2.1 Framework overview

The opposing influences of different dispersal modes on the genetic divergence within a metapopulation can be theoretically explained, but these influences have not been demonstrated to occur along a gradient of river branching in multiple networks. To demonstrate these influences, we followed two successive steps. First, by using a mechanistic model based on evolutionary processes and asymmetric dispersal, we explored the spatial genetic variation among four macroinvertebrate species with flying adult stages that occur in a river network of north-eastern Japan. Second, we evaluated how branching complexity of random virtual networks can differentially affect the broader genetic differentiation throughout catchments in the context of different, asymmetric gene flow modes.

To achieve the first step, we used the empirical data (see the subsequent section "Empirical catchment and genetic data") and developed a metapopulation genetic model based on isolation by distance (i.e. lower gene flow with greater separation in terms of distance along the watercourse) and asymmetric dispersal (i.e. upstream- and downstream-biased movements) between local populations in the river network (see the subsequent section "Development of metapopulation genetic model"). Rather than using their actual locations, the watercourse distances between local populations were used directly as the input data in the model. Consequently, if the spatial connectivity among some local populations is high, then their genetic structure would tend to have consistent differences from that of their metapopulation (their genetic pool). In turn, this model was fitted to these empirical data with its parameters estimated by Bayesian inference (see the subsequent section "Bayesian fitting process"). To validate this inference process using the virtual ecologist approach (Zurell et al., 2010), we performed this metapopulation genetic modelling with known values of parameters to confirm whether the Bayesian estimates matched these known values (Appendix S1). The Bayesian inference applied here successfully provided the reliable estimation of model parameters for two out of three scenarios, as the virtual true values were within the 95% confidence interval of the posterior distributions. The scenario in which the Bayesian inference could not fully identify the virtual true values was when the scale parameters were equal (; Figure S2; see “Development of metapopulation genetic model” for the description of the parameters). Nevertheless, the overall positive evaluation of our inference process provided confidence to apply it to real-world data in our first experimental step.

Bayesian estimation can provide realistic ranges of dispersal asymmetry and other parameters. In our model, these parameters were related to biological features of the four species and were not independent of each other. Because a particular species can be described as having a certain combination of parameters based on a set of trait characteristics and through its systematic clade, comparisons among these species can reveal the opposing influences of branching complexity on metapopulation genetic structure. This can be done by examining the parameter space (i.e. the ranges of possible values of multiple parameters) and how each region of this parameter space reacts to different branching contexts. We used these ranges in our second step to achieve the main objective of this study, which was to evaluate the influence of network branching in a generalizable context, while making our simulations closer to empirical system than to arbitrary conditions. The different parameter combinations (derived from Bayesian fitting process for the four macroinvertebrate species) allowed the creation of virtual species.

For the second step, the metapopulation genetic model constructed above was used to simulate the river-branching influence on genetic divergence in different dispersal modes (i.e. by setting up the virtual species) in 1,000 virtual networks (see the subsequent section "Simulation of river-branching influences"). These virtual networks were randomly created to represent a variety of branching complexity.

2.2 Empirical catchment and genetic data

In an empirical catchment in the Natori and Nanakita rivers located in north-eastern Japan (integrated catchment area c. 1,200 km²; Figure 1), the flow regime exhibits a seasonal pattern, with spring flooding from snowmelt. In the integrated catchment, the rivers flow from the western headwaters at an elevation of 1,350 m on Mount Kamuro to the east and into the Pacific Ocean. Approximately 60% of this area is forested and mountainous. Two major reservoir dams (Kamafusa and Okura dams) are located there on the rivers. The regional lowlands are farmlands (13%, primarily with rice paddy fields) and a mixture of residential and commercial areas (11%).

Genetic data were generated from a previous study (Watanabe, Kazama, Omura, & Monaghan, 2014). We used neutral AFLP markers from four macroinvertebrate species (the mayfly Ephemera japonica, and the caddisflies Hydropsyche orientalis, Stenopsyche marmorata and Hydropsyche albicephala) in this catchment (Appendix S1: Figure S1). These species all have substantially diverse habitat specificity and distributions within the network (Nukazawa, Kazama, & Watanabe, 2015, 2017; Watanabe et al., 2014). In addition, the four species differ in their dispersal symmetry when comparing the difference between their drift and flying potentials (Nukazawa, Kazama, & Watanabe, 2017; Watanabe, Monaghan, Takemon, & Omura, 2010; Yaegashi, Watanabe, Monaghan, & Omura, 2014).

2.3 Development of metapopulation genetic model

This metapopulation genetic model has the two modules that correspond to observation and to evolutionary processes. The observation module describes the statistical inference of the true value of population's genetic frequency based on genetic measurements derived from our sampling of parts of individuals of population. We use Bayesian inference (see the subsequent section "Bayesian fitting process") to fit the output of this module to our genetic measurements and then to calibrate the model parameters. By connecting statistical frameworks with two key processes of gene flow and genetic drift, the evolutionary module allows the genetic frequency of each local population to be equivalent to the frequency of the metapopulation with an estimate of the deviation of the frequency. The deviation is based on the genetic similarity of distance decay among local populations, with the values of deviation tending to be similar if these populations have high ecological connectivity among them along watercourses. The paragraphs below describe the statistical inference used for the descriptions of these two modules.

The observation module describes how genetic measurements and sampling errors are produced. A single AFLP locus has two allelic types, labelled "1" and "0" for the dominant and recessive types, respectively. We denote by the number of individuals for which the dominant type "1" corresponds to genotypes ("1," "1") or ("1," "0") in our diploid species, which was detected at locus l from the total number of individuals (; with types "1" and "0" together) in local population k. Here, was used as the modelling output calibrated by our observed data. This random sampling process can be characterized by a binomial distribution as follows (Guillot, Vitalis, Rouzic, & Gautier, 2014):

(1)

where, in this local population, the frequency of allele "1" is denoted by . The allele frequencies are independent between loci. The detection probability linked to was set up for the allele type "1" corresponding to genotypes.

The evolutionary module describes the genetic structuring of local populations within a metapopulation. For each locus, the frequency of allele "1" in the local populations is determined by their genetic variation (i.e. that related to genetic drift), the watercourse distance between local populations within the network (related to gene flow) and the allele frequencies of the metapopulation. Without genetic drift and natural selection, gene flow leads to genetic homogeneity among local populations, which results in allele frequencies at loci in local populations matching those of their metapopulation (Andrews, 2010). We denote the deviation from the expected frequency of allele "1" in a logit scale as , which was derived from the assumption of local populations being connected at the metapopulation scale. The allele frequency is obtained from the inverse logit transformation as follows:

(2)

where denotes the metapopulation's logit-transformed allele frequency. The deviation can be modelled by a multivariate normal distribution as follows (Bradburd, Ralph, & Coop, 2013):

(3)

where denotes the vector of zero, and the covariance matrix is a function of the watercourse distance between local populations and their spatial relationships. These spatial connections between local populations are either streamflow-connected (populations with downstream–upstream relationships) or streamflow-disconnected (e.g. populations in different headwaters). This statistical process allows similar values of deviation for local populations having high ecological connectivity along watercourses. We modelled the covariance across local populations as a function of the shortest watercourse distance, , along the river network between populations i and j as follows (Ver Hoef & Peterson, 2010):

(4)

where the first or second addend (i.e. for any number that is added to another; for example, and are different addends of their summation ) describes the autocovariance, with the variance or and the scale parameter or related to the downstream (D) or upstream (U) movement, respectively. In the first addend, the autocovariance is set to zero for any two streamflow-disconnected populations. In other words, streamflow-disconnected populations are assumed to be independent and have no gene flow between them via downstream movement. In the second addend, the autocovariance can apply to both situations. The genetic connections between streamflow-disconnected populations are attributed to the upstream gene flow coming from a common source population in the downstream confluences. High values of variance and small values of the scale parameter lead to weak isolation as a function of the watercourse distance.

2.4 Bayesian fitting process

In the Bayesian framework "Stan" (Stan Development Team, 2018), the R interface "RStan" (Stan Development Team, 2019), was used for genetic modelling of the metapopulation and to fit the model using our empirical data (see the above section "Framework overview"). For each species, four Markov Chain Monte Carlo (MCMC) chains (which were used for numerical approximations of Bayesian inference) were run with 20,000 iterations each, and the first half of the iterations for each chain were discarded as “burn-in.” This iteration setting was determined by the convergence of MCMC chains when the R-hat statistic of each parameter approached a value of 1. To estimate the model parameters, 2,000 samples obtained by collecting one sample out of every 20 iterations for each chain were used to build each parameter's posterior distribution. The target average for the proposal acceptance probability and initial discretization interval was set to 0.95 and 0.1, respectively, during Stan's adaptation period.

Prior distributions were set up for the model parameters based on their data type (e.g. continuous or discrete), range of values (e.g. unlimited or positive), adjustments using an informative feature to keep reasonable values but not eliminating those that might be appropriate, parameter validation (through the virtual ecologist approach; see Appendix S1) and model convergence efficacy. The transformed allele frequency has a normal distribution with mean of 0 and variance , and a half-Cauchy distributed hyperprior on the variance with a location of 0 and a scale of 5. The variances ( and ) each have a half-Cauchy distribution with a location of 0 and a scale of 5. The scale parameters ( and ) each have a half-normal distribution with mean of 0 and variance of 2. The four macroinvertebrate species examined (E. japonica, H. orientalis, S. marmorata and H. albicephala) each had both types of relationship between locations (i.e. streamflow-connected and streamflow-disconnected pairs).

2.5 Creation of virtual river networks

We created 1,000 virtual river networks with varying branching complexities (Terui et al., 2018; Yeakel, Moore, Guimarães, & de Aguiar, 2014). These virtual river networks (with scale length e equal to 1 km between local populations) have the branching probability P and metapopulation size N (which is an integer and represents the number of local populations in a river network) randomly drawn from 0 to 1 and from 100 to 500, respectively (Figure S3). This scale length was determined by referring to the local scale of stream macroinvertebrate occurrence, for example a 100-m reach (Stoll, Breyer, Tonkin, Früh, & Haase, 2016). Natural river networks in Japan have a range of P from .3 to .6 (Terui et al., 2018). The full range of P can reflect general conditions and provide insights into networks that, because of human intervention, depart from natural value of P (e.g. a network with p = .1 can represent a human-created canal; Figure S3).

The virtual river networks comprised nodes with scale length e, with each node representing a local population. These nodes were assigned to be either branching (or representing an upstream terminal) or non-branching with a probability of p or 1–p, respectively. A segment was defined as a series of non-branching nodes terminating at a branching (or terminal) node. The number of nodes in a segment is a random variable that is assumed to be geometrically distributed with the probability of success p (equal to the branching probability). To achieve the target number of total nodes in a network, individual segments were formed by consecutively drawing their number of nodes from the geometric distribution. As a statistical reformulation of Horton's laws, a geometric distribution has been used to describe the segment lengths (e.g. Peckham & Gupta, 1999). Before merging the segments to create a river network, the whole drawing process was repeated if a non-equal number of total nodes or a non-odd number of segments was attained. To create the river network, these segments were put together as a hierarchically merged pool in three steps (Figure S3). For Step 1, one segment, consisting of one or more nodes, was randomly selected as the root, and its upstream end was merged with the downstream end of two randomly selected segments. From this procedure, the semi-complete network had two unmerged upstream ends each for the next possible merger. For Step 2, two more segments were then randomly selected, and their downstream ends were merged together to the random draw of one of two (or even more at subsequent steps) unmerged upstream ends of the semi-complete network. For Step 3, Step 2 was repeated until there were no available segments left in the pool.

2.6 Simulation of river-branching influences

We conducted stochastic simulations to illustrate the genetic differentiation and diversity at the metapopulation scale under river branching. Our metapopulation genetic model for each species with the Bayesian median estimates of the parameters added was then used to simulate allele frequencies at loci in local populations in each of the 1,000 random river networks (see the above section "Creation of virtual river networks"). The spatial genetic structure was summarized to the global to be the metapopulation's genetic divergence. In addition, the metapopulation's genetic diversity was calculated as global (Nei, 1973). We performed this simulation using the R packages "stats" and "base" (R Core Team, 2018).

We then built a regression model based on gradient boosting (GB) (Appendix S1) for each of the four macroinvertebrate species, which identified the importance of the following spatial features: (1) the fraction of any two local populations being streamflow-disconnected in all possible combinations; (2) the mean watercourse distance between local populations under different levels of river branching; and (3) the metapopulation size (i.e. number of local populations) for genetic divergence (global ). These features were considered to be summarized aspects of complex spatial configuration of habitat network.

Through further simulations on the same set of river networks, we illustrated how the downstream and upstream dispersal-related parameters of the scales ( and , respectively) influence river-branching effects on the genetic divergence of the metapopulation (global ) and genetic diversity (global ). We considered 3 × 3 (nine total) combinations of the scale parameters, with each having three values, which were the maximum, median and minimum values of range of their Bayesian pooled estimates where the posterior median of both and across the four macroinvertebrate species was pooled together. Because of the shared pool, the maximum, median or minimum values set for the two scale parameters in the combinations are the same. We set the metapopulation's logit-transformed frequency of allele “1” at each locus to zero during each simulation. The number of loci was set as the median across the four macroinvertebrate species. In addition, we set the variance parameter ( or ) to the median value of their Bayesian pooled estimates, with the two variance parameters having the same values. We replicated the nine combinations in each of the 1,000 random river networks.

3.1 Metapopulation genetic modelling

When our Bayesian model was fitted to the empirical genetic data in the Natori and Nanakita catchments, the R² values derived from the residuals (i.e. the differences between the observed and predicted numbers of type "1" alleles at a locus from the number of alleles observed in a local population; see Equation 1) were 0.98, 0.97, 0.97 and 0.93 for H. orientalis, S. marmorata, H. albicephala and E. japonica, respectively (Figure S4).

The metapopulation allele frequencies at loci are species-specific, and the variation of allele frequency was greater in the widespread H. orientalis and S. marmorata than in the other two species that had narrower habitat distributions (Figure S1). Pairwise genetic differences between empirical local populations tended to increase with their watercourse distances throughout the four macroinvertebrate species (Figure 1). Despite substantial variation in the scale parameter that amplified the isolating effect of distance across study species (Figure 2), there was a consistent decline in the genetic correlation between populations (the covariance divided by the variance ; see Equation 4) with the increasing distance between local populations being obvious for the three caddisfly species but not the mayfly having lower value of (Figure S5).

3.2 Simulation of river-branching influences

The increasing branching probability of river networks caused changes in the observed spatial features (Figure 3). For example, the fraction of any two local populations being streamflow-disconnected (e.g. occurring in different tributaries) across the metapopulations occurs at high rates in the heavily branched river networks. However, we found shorter watercourse distances occurred between local populations in river networks with higher branching probability. As the size of the virtual network increases, the metapopulation size (i.e. the total number of local populations of the network) increases as do the values of both spatial configurations under the same level of river branching.

Two spatial features (watercourse distance and fraction of flow-disconnected sites; Figure 3) summarized the spatial configuration of habitat network and were linked to the genetic differentiation of metapopulations (; Figure 4) and total genetic diversity (; Figure S6). However, species-specific responses to the influence of river branching were evident. For example, increased branching probability decreased the genetic divergence and diversity of the metapopulation for one caddisfly (H. albicephala), but the mayfly E. japonica showed the opposite response (higher genetic divergence) before a branching threshold. According to the GB modelling results, the relative importance of streamflow-disconnected habitats, compared with the landscape connectivity via a shorter watercourse distance, was higher for the mayfly than in all the caddisflies examined (Table 1).

Branching complexity has various impacts on genetic divergence and diversity, which is determined by the relative values of the upstream and downstream scale parameters in a genetic covariation function (Equation 4). Stronger dispersal tendency can be associated with a lower value of the scale parameter in either direction, which will result in weak isolation by watercourse distance within a network. When species dispersal was upstream-biased ( < ), branching complexity enhanced genetic homogenization among local populations within a metapopulation. In addition, lower genetic divergence and diversity are related to lower downstream-biased and higher upstream-biased dispersals (relative higher and lower , respectively; Figure 5 and Figure S7).

4.1 Branching complexity and genetic divergence

The role of network branching has been documented, to some extent, in genetic studies of riverscape conducted in situations where higher genetic diversity is observed in downstream populations than in upstream ones (Paz-Vinas et al., 2015). Also, greater river branching can increase the differences between such populations (Thomaz et al., 2016). In dendritic networks, our simulation results showed a species-dependent change in global genetic differentiation levels that occur with increases in network complexity. This finding was also previously observed for two sympatric salmonid species that were found to have remarkably different spawning locations, mating systems and population sizes, and for which biological traits mediated the influences of riverscape features that shape their dispersal and genetic divergence (Whiteley, Spruell, & Allendorf, 2004).

Little or no gene flow can occur with high network branching. This pattern of gene flow is caused by strong genetic isolation from an increase in the number of distal populations (e.g. in headwaters) when the same number of nodes in a habitat network is present. This pattern can generally be observed for some riverine species with high or intermediate levels of downstream-biased vagility, such as fish and molluscs (Osborne, Perkin, Gido, & Turner, 2014; Pilger et al., 2017; Terui et al., 2014). In one, early theoretical study that did not consider dispersal asymmetry (which would be analogous to having equal downstream and upstream dispersals in our study), the dendritic network structure was also documented to promote low genetic distances under conditions of high connectivity (Labonne et al., 2008). However, our results show that the different downstream and upstream gene flows considered in our model can act together to generate similar or even different relationships.

4.2 Importance of species dispersal

The architecture of river networks can be an important extrinsic factor in explaining the observed and simulated genetic patterns in our study, but there was high variation among the species with different dispersal traits in our study. By adopting our mechanistic model that was fitted to empirical data of macroinvertebrate species with flying adult stages, our simulations reveal that their better dispersal abilities can overcome catchment constraints, leading to low downstream-biased asymmetry. The opposite (negative) influence occurs under increased river branching.

In the river systems we examined, asymmetric (either downstream- or upstream-biased) dispersal could determine the effect of branching complexity on genetic divergence of a metapopulation. In our model, the scale parameters for upstream and downstream dispersal tendencies (in the genetic covariation function; Equation 4) can be linked to dispersal asymmetry. These two factors together provide a mechanism behind the countervailing influences of river branching on the genetic divergence of metapopulations. Stronger dispersal can be associated with a lower value of the scale parameter either in downstream or in upstream direction, which will result in weak isolation by distance within a metapopulation.

Our results suggested that, in the mayfly species examined (E. japonica) having high downstream-biased gene flow (based on a higher value of the scale parameter for upstream than for downstream dispersal), river branching has a positive influence on genetic divergence. This phenomenon can be attributed to a higher number of isolated distal branches in river networks (e.g. more headwaters) occurring under this dispersal asymmetry, as typically shown in fish (e.g. Pilger et al., 2017). Larvae of mayflies also are susceptible to high levels of drift during times of increased river flow and therefore have high potential to be strong downstream dispersers (Nukazawa et al., 2017).

In our system, an opposite (negative) influence of the branching network on the three caddisfly species with dispersal symmetry or even upstream-biased dispersal was evident (Figures 2 and 4). The two asymmetry levels were based on the scale parameter for their upstream dispersal being similar to or higher than that for their downstream dispersal, respectively. This result could be attributable to their flying adults generally providing a wide dispersal range and strong terrestrial movement (e.g. at least in the upstream direction). This flying ability would compensate for downstream drift dispersal. This upstream movement dampened the isolation between distal branches in river networks for caddisflies compared with mayflies (and although not examined, likely stoneflies as well), which exhibit restricted distributions to areas very close to their sources of emergence in a stream (Winterbourn et al., 2007).

4.3 Unexplained genetic variations

Based on the results of the virtual ecologist approach (Figure S2), the Bayesian fitting process is more likely to have failed in retrieving the assumed value of variance parameters in the scenario of equal values of scale parameters than in other scenarios. These stochastic effects are derived from evolutionary processes modelled by autocovariances related to upstream and downstream dispersal along the habitat networks. The deviations from stochasticity are not significant when compared to the large difference between the actions of the autocovariances on the genetic structures. Thus, the Bayesian estimations are more easily confounded when both autocovariances have equal or similar influence on the stochastic processes. In general, more sampling of sites in the river network and additional collections of individuals from local populations would result in higher constraints for parameter estimations, which can decrease the stochastic effects. In river systems, this issue can occur from aquatic insects having different aerial dispersal abilities and may not apply in other riverine species (e.g. fish and mollusc species) having downstream-biased dispersal.

Some bias in our modelling results could result from not considering overland dispersal, particularly when asymmetric dispersals along the watercourse in our model explain a large part of the genetic variation at a metapopulation scale. Overland dispersal across tributaries or catchments has been demonstrated to influence genetic differentiation (Alp et al., 2012; Chaput-Bardy, Fleurant, Lemaire, & Secondi, 2009). However, this type of dispersal may not have made an important contribution in our empirical system when compared with in-stream dispersal. Overland movement can be particularly important to strong fliers such as damselflies and riverine birds, for example dippers (Hernández et al., 2012). However, aquatic insects with weaker flying strength (e.g. mayflies or caddisflies) have larger longitudinal tendency to use stream corridors than to use lateral areas away from the stream channel (Petersen et al., 2004). However, empirical genetic evidence indicates that in-stream dispersal was the main dispersal pathway (Chaput-Bardy, Lemaire, Picard, & Secondi, 2008).

4.4 Overview and management implications

From a conservation and management perspective that attempts to include considerations of the spatial genetic structure of metapopulations (e.g. Luque, Saura, & Fortin, 2012), an understanding of the branching structure's role in driving metapopulation genetic divergence is critical. Dispersal can dictate differences in genetic diversification in landscapes (Medina, Cooke, & Ord, 2018), and our predictive modelling based on asymmetric dispersals across networks sheds light on better management practices. For example, in the context of managing native or even invasive species (Chaput-Bardy, Alcala, Secondi, & Vuilleumier, 2017), our simulation results provide insight into the evolutionary importance of dispersal abilities and modes, suggesting that these intrinsic factors should be considered in decision-making processes where one management strategy does not fit all species (Figure S9). For example, the same management and conservation practices can produce different or even opposite results for species with varying levels of asymmetric gene flow and genetic drift (e.g. in dendritic river systems). Moreover, by adopting a Bayesian-fitted mechanistic framework into local landscapes, simulations in virtual networks could result in insights gained via experimental or empirical studies that can assist managers build specific strategies for different regions. For example, the potential genetic influences from past or planned human alterations of river systems (e.g. dam construction, loss of headwater habitats from pollution, branches diverted for agriculture or creation of flood-bypass channels) may be incorporated into assessments of branching complexity. In doing the assessments, ecological connections rather than purely hydrological linkages are considered to be the elements forming the structure of habitat networks. In addition, this approach can be considered for incorporation into management strategies (e.g. in-stream barrier removal and habitat restoration placed in the context of the river network) and anticipated responses to environmental changes (e.g. drying out of segments due to reduced precipitation under climate change).