Ecology, 93(1), 2012, pp. 29–34 Ó 2012 by the Ecological Society of America Optimum swimming pathways of fish spawning migrations in rivers BRANDON MCELROY, 1 AARON DELONAY, AND ROBERT JACOBSON U.S. Geological Survey, Columbia Environmental Research Center, 4200 New Haven Road, Columbia, Missouri 65201 USA Abstract. Fishes that swim upstream in rivers to spawn must navigate complex fluvial velocity fields to arrive at their ultimate locations. One hypothesis with substantial implications is that fish traverse pathways that minimize their energy expenditure during migration. Here we present the methodological and theoretical developments necessary to test this and similar hypotheses. First, a cost function is derived for upstream migration that relates work done by a fish to swimming drag. The energetic cost scales with the cube of a fish’s relative velocity integrated along its path. By normalizing to the energy requirements of holding a position in the slowest waters at the path’s origin, a cost function is derived that depends only on the physical environment and not on specifics of individual fish. Then, as an example, we demonstrate the analysis of a migration pathway of a telemetrically tracked pallid sturgeon (Scaphirhynchus albus) in the Missouri River (USA). The actual pathway cost is lower than 10 5 random paths through the surveyed reach and is consistent with the optimization hypothesis. The implication—subject to more extensive validation—is that reproductive success in managed rivers could be increased through manipulation of reservoir releases or channel morphology to increase abundance of lower-cost migration pathways. Key words: energetics; energy expenditures; least cost; least-cost migratory pathways; fluvial velocity fields; migration; Missouri River, USA; pallid sturgeon; Scaphirynchus albus; spawning migration. INTRODUCTION Many fish species that spawn in fluvial habitats migrate long distances upstream. To the degree that they use the same resources to swim and to develop gametes, there is a clear evolutionary advantage to traversing a migratory pathway that minimizes energy expenditure. Specifically, as has been demonstrated for brown trout (Salmo trutta), chinook salmon (Oncorhyn- cus tshawytscha), and sockeye salmon (Oncorhynchus nerka), the fecundity of spawning fishes decreases with migratory distance and difficulty (Kinnison et al. 2001, 2003, Crossin et al. 2004, Jonsson and Jonsson 2006). While the migratory cost in the study of Kinnison and others (2001) is singularly a function of total migration distance, in large rivers with a great variety of possible paths through a range of water velocities, the cost should be a function of the total path distance and velocities experienced along it. This assumes that there are sufficient cues for fish to start migrating and that migration is its primary behavior. In this paper we develop the theoretical framework for analyzing least- cost migratory pathways of fluvial fish including a cost function for comparing pathways, and apply it to an example of telemetry-based migration pathway data for the endangered pallid sturgeon (Scaphirhynchus albus) in the Lower Missouri River, USA. Pallid sturgeon is a federally listed, endangered species endemic to the Mississippi Basin (Mayden and Kuhajda, 1997). In spring reproductive females migrate distances of 100–150 km or more (DeLonay et al. 2009) to spawn upstream. Because flows in the Missouri River are highly managed for purposes including navigation, flood control, power generation, and ecosystem needs, the effects of reservoir releases and associated flow velocities during migration and spawning periods could have implications for successful reproduction of pallid sturgeon. Moreover, channelization for navigation has substantially altered the river’s velocity distribution along its lower 1200 km (Jacobson and Galat 2006). Within this context it has been observed that adult pallid sturgeon utilize relatively slow-velocity regions along the inner banks of bends during migration (DeLonay et al. 2010). Possibly these fish are selecting pathways in order to optimize their migration. For these reasons, we seek to elucidate the physical controls on energy expenditure of fishes during spawning migrations in rivers. The general problem of finding a most efficient or otherwise optimal path from a starting location to an ultimate location or series of locations is well developed in the context of networks such as roadways, railways, electric transmission lines, the internet, or other features of physical infrastructure (Ahuja et al. 1993, Stoll 1989, Huitema 2000, Brandimarte and Zotteri 2007). Regular lattices such as a digital elevation models (DEM) can also be viewed as a series of interconnected locations (Rees 2004). Here, we employ a discretized three- Manuscript received 15 June 2011; revised 24 August 2011; accepted 1 September 2011. Corresponding Editor: S. J. Simpson. 1 Present address: University of Wyoming, Department of Geology and Geophysics, Department 3006, Laramie, Wyoming, USA. E-mail: bmcelroy@uwyo.edu 29 R eports