## Introduction

While movement is one of the most fundamental attributes of animals, our weak understanding of the spatial scale over which animals move has limited our ability to define a population unit (stock identification) and to manage and conserve populations effectively (Gowan *et al.* 1994). These problems are especially perplexing in fisheries management, particularly for salmonids. Moreover, an increasing number of studies are challenging the commonly held view (Gerking 1959) that stream fish move very little (Gowan *et al*. 1994; Gowan & Fausch 1996; Kennedy *et al*. 2002).

Understanding movement is also important at the ecosystem level where it contributes to flows in nutrients and energy on the landscape (Polis & Strong 1996; Finlay, Khandwala & Power *et al.* 2002), and determines the spatial scale of predator–prey interactions, which is key to our understanding of stability and function (McCann, Rasmussen & Umbanhowar 2005). Information on the scale of movements of fish is also vital to ecotoxicological studies on effects of effluents, and environmental programmes require information on the scale of movements of fish to assess impacts (e.g. Galloway *et al*. 2003; Gray & Munkittrick 2005).

While a great deal of spatial information has been obtained from studies that have marked and tagged individual organisms, and more recently tracked them with telemetry and various types of weirs (Cunjak *et al.* 2005; Gowan & Fausch 1996), developments in chemical tracer technology (e.g. stable and radioisotopes) makes it possible to infer the spatial scale of many ecological processes through indirect means (Hobson 1999; Tucker *et al*. 2000; Finlay *et al*. 2002; Kennedy *et al*. 2002; Cunjak *et al*. 2005). Chemical tracer approaches can potentially be applied to landscape studies in any situation where a spatial gradient exists; but, while this type of approach is applied routinely in geochemistry, it has seen limited application to ecological problems.

River geochemists have shown that the inorganic δ^{13}C signature in rivers undergoes a very predictable shift from headwater signatures in equilibrium with soil carbon dioxide signatures and weathering processes to heavier and more atmospherically equilibrated signatures downstream (Telmer & Veizer 1999; Finlay 2003). This gradual transition is attenuated as a result of the slow release of isotopically depleted carbon dioxide into the river from decomposition of refractory organic carbon, combined with the gradual downstream increase in alkalinity and pH (Yang, Telmer & Veizer 1996). Thus, inorganic δ^{13}C signatures in small tributary streams are ~−16‰, and become gradually more enriched to ~−7‰ or heavier in lower reaches.

As a result of this geochemical gradient, river biota acquires a distinct gradient in the δ^{13}C. Additional factors such as stream velocity and periphyton biomass influence signature fractionation (Trudeau & Rasmussen 2003; Finlay 2004; Rasmussen & Trudeau 2007) through their effects on boundary layer carbon dioxide availability and diffusion, and add ‘noise’ to along-stream signature gradients of periphyton and invertebrates. Along most mainstem river courses, invertebrate signatures predominantly reflect authochthonous carbon sources (Finlay, Power & Cabana 1999; Finlay 2001; McCutchan & Lewis 2002) and all but the specialist ‘leaf shredders’ tend to acquire the along-stream carbon signature gradient described above. The signatures of relatively sedentary fish (e.g. sculpins, Gray *et al.* 2004; and small cyprinids, Rosenfeld & Roff 1992; Doucett *et al*. 1996), usually reflect local carbon sources; however, those of more mobile fishes such as juvenile salmonids should reflect their feeding movements and likely remain out of signature equilibrium with local food sources. Thus, signature interpretation and estimation of diet-tissue isotopic fractionation (Vander Zanden & Rasmussen 2001) can be complicated for mobile fish, since their movements can result in substantial signature deviations from local sources. Such signature deviations can, however, be useful in inferring the spatial scale over which consumers move and feed, and thereby, integrate resource signatures. In this paper, we outline a framework for analyzing the signatures of mobile consumers on a linear resource signature gradient. We then demonstrate the applicability of this framework using data from published and unpublished sources, and use it to compare the movement estimates obtained for free-ranging juvenile fishes, with those of stream resident salmonids as well as more sedentary fish taxa.

### theory

A consumer in isotopic equilibrium with a resource whose signature has an along-stream gradient will tend to acquire the same signature gradient, with its signatures displaced by an amount* f* (either + or −). This trophic shift can be due to biased consumption, assimilation or tissue fractionation (Fig. 1a). Alternatively, a consumer that is mobile, i.e. does not feed in one place long enough to achieve isotopic equilibrium with local resources, can have signatures that deviate from local resources and such deviations should tend to be most pronounced near the boundaries of the resource gradient since movement history is likely to be biased away from the boundary in the direction of the gradient average, *Y*_{AVG} (Fig. 1b). We can approximate such a shift in the consumer signature gradient as a straight line (small dashed line Fig. 1b) whose slope is reduced relative to that of the resource gradient; the more mobile the consumer the greater the shift. If consumer feeding movements are so pronounced that they span the entire resource gradient, all of these fish may acquire signatures close to the gradient average (*Y*_{AVG}) leaving no detectable gradient in the consumer signature. For simplicity, in Fig. 1b, *f* = 0. Shifts in gradient slope resulting from feeding movements can also complicate the estimation of *f*, especially if the signature gradient is not well characterized, and we will return to this point below.

In this paper, we demonstrate that signature shifts along a gradient, of the type shown in Fig. 1b occur, and that they likely reflect movement history. We propose to use the signature deviation as a quantitative estimator of the spatial scale of that movement history. It is important to specify that our proposed estimator will detect the spatial scale over which feeding occurs, and should be insensitive to movements where feeding does not occur, or occurs to a reduced extent (e.g. spawning migrations or movements to overwintering habitat). Moreover, the estimate would be weighted by the extent to which different feeding areas contributed to feeding and ultimately growth.

We make the simplifying approximation that the deviation of the consumer signature from local sources (α) results from averaging across a uniform distribution range corresponding to 2α, produced by spatial averaging across a feeding range β (Fig. 1c). Let the resource signature (*Y*_{R}), change with river distance (*X*) according to

*Y*

_{R}=

*Y*

_{1}+

*SX*(eqn 1)

where *S* is the slope along the river gradient and *Y*_{1} is the intercept at *X* = 0. Also, let the consumer signature (*Y*_{C}) be weighted in the direction of the gradient average (*Y*_{AVG}) in proportion to *p* (= β*/*Δ*X*) the proportion of the gradient over which an individual feeds and averages over, and weighted in the direction of the resource signature (*Y*_{R}) in proportion to *q* (= 1 − p). Thus, *p* reflects the degree to which consumer signature slopes are shifted relative to resources; however, it must be noted that this is a relative shift which must be scaled to the spatial scale of the gradient (Δ*X*) to provide a movement estimator (β). The equation for the consumer signature obtained by this weighting is

*Y*

_{c}=

*Y*

_{AVG}

*p*+

*Y*

_{1}

*q*+

*SqX*+

*f*(eqn 2)

and has the properties illustrated in Fig. 1c, in that its slope is reduced relative to *S*, and when *f* = 0, its intercept is shifted towards *Y*_{AVG} in proportion to *p*. Moreover, its boundary conditions are also appropriate; that is, if *p* = 0, and *q* = 1 (local feeding), the consumer signature will equal the resource signature +*f*, and if *p* *=* 1, and *q* = 0 (feeding over the whole gradient) then *Y*_{C} = *Y*_{AVG} + *f*. The consumer signature deviation at any point in the gradient (α) from the resource signature

*Y*

_{AVG}

*p*+

*Y*

_{R}

*q*] −

*Y*

_{R}+

*f*

and

Thus, the slope of the consumer signature gradient (equation 2) divided by the slope of the resource signature gradient (equation 1), will be an estimator of *q*, and β can be obtained by multiplying the slope reduction factor (*p*) by the spatial scale of the gradient. Although *f*, the trophic shift, will affect the absolute value of the consumer signature, it will not affect the slope and therefore have no effect on the estimate of β (Fig. 2). *f* can be estimated by comparing the intercepts of the consumer and the resource equations (equations 1 and 2) once *p* and *q *have been determined from the slope ratio.

An alternative approach to estimating β that may be statistically more robust is to plot consumer signatures directly against resource signatures, i.e. *C*(*R*), as shown in Fig. 1d and thus remove the common variable *X*. In this case, movement can be assessed by comparing the slope of the consumer-resource signature plot to the value of 1, which would be expected if each consumer were in signature equilibrium with resource signatures in the immediate locality in which it was captured. Because we can write equation (2) as

*Y*

_{c}=

*Y*

_{AVG}

*p*+

*q*[

*Y*

_{1}+

*SX*] +

*f*

*Y*_{C} can be written as a function of *Y*_{R}

*Y*

_{c}=

*Y*

_{AVG}

*p*+

*q*

*Y*

_{R}+

*f*(eqn 4)

The slope of the consumer-resource signature therefore provides a direct estimate of *q* = 1 − *p*, and β can be obtained without having to compare consumer and resource gradient slopes.

Thus, β can be estimated in two different ways, the first, outlined in equation (3) is based on the degree of reduction of the slope of the consumer's signature with respect to river distance (*X*), and the second, outlined in equation 5 is based on the deviation of the consumer resource signature plot from 1. Standard error estimates can easily be derived for the β estimates (*SE*_{β}) based on equation (5) since they will equal the SE of the *C*(*R*) slope multiplied by Δ*X*; however, for β estimates based on equation (3) (*SE*_{β}), estimates would need to be based on ratio algorithms (see Cochrane 1977).

The estimation of *f* for a mobile consumer is not trivial since it can be confounded by movement (Fig. 2); it can be calculated from the intercept of the consumer-resource signature plot, which equals *Y*_{AVG}*p* + *f*, once *p* has been determined from the slope. Similarly, *f* can be estimated from the intercept of the river gradient model for the consumer signature since the intercept of this plot =*Y*_{AVG}*p* + *Y*_{1q} + *f*.

In this model, we have assumed that the consumer feeds entirely on aquatic food sources, and when this is not the case, we would expect along-stream slopes to be influenced by terrestrial signatures. Since the signatures of terrestrial organic matter tends to be relatively constant and in the mid-range of the along-stream autochthonous signature gradient (~−28‰Finlay 2001), consumption of terrestrial material should tend to reduce the along-stream slope of the consumer's signature, since signatures both upstream and downstream will tend to be deflected in the direction of the terrestrial signature (France 1995). Therefore, terrestrial consumption would be expected to confound estimates of mobility based on along-stream signature slopes. The model introduced above can be extended to include the combined influence of consumer mobility and terrestrial consumption, to develop algorithms for estimating the scale of movement adjusted for the proportion of the diet made up by terrestrial material.

In this paper, the following questions are addressed:

- 1Do the dissolved inorganic carbon (DIC) δ
^{13}C signature gradients that have been shown by geochemists to occur in a wide range of rivers, lead to simple gradients in river biota? - 2Are the signature gradients for free ranging mobile consumers such as juvenile Atlantic salmon (
*Salmo salar*L.) that eats mainly aquatic invertebrates (Thonney & Gibson 1989; Mookerji, Weng & Mazumder 2004), significantly less steep than those of primary producers and invertebrate primary consumers? - 3Do fish that are resident in small streams that are constrained spatially by movement barriers (e.g. waterfalls, culverts, beaver dams), or sedentary species, have signatures that more closely resemble those of their resources than free-ranging fish, that are free to move throughout the river system?

It is important to clarify at the outset that the data assembled in this paper to test the hypothesis were not collected for this purpose, and that the model was elaborated post hoc. As a result, the data serve more to demonstrate that the approach is promising and that rigorous tests are possible. To test the idea rigorously will require detailed descriptions of the signature gradients for a number of individual systems, and the purpose of this paper is to motivate studies on a wide range of systems so as to fully explore the spatial applications of stable isotopes to studies on movement.