extinctions of the former. In other words, for sedentary individuals, low quality patches may act as stochastic sinks, i.e., they exhibit negative local stochastic growth rates but positive intrinsic growth rates (Schreiber 2012). Indeed, Schreiber (2010) showed that under the assumption that both dispersal types persist at stationarity in landscapes with spatiotemporal uncorrelated fluctuation, not all patches can be occupied by sedentary individuals. This follows from the fact that, at stationarity, both local growth rates of non-dispersers and spatially averaged growth rates of dispersers are zero. This results in negative growth rates in patches without sedentary individuals, which can only be occupied by dispersers. Third, the extinction of sedentary individuals in these smaller patches was not observed when we modeled a sexual species. This can be attributed to the sequence of life history events that we implemented. As individuals mate in their natal patch, but reproduce after the dispersal event, local extinction of sedentary individuals is prevented by the repeated production of sedentary offspring by dispersing individuals. This mechanism also explains the substantially lower number of dispersers in each patch compared to the asexual model.

Distribution of fixed dispersal phenotypes: empirical data

Subsuming long and short winged individuals in Pterostichus vernalis as dispersing and sedentary individuals, results from our empirical system are in strong agreement with these IFD predictions, with a significantly lower proportion of long-winged individuals observed in high density populations. To verify if this negative relationship can be attributed to the homogeneous distribution of dispersers throughout the landscape, we compared the variation in their frequency with those expected from a Poisson distribution. This revealed that the distribution of dispersers matches such a random distribution much more closely compared to variation in the number of sedentary individuals. The somewhat higher variance in number of dispersers than expected under a Poisson distribution can be attributed to several issues. First, the data at hand originate from a single season sampling campaign and stochastic variation in population numbers due to both extrinsic and intrinsic factors as well as sampling variation might result in a slight deviation from a random distribution of macropterous individuals. Second, the genetic system that determines wing size might result in a slower response in the distribution of dispersal types than expected under a mere ecological model (Roff 1994, Denno et al. 1996, 2001, Roff and Fairbairn 2007). This was confirmed by our simulations of the sexual species showing that even under global dispersal the variance in the frequency of dispersing individuals is about three times larger compared to the expected variance under Poisson expectations and agreed reasonably well with the variance observed for our empirical data. These factors are however unlikely to cause the observed differences in the total number of brachypterous individuals in each local population as this would result in an equal degree of overdispersion in the numbers of both dispersal types among patches. Hence, the more than four times higher degree of overdispersion in number of brachypterous individuals among populations strongly suggests that this primarily reflects variation in carrying capacity among populations. This was further confirmed by the positive relationship between habitat suitability and the total number of captured individuals within each trap set.

The mechanism that determines the number of dispersing individuals in each patch and, hence, the minimum patch capacity, is unfortunately hard to test directly in natural populations. First, a minimum patch capacity that is larger than zero is most likely determined by a minimal degree of habitat suitability before dispersing individuals settle (i.e., source habitats), although life history traits (Legendre 2004) or Allee effects (Stephens and Sutherland 1999) might have additional influence. As the conditions that determine this smallest local population size are in this case an intrinsic property of the species (Colas et al. 2004), they are not expected to differ substantially among landscapes. In line with this, the average number of macropterous individuals captured in each trap appeared to be remarkably similar among landscapes. This contrasted strongly with the number of brachypterous individuals, for which the average capture densities at each trap set varied from zero in landscape N-Ben to almost 14 in landscapes B-Kap and B-Voe. Second, if the species would not be able to discriminate between source and sink habitats, resulting in frequent settlement of dispersers in sink habitats, the factors that determine the number of dispersing individuals are more complex and determined by the magnitude of temporal and spatial autocorrelation in local population fluctuations (Schreiber 2010). In turn, such fluctuations are properties of the individual landscapes, and therefore less likely to result in an equal number of dispersing individuals per trap in each landscape.

The analogy between IFD and the evolution of dispersal has primarily been introduced to explain the evolution of conditional dispersal strategies (Lemel et al. 1997a, Holt and Barfield 2001, Clobert et al. 2008). As originally defined, it is assumed that individuals are able to perceive the negative density-dependent effects caused by immigration. Such conditional dispersal strategies are predicted from theoretical studies (Travis and Dytham 1999, Poethke and Hovestadt 2001, Kum and Scheuring 2006, Bocedi et al. 2012) and confirmed by a wealth of empirical observations (Doncaster et al. 1997, Bowler and Benton 2005, Clobert et al. 2009). Here, we show in agreement with other studies (McPeek and Holt 1992, Lemel et al. 1997a, Holt and Barfield 2001, Cressman and Krivan 2006, Massol et al. 2011) that this strong “ideal” assumption is not a necessary condition for a population to attain an IFD (McPeek and Holt 1992, Lemel et al. 1997a, Holt and Barfield 2001). However, until now no empirical data were available to support these theories.

Expected distribution of fixed dispersal phenotypes under the IFD vs. the competition–colonization model

Although empirical data on wing polymorphic species played a historic and pivotal role in developing some key principles of dispersal evolution theory and in explaining dispersal polymorphisms (cf. den Boer 1968, Roff 1986), they were until now only interpreted in the context of differences in habitat persistence and local extinction of patches. In particular, the theoretical arguments and models developed by den Boer (1968) and Roff (1994) proved to be highly valuable in explaining the relative increase of long-winged individuals in recently colonized habitats (den Boer 1968, Denno 1994, Denno et al. 1996). Alternatively, the adaptive value of the wingless morph in persistent habitat patches is primarily attributed to an expected higher fecundity or earlier maturation achieved due to a lack of investment in wing muscles and other dispersal-related traits (Aukema 1991, Zera and Denno 1997, Roff and Fairbairn 2007). Under the present model and those developed by (McPeek and Holt 1992, Doebeli and Ruxton 1997), the higher fecundity of the sedentary morph is not a necessary condition to explain the coexistence of both morphs. Selection for short-winged individuals rather stems from the unconditional dispersal rate of long-winged individuals, which is selected against when source–sink dynamics are reached.

If habitat persistence, rather than the density-dependent fluctuations proposed here, would be the sole factor explaining the dispersal polymorphism in P. vernalis the number of dispersers per patch would be expected to be dependent on the local population size and a much larger degree of overdispersion in the number of dispersers among patches and landscapes would be expected. To confirm this we adapted the model previously developed by Roff (1994) allowing for spatial variation in K_i and investigated the relationship between the number of brachy- and macropterous individuals and patch capacity. This indeed shows that under patch extinction dynamics (i) the frequency of dispersers increases strongly with increasing patch capacity and (ii) that the variance in the number of long-winged individuals is much larger than expected from a random distribution (see the Appendix and Supplement 2 for the results of the simulation and the code). It nevertheless remains very reasonable that if local extinctions occur in the landscape, they will first be occupied by long-winged individuals that will decrease in proportion after the colonization event. However, our data suggest that local extinctions and dispersal related fitness trade-offs (cf. Roff 1986, Haag et al. 2005, Roff and Fairbairn 2007) are not a necessary prerequisite to explain the evolution and persistence of dispersal dimorphisms.

Frequency of dispersal phenotypes vs. dispersal rates

Another important conclusion that arises from this work is that concentrating solely on the average dispersal rate, or the proportion of both dispersal polymorphisms, may lead to spurious conclusions concerning a population's colonization ability in response to habitat deterioration. An increase in the proportion of macropterous individuals due to decreasing local population sizes, and consequently increased local extinction rates (Hanski 1998), could be interpreted as an evolutionary “rescue effect” against metapopulation extinction (Heino and Hanski 2001, Parvinen 2004). In this study, an increase in the proportion of macropterous individuals in response to these factors is only determined by a decrease in the number of brachypterous individuals. As the number of macropterous individuals in the landscape is much less affected, colonization ability within the landscape remains unaltered despite the observed increase in average dispersal rate.

Besides the fact that this is to our knowledge the first empirical verification of the distribution of a dispersal polymorphism within an IFD framework, our results further emphasize that interpretation of the frequency of individuals expressing discrete dispersal morphs in a landscape, rather than simply mean dispersal rates, could greatly enhance our understanding of the mechanisms that underlie interdemic variation in dispersal rates and such consideration is therefore required in future empirical and theoretical studies of dispersal evolution and in conservation research.

Appendix

Expected distribution of fixed dispersal phenotypes under the competition–colonization model described in Roff (1994) (Ecological Archives E094-228-A1).

Supplement 1

C++ source code to model the spatial distribution of fixed dispersal phenotypes as described in this paper and a short description of the program and its options (Ecological Archives E094-228-S1).

Supplement 2

R code to model the spatial distribution of fixed dispersal phenotypes under competition-colonization dynamics as described in Roff (1994), but allowing for spatial variation in K (Ecological Archives E094-228-S2).