Watersheds are drained by river networks, which route materials and energy from headwaters to terminal water bodies. River networks likewise perfuse the terrestrial portion of watershed ecosystems and reroute some of these materials upslope via material exchange between rivers and land. Here we develop a model of resource exchange between rivers and watersheds to predict the spatial extent of material and nutrient fluxes from aquatic portions of watershed ecosystems. The model is based on a geomorphic template that includes river network structure, topography, and channel sinuosity as well as important biological attributes (productivity and dispersal ability). Analysis of this model suggests that the geomorphic template strongly influences the spatial extent of resource flows in watershed ecosystems. The geomorphic template also predicts the location of areas of concentrated resource exchange, typically at ridge crests, in meander bends, and tributary junctions. We contend that these areas represent hotspots of foraging activity for terrestrial consumers, especially those at the reach scale (meander bends). More generally, our model suggests that the spatial extent of aquatic resource flow equal in magnitude to 20% or greater of terrestrial production may encompass as much as 20%–50% of terrestrial portions of watersheds. Resource flow from rivers to terrestrial ecosystems is not merely an edge effect. Instead, the river network may reroute a substantial flux of materials into watershed ecosystems.
 The flow of energy, material, and organisms (hereafter, referred to as resources) across aquatic-terrestrial (AT) boundaries is a ubiquitous process in many ecosystem types [Hynes, 1975; Summerhayes and Elton, 1923]. In watershed ecosystems, most research has focused on the flow of terrestrial derived resources (e.g., leaves) to recipient aquatic ecosystems [e.g., Fisher and Likens, 1973; Wallace et al., 1997]. This focus emerged from the logical assumption that gravity-fed resource flows are paramount, especially when considering the substantially larger surface area of the terrestrial component of watersheds in comparison to the aquatic component. This historical view is changing though, and the more recent focus has been on the flow of aquatic-derived resources to terrestrial ecosystems [e.g., Baxter et al., 2005; Marczak et al., 2007]. The flow of energy, detritus, and organisms from the water to the land it turns out, can have large effects on terrestrial community structure [Polis and Hurd, 1996]. Many studies have now shown that aquatic insects have strong effects on terrestrial consumer abundance [Paetzold et al., 2005; Sabo and Power, 2002a], distribution [Nakano et al., 1999; Sabo and Power, 2002b], and behavior [Gray, 1993]. These studies have shown that aquatic insects can represent a substantial portion (25%–100%) of energy supplied to terrestrial consumers [reviewed by Baxter et al., 2005]. Nevertheless, it is our contention that many of these studies, including our own [Bastow et al., 2002; Sabo and Power, 2002b], have focused too narrowly on these phenomena immediately at the aquatic-terrestrial (AT) boundary.
 To support this contention, we reanalyzed papers included in a meta-analysis conducted by Marczak et al. . In this meta-analysis, Marczak et al.  used 32 studies to examine the effects of resource flow on changes in consumer density or biomass in recipient ecosystems. We extracted information describing the spatial extent at which consumer response to resource flow was measured from the donor ecosystem from 27 of these papers (Table 1). Here we defined “spatial extent” broadly as the linear distance between the donor system and the closest and farthest experimental replicate of measurement in the recipient system for each study. Papers were excluded if we could not extract data characterizing the distance from the donor ecosystem at which measurements were made. We found that most (59%) of studies on spatial resource flow measured consumer response to donor subsidies immediately at the interface between donor and recipient ecosystems (i.e., 1–10 m; Figure 1). If we assume that the location of these studies represents the typical maximum spatial extent of resource flows, then the extent of resource flows from donor habitats on recipient ecosystems is typically unsubstantial. Marczak et al.'s  meta-analysis was not limited to aquatic insect resource flows, but included numerous types of resource flows including marine detritus, leaf litter, salmon, and terrestrial insects. If we limit our analysis to only those studies that examine the effects of aquatic insects (from rivers) on terrestrial consumers (n = 8), we see a vaguely similar pattern. The majority of studies (5/8) examine the response of terrestrial consumers to aquatic insects within 10 m of the AT boundary and the frequency distribution for the spatial scale of stream studies is similar to that for all studies combined, albeit not as steeply declining (Figure 1). These results either suggest that the extent of river watershed exchange in watershed ecosystems is truly limited to terrestrial consumers distributed immediately at AT boundaries, or that the focus of studies on river-watershed resource flow has been overly narrow.
Table 1. Spatial Extent of Resource Subsidies to Consumers in Recipient Ecosystema
 Thus, the main question we ask is: Are the effects of aquatic resource flows across AT boundaries exclusively an edge phenomenon? Or conversely, does the influence of aquatic resource production perfuse far into watershed ecosystems by virtue of contact of the watershed with the river network? We suspect that the spatial extent of aquatic resource fluxes in watershed ecosystems will be driven by characteristics of the river network structure and planform geometry. Specifically, we hypothesize that watershed geomorphology will exert strong controls on the spatial extent of river watershed exchange. In this vein, we derive a series of equations based on classic studies in fluvial geomorphology [Horton, 1945; Leopold and Maddock, 1953; Leopold and Wolman, 1960; Schumm, 1963; Stolum, 1996; Strahler, 1952] that allow us to estimate the proportion of a theoretical watershed influenced by aquatic resource flows. Specifically, we examine the proportion of the watershed in which we can detect a flux of aquatic resources equivalent to 20% of terrestrial secondary production. We then use these equations to understand two aspects of the spatial extent and pattern of aquatic resource concentration within watersheds. First, we predict the spatial extent of aquatic resource flows in watersheds using a realistic variation in geomorphology (network density and bifurcation, topography, and planform geometry) and biology (resource production and dispersal ability). Second, we identify areas where overlap in aquatic resource flow from adjacent sub-basins, confluent tributaries, and parallel sides of meander bends produce areas of putatively high local aquatic resource supply. We hypothesize that these “hotspots” constitute critical foraging habitats for terrestrial consumers, and are predictable from landscape form.
 Our work is relevant to a hydrologic sciences audience for several reasons. First, hydrology determines river network architecture, planform geometry, and other aspects of geomorphology that enhance the contact of the river channel with the terrestrial portions of the watershed. We review and synthesize these relationships as they relate to our ecological questions. Second, hydrology determines the relative abundance, community composition, and trophic structure of aquatic organisms [Power et al., 2008; Sponseller et al., 2010; Hagen, 2010; Sabo et al., 2010]. In a similar vein, hydrologic variability determines primary and secondary production of river ecosystems [Fisher et al., 1982; Cross et al., 2011] and the export of river-derived carbon and nutrients to terrestrial portions of the watershed [Hagen, 2010]. Hence, hydrology influences the spatial and temporal variability in insect emergence and the exchange of energy between rivers and their watersheds indirectly (via geomorphology) and directly (via effects on aquatic primary and secondary production). The framework we develop will provide a theoretical foundation for future empirical work on both of these important topics in the field of ecohydrology.
2. Network Approach to Predicting River-Watershed Exchange
2.1. Counter-Gravitational Energy Flow in Watersheds
 We took the well-documented example of movement of mobile stream insects into terrestrial portions of the watershed as a motivating example while developing a framework for predicting the spatial extent of energy flow between channels and uplands. Riverine macroinvertebrates begin their life cycle as eggs and larvae in the aquatic portion of the watershed ecosystem where they grow and develop by eating organic matter, some portion of which is derived from primary production by riverine algae, phytoplankton, or macrophytes [Merritt and Cummins, 1984]. Many of these animals eventually metamorphose into air breathing, terrestrial adults, some of which can disperse by flight over large distances [MacNeale et al., 2005]. Many larval aquatic invertebrates drift passively downstream [Waters, 1972] and this net displacement is typically counteracted by net upstream movement by flight as adults [MacNeale et al., 2005]. Very little is known about the extent and directional bias of lateral (upslope rather than within channel) movement of adults from the stream into the surrounding (dry) portions of the watershed. Three observations are relevant to our work below. First, completion of the lifecycle requires most of these macroinvertebrates to oviposit in or near the stream. Second, as much as 96% of the biomass from emergence is eventually exported to the terrestrial system [Jackson and Fisher, 1986]. This biomass may either be consumed by terrestrial consumers, including insects, spiders, lizards and bats [see Baxter et al., 2005 for review], or incorporated into detrital channels of the terrestrial food web via decomposition. Finally, a recent study on a single taxonomic group with excellent relative flight capacity suggests that adult macroinvertebrates do indeed disperse across boundaries of subwatersheds [MacNeale et al., 2005] but this is rare. Given the observation that much of the emergent insect biomass is lost to the terrestrial system, we assume diffusive lateral movement (flight) below. There are many reasons why diffusion may not hold, none of which have been adequately measured in any systematic fashion. The most important of these reasons is advection, which we discuss in more detail below. Nevertheless, empirical evidence suggests that the spatial pattern of abundance and biomass of aquatic insects from river to ridge is well described by an exponential decay process [Power et al., 2004; Hagen and Sabo, 2010].
 To enhance the generality of our approach we adopted a simplified geometry with readily apparent assumptions, and believe that this enables inference to various other specific examples of channel-upland flux. By flux we mean the instantaneous carbon input that is contributed to any point in the watershed from the channel. We propose that the spatial extent of this flux (i.e., resource exchange) depends on a complex set of landscape factors including: river network geometry, topographic relief, the planform geometry of the river channel, longitudinal changes in aquatic primary production throughout the river network, and the functional form of the relationship between the flux of aquatic resources (f, a flux in kg ∗ m−2 ∗ d−1; Table 2) and distance from the river. Before considering these complexities we first define the quantity (b) as the bandwidth of lateral exchange, or the horizontal distance to which river-derived resources penetrate terrestrial landscapes perpendicular to the path of the river. We further narrow the scope of this bandwidth by defining a perpendicular distance at which aquatic resource fluxes equal >20% of terrestrial production and use this critical distance ( ) to identify the proportion of the watershed receiving a significant flux of aquatic production. We hypothesize that the overall spatial extent of exchange may be high even when bandwidth is low, because the dendritic form of many drainage networks may correspond to high exchange efficiency. After deriving a basic equation relating b to the spatial extent of resource flow, we examine the effect of incorporating progressively more detailed elements of landscape complexity on this spatial extent.3
Table 2. Abbreviations, Descriptions, and Units for Model Parameters
Table 3. Properties of Rivers Used in Confronting Model With Data
Drainage Area (km2)
River Length (km)
Buffer Area (km2)
South Fork Eel River, CA
San Pedro River, AZ
2.2. Hortonian River Networks
Horton  devised a scheme for “ordering” stream segments in river networks using the following three simple rules:
 1. Source streams (with no tributaries) are assigned a Horton order of 1;
 2. Tributaries of equal order ν, join to make a stream or order ν + 1; and
 Tributaries of lower order (ν-1) that join stream segments of order ν do not alter the order of the recipient stream segment.
 Theoretical Hortonian networks are simple, they are self-similar [Dodds and Rothman, 2001], and for our purposes they lend themselves to easy analysis in exchange for a crude approximation of the network structure of real rivers.
3.1. Horton's Laws
 We start with a simple Hortonian river network (Figure 2) [Horton, 1945]. In this network, stream number (Nν), length (Lν), and contributing area (Aν) scale with Horton order ν such that the ratios of stream numbers , lengths , and contributing areas are constant across a wide range of stream orders [Horton, 1945; Leopold and Miller, 1956; Schumm, 1956; but see, Dodds and Rothman, 2001]. Here ν is the order of the stream, N is the number of streams of order ν in the entire network, is the average length of a stream segment of order ν, and is the average area contributing exclusively to a segment of order ν. In this paper we use empirical values measured by Horton  as the basis of our analysis (e.g., Rn = 3.78, Rl = 2.64, Ra = 4.75).
3.2. Effects of River Network Structure on River-Watershed Exchange
 Given Horton's laws, we can derive a measure of the spatial extent of resource flow that allows us some flexibility to account for the effects of the complexities in geomorphology such as network density, the effects of valley slope, tributary junctions, and meander bends. We can also add in some biological details such as systematic variation in aquatic resource production throughout the river network and spatial attenuation of lateral exchange in resource flows into terrestrial portions of the landscape. We start with a very basic description of exchange that initially ignores these biological complexities, but that will eventually accommodate all of them. All variables and parameters in this model are defined in Table 2.
 First, we define the total valley length of all river segments in the network as (in km), the bandwidth of lateral exchange along one side of this river length as b (in km), and the total areal extent of aquatic influence on terrestrial landscapes is hence:
The units of are in km2. The proportion of the terrestrial landscape influenced by aquatic resource flow is thus:
where is the drainage area of the entire watershed. For the remainder of the paper, we use this metric (and various modifications of its basic derivation in equation (2)) synonymously with the spatial extent of aquatic resource flow in watersheds, or more simply, the spatial extent. is a proportional measure of the spatial extent of aquatic resource exchange with terrestrial landscapes. has a natural link to the study of drainage networks in geomorphology. Fluvial geomorphology defines the ratio of total stream length to drainage area as drainage density, a quantity that is influenced strongly by hydrology via the interaction between hydroclimate, lithology, and land cover. Drainage density serves as a proxy for the topological complexity of river networks [Knighton, 1998]:
Substituting in equation (2) we get an expression for the proportional extent of aquatic resource flow to watershed ecosystems in terms of the complexity of the river network:
 For watersheds of equal area, the spatial extent of resource flow from rivers to watersheds should increase with drainage complexity as (Figures 3a–3b) as well as with the distance of lateral exchange, b (Figures 3c–3d). The total river length of a basin is determined in large part by hydrology; increases with decreasing mean annual precipitation. This is because a greater proportion of precipitation becomes runoff and erodes channels in arid catchments [Knighton, 1998]. Representative values for range from 1 to 20 km km−2 in arid and semiarid biomes and 1–6 km km−2 in biomes with higher annual precipitation [Gregory, 1976]. Increases in the distance of lateral exchange should lead to twofold higher increases in the spatial extent of resource flow than comparable (e.g., per km) increases in total stream length by virtue of two sides of contact between the river and the adjacent watershed. Thus, assuming values of b on the order of 0.01–1 km and a uniform distribution of that flux laterally (f ∼ constant from 0 → b), Px ranges from . Where (i.e., ), b is sufficiently large to produce areas of exchange overlap at tributary junctions, in meander bends, and most importantly (as we will document below), across divides separating smaller sub-basins in the headwaters of the watershed. Even assuming low values for (∼0.5 km km−2), for b > 1 km. Thus, the spatial extent of aquatic resource flow can be nearly complete even for sparsely drained river networks when resource exchange is high or conversely, when resource exchange is low in watersheds with high-drainage density. Below, we focus on estimating the overlap of zones of lateral exchange at tributary junctions, in meander bends, and across sub-basin divides to ascertain whether Px is still substantial after correcting this measure of the spatial extent of aquatic resource flow for these double-counted areas. This corrected estimate of the spatial extent of aquatic resource flow can be approximated by the following general equation:
where O is the overlap at tributary junctures (trib), meander bends (m), and across ridges separating sub-basins (sub). Below we develop expressions for , , and , as well as refining to reflect elongation of the stream length by channel meandering.
3.3. Predicting Tributary Overlap
 Because of the dendritic structure of the networks, we consider overlap in the resource exchange at tributary junctions, which is a potentially pervasive phenomenon at the watershed scale (See Auxiliary Material).1 Thus, more accurate expressions of that account for these zones of overlap are:
where is the overlap of resource exchange at the tributary junctions (in km2), is the number of tributary junctions (nodes) in the river network, and j is the overlap area on one side of a single tributary-main stem junction. There are two such regions of overlap of equal area on either side of the junction at each node (hence, 2j). From Horton's law of stream numbers we know that the number of streams of order ν, is:
From this relationship we can deduce the number of tributary junctions as:
where is the order of the largest stream segment in the network (see Auxiliary Material for details). Finally, we can define the area, j, as a parallelogram with sides of length, b, and a tributary junction angle of (see Auxiliary Material), namely:
 Substituting expressions (6b) and (7) into (5a) and (5b) we arrive at an expression for the proportional extent of aquatic resource exchange across the entire watershed corrected for overlap at tributary junctions:
Thus, the spatial extent of resource flow will decrease with branching given constant area and total stream length. Yet, the total area of overlap of lateral exchange between merging stream segments increases with branching. The spatial extent of energy exchange increases and overlap area decreases as a function of .
3.4. Effect of Valley Form and Steepness on River-Watershed Exchange
 Our expression for the spatial extent of resource flow (equation (9)) assumes a perfectly flat landscape such that the bandwidth of lateral exchange, b, can be measured as a planar or an azimuth distance. In most studies of resource exchange in watershed ecosystems, the bandwidth is measured as a slope distance (see Auxiliary Material), such that the spatial extent of resource exchange is overestimated by this measure. Topography should reduce lateral exchange, b (measured in azimuth m). To make predictions about the influence of topography on the spatial extent of resource flow we correct the distance of lateral exchange for the azimuth (m) distance as,
where is the slope distance and is the slope angle (see Auxiliary Material for an illustration). Substituting the expression for (equation (10)) for b in equation (9), we arrive at a new expression for the spatial extent of resource flow that includes slope and tributary complexities:
3.5. Effects of Channel Form on River-Watershed Exchange
 In section 3.5 we focus on reach scale variation in channel planform geometry and how this may influence the spatial extent of aquatic resource exchange at both the reach and the watershed scale. River channels have many different forms, including straight, sinuous, braided, and anastomosing. These forms have strong relationships with discharge (Qb), slope (s), the sediment load and size (G, D), and local soil composition (e.g., silt-clay content of bank, M) [Leopold and Maddock, 1953; Leopold and Wolman, 1957; Schumm, 1963]. We focus on the sinuosity of channels for two reasons: sinuosity occurs across a broad range of channel widths (w) [Leopold, 1994], and sinuosity is somewhat more tractable mathematically than other variations of planform geometry.
 Sinuosity (S) is defined in terms of the meander wavelength ( ) and the path wavelength ( ). The meander wavelength is the straight line (valley) length of a reach of river, whereas path length is measured along the river's course in the same reach (Figure 4). Sinuosity is the ratio of path to meander lengths:
 Channel meandering increases the surface area or the edge of river in contact with the terrestrial landscape. The corresponding increase in the spatial extent of aquatic resource flow is proportional to S, assuming no overlap in lateral exchange between adjacent portions of the reach within a single meander. As a result, the simplest approximation of the spatial extent of aquatic resource flows to terrestrial portions of the watershed that takes channel planform geometry into account is:
Equation (13) holds for rivers in which the bandwidth of exchange ( ) is less than , the average radius of curvature of meanders (see Figure 4a). In cases in which (i.e., Figure 4b), the spatial extent of resource flow is complicated by overlap within single (half cycle) meanders (Figures 4e and 4f) and between adjacent meanders (Figures 4b, 4c, 4e, and 4f).
 For most real rivers with irregular meanders, it would be intractable to predict the associated overlap in a deterministic framework. Empirical estimates of lateral extent and overlap may be more easily extracted from analysis of digital elevation models using geographic information systems (GIS). In this paper, we assume for simplicity that meanders are regular and semicircular in shape. Assuming semicircular meanders, we can predict the spatial extent of aquatic resource exchange as the product of the meander wavelength (λ) and , where is the average valley width of aquatic resource exchange (see Auxiliary Material). This width, , can be defined as the sum of one half of the average meander amplitude, , and the bandwidth of aquatic resource exchange ( ), or:
And assuming constant throughout the river network, the expression for the spatial extent of aquatic resource exchange then can be expressed as,
A more complete derivation of and its relationship to the spatial extent of aquatic resource exchange is given in Auxiliary Material. We note here that our derivation of is similar to the concept of meander belt width (equivalent to 2h in Auxiliary Material by Camporeale et al. ), with the exception that includes the biological attributes of resource diffusion and/or dispersal laterally. Equations (13) and (15) represent the endpoints of a continuum of varying effect of planform geometry on the spatial extent of aquatic resource exchange. Specifically, at low bandwidths of lateral exchange ( ), planform geometry increases Ps by a factor of S; however, as becomes larger ( ), the effects of sinuosity become swamped by the sheer magnitude (measured in terms of dcrit) of resource exchange and lateral overlap in aquatic resource exchange can occur among adjacent meander bends. This overlap leads to the overestimation of the lateral extent of aquatic resource exchange via equation (13). Thus, is a better predictor of resource exchange than the river's path length ( ) when is large. Finally, S and can be used together to predict the overlap in resource exchange (within or across adjacent meander bends). Specifically, for cases in which , overlap in meanders can be expressed as,
where includes overlap and does not (see Auxiliary Material for a more detailed derivation).
3.6. Sub-Basin Overlap
 Overlap in resource exchange can occur at the reach scale across meanders (equation (16); Figure 4) and at the watershed scale as the number of tributary junctions increases with (equation (6b); Figure 3). Overlap in resource exchange can also happen across sub-basins within the watershed where the bandwidth of exchange ( ) exceeds half the sub-basin (valley) width (see Auxiliary Material). This is an important consideration in small sub-basins feeding the smallest tributary streams high in the river network. We can again estimate the overlap area indirectly as a difference, in this case between the spatial extent of resource exchange P and the proportion of the total watershed area contained in the sub-basin drainage. Specifically, we first define the spatial extent of aquatic resource flow in a sub-basin for a stream segment of order ν as and the proportion of the area of the total drainage represented by that same sub-basin (of a stream segment of order, n) as . Given these definitions we can estimate cross-basin overlap as the magnitude by which spatial resource flow exceeds the sub-basin's drainage area, namely:
Here we use , to avoid double-counting the overlap in meander bends. In this way, we treat any area of resource exchange above and beyond the area of the sub-basin as overlapping with a neighboring sub-basin (see Auxiliary Material for more detail). Unbiased estimates of the total spatial extent of aquatic resource exchange can thus be estimated as the sum of Ps estimates corrected for all three forms of overlap in resource exchange, summed across all sub-basins in the watershed, or:
3.7. Longitudinal Variation in Sinuosity Through the River Network
 We have assumed until now that planform geometry of river channels is constant throughout the network. However, we know that S, , and can vary systematically with stream order, channel width, discharge, and drainage area (see Auxiliary Material). As a result, Ps, Om, and Osub also vary with stream order and sub-basin size. Many of these relationships can be quantified via regionally based, empirically measured power laws using hydraulic geometry [Carlston, 1965; Leopold and Maddock, 1953].
 To implement a systematic variation in planform geometry of channels throughout the river network, we make use of well-established relationships between drainage area ( ), bankfull discharge ( ), and stream width (w). We then use the estimates of channel width (specific to the stream order) to estimate three key metrics of the average channel planform for segments of each order in the network: the radius of curvature, , the meander wavelength, , and the meander amplitude, . We estimate then the angle of the meander bend, , indirectly from and . Together these metrics allow us to describe the systematic variation in planform geometry of channels as they relate to the drainage area. These relationships lead us to the following, more detailed, equation describing the spatial extent of aquatic resource flow to the terrestrial portion of the watershed in which a variety of details describing the watershed's geomorphology are considered (see Auxiliary Material for details):
where , is the order specific sinuosity (see Auxiliary Material for details), is the area contributing to a stream segment of order ν, from its exclusive sub-basin and all tributaries upstream, and all other variable are defined as above.
3.8. Longitudinal Variation in Secondary Productivity Through the River Network
 Aquatic primary and secondary production should vary between small tributaries and larger rivers within the watershed. The river continuum concept (RCC) [Vannote et al., 1980] provides a foundation for understanding this variation. The RCC posits a unimodal relationship between stream order (ν) and aquatic primary production. In this paper we use a unimodal relationship for an instream secondary production of resources. To do this we determine order-specific secondary productivity values using the polynomial relationship:
where is the baseline resource production in source streams, and β1–β3 are shape parameters relating to stream order.
3.9. Lateral Variation in the Bandwidth of Exchange,
where is the azimuth distance (perpendicular to river edge). However, equation (21) implies no lateral bound to lateral exchange as the exponential decay functions are continuous, but decreasing functions of . Thus, in these situations, the spatial extent of aquatic resource flow may be best described in terms of contours representing some critical magnitude of riverine production exchanged (laterally) with the watershed. For example, one might ask what proportion of the area of the entire watershed receives some critical level of aquatic resource flow (e.g., ). To estimate this, one can set equal to and solve for the appropriate distance, in terms of the remaining parameters. Thus, for simple exponential decay, we have:
where is the azimuth distance at which a critical flux of aquatic resources can be measured. Now, we can substitute for (e.g., in equation (21)), giving:
where is defined in terms of instead of (see Auxiliary Material) and . We can similarly define overlap ( , , ) by substituting for (as in Auxiliary Material), and summing across streams of order ν. We denote these estimates of overlap that account for network level variation in secondary production (in the stream) and decay in with distance from river (via equation (22)), as , , and .
3.10. Parameterization and Sensitivity of the Model
 We use the concepts developed in equations (1)–(23) to predict the spatial extent and overlap in aquatic resource flows in the terrestrial portions of watersheds for a set of Hortonian river networks that vary in drainage density ( ) and sinuosity (S). To do this, we first constructed a base network consisting of an 11th order stream in which = 3.78, = 2.64, and = 4.75. We then created networks with drainage densities ranging from 1 to 30 by adjusting the average width, and thus area ( ) of the contributing watershed to the largest stream segment in the drainage ( , exclusive area, = 500–75,000 km2). Thus, we use the same shape of the river network, but a range of to achieve variation in . Variable also allowed us to create gradients in planform geometry (e.g., S ∼ 1–6) as S can be estimated by scaling relationships between , stream discharge ( ), stream width (w), and several measures of planform geometry (see Auxiliary Material).
 In this paper we set (g m−2 d−1), β1 = 0.75, β2 = 1, and β3 = 0.005. These parameters produce a strongly unimodal relationship between stream order and secondary productivity (g m−2 d−1) and a saturating relationship between stream order and total resource availability per meter of shoreline (i.e., , in units of g m−1 d−1; see Auxiliary Material). Using these values in equation (20) led us to generate a range of values for secondary production, , of 0.16–0.35 g m−2 d−1. These values are within the range of observed values (0.008–0.012 g m−2 d−1 in Hubbard and Bear Brook, and 0.418–0.696 g m−2 d−1 for Sycamore Creek) (Hall et al.  and Jackson and Fisher , respectively). Secondary production (pν) may overestimate the proportion of aquatic productivity that enters the terrestrial watershed because pν includes both the production of aquatic insects without a terrestrial phase, as well as the proportion of insects that are transported downstream as drift. This effect will be minor, as the magnitude of aquatic secondary production does not actually figure significantly in the calculation of because is scaled by a constant value for terrestrial secondary production (see section 3.9). We set = 0.1–1.0 such that aquatic resource flux to terrestrial portions of the watershed decays relatively rapidly with (perpendicular) distance from the river. Finally, we fixed relative secondary production in the surrounding forest watershed (hereafter, ) as a constant throughout the watershed, and equal to 1–3 times the maximum level of aquatic production in the drainage. Thus, streams are most productive in mid- to large-order segments, the stream environment is less than or equal to the surrounding watershed in secondary productivity (per area), and the lateral exchange of aquatic resources drops steeply with the distance from river. All three conditions could be considered conservative when assessing the importance of exchange of resources between aquatic and terrestrial portions of watersheds. Given these parameter values, we then ask over what proportion of the watershed can we detect a flux of aquatic resources equivalent to 20% of terrestrial (secondary) production (e.g., )? Here lateral exchange distances ( ) range between 0 and 60 m, assuming values for , , and given above. Thus, we estimate for the very realistic case in which aquatic resources have their strongest effects on terrestrial systems near the edge of aquatic-terrestrial ecosystems.
 We explore the sensitivity of our model estimates of to changes in four key parameters in our model: S, , , and . To do this we first plotted changes in across gradients in sinuosity (S = 0–6) and drainage density ( = 0–25) assuming four combinations of values for and that produced a gradient of high to low : and , and , and , and and . Second, we plotted the same response surface of across gradients in the decay constant of lateral exchange ( ) and relative terrestrial secondary productivity ( ) assuming four combinations of values for S and that produced a gradient of low to moderate river network complexity: S = 1.9722; = 2.1808, S = 1.9523; = 6.5424, S = 4.5885; = 2.1808, and S = 4.5495; = 6.5424.
4.1. Effect of Valley Form and Steepness on River-Watershed Exchange
 Valley slope has a minimal effect on the lateral exchange of aquatic resources in the terrestrial watershed when valley slope is <30°. Azimuth distances, even for relative steep slopes (σ ≈ 45°, which is equivalent to 100% grade), are only ∼30% less than their corresponding slope distance (see Auxiliary Material). For the steepest slopes near 90°, found in deep canyons, is zero. Thus, lateral exchange and the spatial extent of resource flow are under strong geomorphic control, but this control is weak for all but very steep (e.g., high σ) canyons.
4.2. River Network Structure and Resource Exchange in Watersheds
 Model estimates of ranged from nearly 0 to1, demonstrating that the spatial extent of resource exchange from rivers to surrounding terrestrial ecosystems can include the entire watershed (Figures 5 and 6). Within the range of commonly observed values for sinuosity and drainage density (S = 1.5–3.5 and = 1–6) [Leopold et al., 1992; Stolum, 1996, Gregory, 1976], ranges from ∼0.2 to 0.5. Thus, over a realistic range of geomorphic variations in river network structure and planform geometry, we predict that the spatial extent of aquatic resource flow to terrestrial portions of watersheds is measurable, permeating as much as 50% of the entire area of the drainage (gray boxes in Figure 5).
4.3. Sensitivity of Pcorrect to Key Parameters
 Our simulation results further suggest that the spatial extent of aquatic resource flows to terrestrial ecosystems ( ) depends strongly on the density of the river network ( ; Figure 5) and to a lesser extent on the decay constant of aquatic resource exchange ( ; Figure 6). Sinuosity (S) and the ratio of terrestrial to aquatic secondary production ( ) had relatively small effects on compared to and . Specifically, contours of plotted as functions of and S were only minimally diminished by decreases in brought on by increases in (Figure 5). Similarly, contours of were relatively insensitive to increases in (Figure 5). By contrast, contours of plotted as a function of and were insensitive to changes in S but highly sensitive to changes in (Figure 6).
4.4. Predictions of Overlap in Resource Exchange
 Our model predicts substantial overlap in resource exchange at the reach as well as the network scale. An overlap at tributary junctions increases only with drainage density and the number of tributary nodes in the network (Figure 7). Cross-basin overlap also increases only with drainage density reflecting increasing proximity of river network segments to basin edges in denser river networks. In contrast to tributary junctions and cross-basin sources of overlap, overlap in meander bends increases with both drainage density and sinuosity reflecting the effects of both of these factors on total stream length, and thus the number of meander bends in the network. Total overlap (i.e., between adjacent sub-basins, meanders, and tributaries combined) can range from ∼0 to nearly 50% of the entire drainage area when we consider streams with parameter values for sinuosity and drainage density within the range commonly observed (S ∼ 1.5–3.5 and Dd ∼ 2–6, respectively). Most of this overlap in resource exchange areas occurs among neighboring meanders and along ridges (dividing neighboring sub-basins; ∼ 0.5%–10% and ∼ 50%–100%, respectively). Finally, sub-basin overlap is most common (though rare relative to meander overlap) in low order tributary streams, whereas meander overlap is most common in larger-order rivers (Figure 8).
 Overlap at tributary junctions is minor compared to that between sub-basins and meander bends on a per-area basis ( < 0.5%). Overlap in meander bends and at ridge crests (cross-basin overlap) is much more substantial. However, our estimates of overlap do not take into consideration changes in aquatic resource concentration with distance from the river, and how these changes determine concentration in overlap areas. Specifically, we suspect that though tributary junctions represent a minor source of overlap in resource exchange in terms of area, tributary junctions and meander bends (especially in low order streams) should have some of the highest resource concentrations due to overlap of lateral fluxes that have been diluted very little being close to the source (river). These areas of overlap may represent “hotspots” of resource concentration and/or energy exchange between aquatic and terrestrial ecosystems.
4.5. Confronting the Model With Data
 We collated data on the spatial abundance of aquatic insects in two watersheds (Table 3) to quantify , dcrit, and hence, and the potential for sinuosity and tributary junctions to create hotspots of resource exchange between aquatic and terrestrial ecosystems. Specifically, we used data from separate efforts spanning 5 yr (1998, 2006–2009) of aerial insect sampling on the South Fork Eel River (Mendocino County, CA, U.S.A.) and the San Pedro River (Cochise and Pima counties, AZ, U.S.A.). Insect sampling consisted of monthly or seasonal deployment of sticky traps hung at regular intervals perpendicular to the river flow path (typically, 0, 1, 5, 10, 25, 50, and 100 m azimuth distance from the river) to estimate and the biomass of aquatic insects directly above the wetted river channel (for detailed methods see Power and Rainey ; Hagen and Sabo ; Hagen ; Hagen and Sabo ). Using these data we ask two questions: How well do empirical observations of compare to estimates from our model for a representative watershed (here the South Fork Eel River)? and Do tributaries and meander bends create “hotspots” of increased aquatic-terrestrial energy exchange in the form of denser swarms of adult aquatic invertebrates over the active channel?
 To address the first question, we downloaded the watershed boundary and flow line for the eight-digit Hydrologic Unit Code (HUC) basin corresponding to the SF Eel River (HUC: 18010106) and San Pedro River (HUC: 15050602) from the National Hydrography Data set (NHD Plus). We then estimated and dcrit based on sticky trap data from sticky trap transects from the SF Eel and San Pedro rivers [Power and Rainey, 2000; Hagen and Sabo, 2011; Hagen, 2010]. Values for were −0.1 on average (range: −0.24–0.004), identical with the value used in simulations generating Figures 5–7. To be consistent with simulations in these same figures, we estimated dcrit assuming that average terrestrial production was equivalent to aquatic production (a conservative assumption for both river basins)[Power and Rainey, 2000; Hagen, 2010], producing dcrit values of ∼14–68 m (median ∼45 m). was then estimated via equation (2) with and using the NHD Plus data to estimate drainage density . The estimated drainage density for the SF Eel and San Pedro rivers was <1 suggesting underestimation of by NHD Plus. Using these estimates of drainage density for the two basins is ∼0.105 for dcrit = 65 m (range 0.10–0.11). This value for is nearly identical to the estimate from our model for rivers with realistic sinuosity (S ∼ 2–6) and drainage density of ∼1 (Figure 5).
 To assess the potential for tributaries and sinuosity to create hotspots of resource exchange between rivers and watersheds, we again used our sticky trap data, but in this case to quantify invertebrate swarm density directly over the river (biomass at 0 m). We then estimated the river path distance to the nearest tributary and reach scale sinuosity for each trap location. Since some of our sticky trap transects were set in steep canyons (some transects in the SF Eel River basin), we characterized the linear relationship between sinuosity (S) or proximity to tributary and aquatic invertebrate biomass in the airspace directly above the channel (0 m) separately for confined versus unconfined channels (see Hagen and Sabo  for details). Tributary distance had no effect on aquatic invertebrate biomass in either confined or unconfined channels. By contrast, aquatic invertebrate biomass increased positively with sinuosity, but only in unconfined channels (Figure 9). These data confirm that meander bends in rivers define hotspots of resource exchange in watershed ecosystems; and that valley form may constrain the effect of planform geometry on hotspot formation.
 Resource flows between aquatic and terrestrial compartments of watershed ecosystems are central to understanding ecosystem energetics within watershed landscapes [Vannote et al., 1980; Nakano and Murakami, 2001; Baxter et al., 2005]. Gravity forces energy downslope from the terrestrial to the aquatic compartment and organisms move this energy back upslope. The importance of this countercurrent energy flow depends on biological attributes of rivers (productivity and dispersal ability) as well as the perfusion of the watershed by the river network (planform geometry and network complexity). Although there is much agreement that aquatic resources are important for terrestrial species living in edge (riparian) environments, very little is known about how extensive the influence of this countercurrent exchange is in watershed ecosystems. Here we provide the first theoretical treatment of the spatial extent of aquatic resource flows to terrestrial landscapes. Our review of the literature on resource flows between rivers and their watersheds suggests that a great majority of research has focused very narrowly on how aquatic resource flows affect food webs only very near a river's edge. Despite this narrow focus, our model of resource exchange in river networks provides support for three ideas. First, the spatial extent of aquatic-terrestrial resource flows in watersheds is under strong control by the geomorphic template of the river network ( ) and channel planform geometry (S). Second, given this control, even modest levels of flux at the channel-bank interface may still affect a meaningful portion of the watershed's terrestrial area (20%–50%) at a level equivalent to >20% terrestrial secondary production. And third, within this landscape of resource exchange the river's planform geometry may create overlap zones that constitute potential “hotspots” of foraging activity for terrestrial consumers.
5.1. Geomorphic Control of Pcorrect and “Hotspots” of Resource Exchange
 Drainage density can enhance the exchange of resources between aquatic and terrestrial compartments in watershed ecosystems by effectively increasing the surface area for exchange relative to the total watershed area. In fact, drainage density was paramount in predicting the spatial extent of aquatic resource flows in terrestrial watersheds, while sinuosity and relative secondary productivity had little effect (Figure 6). Similarly, a more meandering channel exposes a greater area of a reach of the river valley to exchange between aquatic-terrestrial ecosystem compartments; however, much of this additional exchange area is overlapping such that sinuosity increases overlap in the areas of exchange ( ) much more than the spatial extent of resource flow ( ). Finally, terrestrial overlap of aquatic resource fluxes originating from different stream segments, adjacent meander bends, and two streams joining at confluences are regular and predictable features of watershed ecosystems. These overlap areas constitute potential “hotspots” of resource availability for terrestrial consumers.
 Our analysis of a landscape model for resource exchange from rivers to terrestrial portions of watershed ecosystems suggests that a pressing challenge in watershed science is to empirically quantify the spatial extent of aquatic resource flows to terrestrial ecosystems. This task will require the application of new tools, some unfamiliar to stream ecologists. For example, while tracing and remote sensing technologies are commonplace in oceanographic studies [e.g., Frouin et al., 1996; Longhurst et al., 1995], the level of high-resolution mapping technology to detect insect swarms within forested watersheds is still in the early stages of development. Even so, current weather Doppler radar systems are sensitive enough to detect insect swarms at the scale of tens of kilometers (available at http://www.cirrusimage.com/ephemeroptera_mayflies.htm). High-resolution tracing and remote sensing technologies would provide useful information on the spatial distribution of aquatic resource flow within watershed ecosystems. But, rather than investing in extensive remote sensing networks to detect the extent of aquatic resource flow, we argue that a more fruitful expenditure of research time and money would be to concentrate research efforts at hypothetical locations of resource overlap. We predict that areas of overlap represent hotpots in energy exchange between aquatic and terrestrial realms of watershed ecosystems. These hotspots will be most relevant to food web ecologists because they represent areas of the watershed where one would expect to find concentrated abundances of terrestrial consumers, intensified species interactions, as well as altered food web dynamics. Moreover, our field data suggest that channel meanders do indeed lead to hotspots of aquatic insect biomass concentration in rivers in the western United States.
 The importance of hotspots could be further tested by comparing the relative abundance of mobile consumers and aerial aquatic insects in predicted areas of resource overlap (ridges between adjacent sub-basins, meanders) versus areas with minimal overlap (along linear sections of river). Our model predicts that aerial insects should be more concentrated in hotspots at tributary junctions, in meander bends, and at ridge crests. If consumers are able to track resources at these spatial scales, we predict that they should likewise concentrate their foraging activities in these hotspots of resource exchange. In this vein, bats and birds represent ideal study organisms as they are abundant consumers along river ecosystems, are highly mobile consumers that are able to sample widely spaced habitats within a watershed, and aquatic insects represent a significant prey item [Belwood and Fenton, 1976; Brigham et al., 1992; Nakano and Murakami, 2001].
5.2. Caveats and Model Assumptions
 Our model of aquatic resource flows in terrestrial landscapes is very basic, and its application requires a number of simplifying assumptions that may in some cases, limit its realism. First, in our model the “resource” is an adult aquatic insect flying through the terrestrial landscape. Hence, we assume that aquatic insects are capable of dispersing long distances, perhaps even over basin ridges, and recent studies validate this assumption [MacNeale et al., 2005]. Moreover, our model assumes that resource flows from the stream source into the terrestrial landscape can be approximated by a simple one-dimensional diffusion process [Turchin, 1998]. More specifically, we assume that flows of resources occurring in an oblique (rather than perpendicular) direction with respect to the channel's flow, average out such that unidirectional, lateral flows give a reasonable approximation of two-dimensional resource availability. Similarly, we assume pure diffusion without advection, and thus ignore the potentially important effects of wind in determining both the spatial extent of aquatic resource fluxes in terrestrial ecosystems, and the pattern of overlap in fluxes from different portions of the river network.
 Wind may increase the spatial extent of aquatic resource flows. For example, aquatic insect swarms may be carried by wind further and more completely into the terrestrial landscape, especially when steep valleys or riparian vegetation do not constrain swarms to riparian corridors. By contrast, wind could also decrease the spatial extent of aquatic resource flows to terrestrial portions of the landscape if wind moves primarily longitudinally through river networks bounded by steep canyons and dense riparian forest vegetation, thereby constraining aerial aquatic resources to the river valley. Despite these important examples, we suspect that wind exerts a potent, but more transient effect on the pattern and spatial scope of aquatic resource fluxes to terrestrial ecosystems. Over larger temporal and spatial scales, we hypothesize that the geomorphic template exerts a stronger influence on patterns of aquatic-terrestrial resource exchange. Evidence from mass mark recapture studies [MacNeale et al., 2005] and in this paper (Figure 9) support this hypothesis.
 Second, we make a number of simplifying assumptions about the terrestrial landscape. These assumptions include: square-shaped drainages of sub-basins, constant 45° tributary angles throughout the river network, constant, and moderate topographic relief (e.g., slope angles of 15°), and regular sinusoidal meanders that are circular in shape (not tortuous). Clearly, real landscapes have much greater complexity: irregularly shaped drainage areas that may lengthen with drainage area (i.e., Hack's Law, see Rodriguez-Iturbe and Rinaldo ), variable confluence angles controlled by slope and geology, regular changes in topographic relief from headwaters to main stem [Dodov and Foufoula-Georgiou, 2005], and irregular (tortuous) meander bends. Moreover, we ignore the complexities of braided and anastomosing channels which may serve to increase in larger rivers. Future versions of this model might include relationships between watershed area and slope/sediment size which could allow one to estimate certain properties of these more complex forms of geometric channel variation [e.g., Eaton et al., 2004, 2010; Kleinhans and van den Bergh, 2011].
 In these and other ways our model is but a crude approximation of the landscape and how this landscape controls aquatic resource fluxes. We suggest that future studies use digital elevation models (DEM) of real landscapes and empirically measured resource fluxes in representative stream segments within these landscapes to obtain a more realistic measure of the spatial extent of aquatic resource flows to watershed ecosystems.
 Third, we ignore potentially important seasonal variation in aquatic resource production (e.g., insect emergence). If the flux of aquatic resources is seasonally pulsed, our estimates of the may not reflect accurately the annual influence of aquatic resources on food webs in the terrestrial compartment of watershed ecosystems. During times of low riverine productivity, our estimate of would overestimate the proportion of the watershed influenced by aquatic resources. On the other hand, during periods of peak riverine productivity, our estimate of may underestimate the spatial extent of aquatic resources, particularly because periods of peak riverine productivity often coincide with times of low terrestrial productivity [Nakano and Murakami, 2001].
 Finally, our calculations of overlap area are not weighted by resource concentration in a way that reflects natural decay in resource flux with distance from the river source (via, in equation (25)). In this way, overlap areas in confluences, meander bends, and across basin divides are weighted equally. This is clearly unrealistic because overlap in small confluences or small meander bends likely leads to much higher concentrations of aquatic resources than overlap at basin edges where resource fluxes may be considerably diluted. Overlap close to the river may actually increase resource concentration above what we would expect for a single or straight river channel.
5.3. Synthesis and Future Direction
 The concept of trophic subsidies in ecology has garnered a lot of empirical support since its introduction [Polis and Hurd, 1995, 1996; Polis et al., 1997; Baxter et al., 2005; Marczak et al., 2007]. This empirical work suggests that physical boundaries like those between land and the ocean, lakes, or rivers are boundaries of convenience to ecologists, not boundaries of significance to ecological processes. In this paper we provide a quantitative illustration of the consequences of incorporating this growing awareness of the notion that biological processes are influenced by, but do not strictly adhere to changes in obvious physical features of the landscape. The major take-away messages from this work are threefold. First, the boundary between land and water in watersheds is not sharp; instead, the influence of the rivers on terrestrial food webs is significant but decays steeply with distance. Second, in spite of the decaying influence of aquatic production on terrestrial food webs, the topology of river networks should have a large bearing on the spatial extent of resource exchange between water and land and this topology, and the planform geometry of rivers at the reach scale likely dictate the location and magnitude of exchange within the watershed. These ideas are imminently testable in the field and represent a critical frontier for the continued synthesis of landscape and food web ecology. Finally, hydrology is paramount in determining the influence of riverine production on terrestrial landscapes. Hydrology determines river network topology and planform geometry via the influence of basin size and discharge on these geomorphic properties. Hydrology also influences primary and secondary production via influences of extreme events on nutrient availability and population dynamics. This latter effect of hydrology has yet to be thoroughly explored; however, the framework we develop in this paper would provide an excellent starting point for estimating the effects of hydrologic variability on the spatial extent of aquatic resource exchange at the watershed scale.
 We thank D. Auerbach, P. Dockens, S. Fisher, T. Harms, J. Heffernan, J. Maron, K. McCluney and six anonymous reviewers for comments on previous drafts of this manuscript. The ideas in this paper emerged from valuable discussions we had with S. Fisher, J. C. Moore, A. Paetzold, M. Power, and R. Sponseller.