Florida International UniversityFIU Digital CommonsCenter for Coastal Oceans Research FacultyPublications Institute of Water and Enviornment10-26-2016Relations between morphology, buoyancy andenergetics of requiem sharksGil IosilevskiiTechnion-Israel Institute of TechnologyYannis PapastamatiouDepartment of Biological Sciences, Florida International University, ypapasta@fiu.eduFollow this and additional works at: https://digitalcommons.fiu.edu/merc_facPart of the Life Sciences CommonsThis work is brought to you for free and open access by the Institute of Water and Enviornment at FIU Digital Commons. It has been accepted forinclusion in Center for Coastal Oceans Research Faculty Publications by an authorized administrator of FIU Digital Commons. For more information,please contact dcc@fiu.edu.Recommended CitationGil Iosilevskii, Yannis P. Papastamatiou R. Soc. open sci. 2016 3 160406; DOI: 10.1098/rsos.160406. Published 26 October 2016rsos.royalsocietypublishing.orgResearchCite this article: Iosilevskii G, PapastamatiouYP. 2016 Relations between morphology,buoyancy and energetics of requiem sharks.R. Soc. open sci. 3: 160406.http://dx.doi.org/10.1098/rsos.160406Received: 16 June 2016Accepted: 29 September 2016Subject Category:PhysicsSubject Areas:biomechanicsKeywords:cost of transport, active metabolic rate,optimal swim speed, sharksAuthor for correspondence:Gil Iosilevskiie-mail: igil@technion.ac.ilElectronic supplementary material is availableonline at https://dx.doi.org/10.6084/m9.figshare.c.3517521.Relations betweenmorphology, buoyancy andenergetics of requiemsharksGil Iosilevskii1 and Yannis P. Papastamatiou21Faculty of Aerospace Engineering, Technion, Haifa 32000, Israel2Department of Biological Sciences, Florida International University, Miami, FL 33181,USAGI, 0000-0002-4114-3214Sharks have a distinctive shape that remained practicallyunchanged through hundreds of millions of years of evolution.Nonetheless, there are variations of this shape that varybetween and within species. We attempt to explain thesevariations by examining the partial derivatives of the costof transport of a generic shark with respect to buoyancy,span and chord of its pectoral fins, length, girth andbody temperature. Our analysis predicts an intricate relationbetween these parameters, suggesting that ectothermic speciesresiding in cooler temperatures must either have longerpectoral fins and/or be more buoyant in order to maintainswimming performance. It also suggests that, in general, thebuoyancy must increase with size, and therefore, there must beontogenetic changes within a species, with individuals gettingmore buoyant as they grow. Pelagic species seem to have nearoptimally sized fins (which minimize the cost of transport),but the majority of reef sharks could have reduced the cost oftransport by increasing the size of their fins. The fact that theydo not implies negative selection, probably owing to decreasedmanoeuvrability in confined spaces (e.g. foraging on a reef).1. IntroductionWithin marine environments, sharks represent a wide range ofupper and mid-level predators. They can be found in most marinehabitats from coastal to pelagic and deep sea, and encompass avariety of feeding modes, including those specializing on marinemammals and filter-feeding [1]. These habitats also span a widerange of temperatures from arctic to tropical conditions. All sharkslack a swim bladder and therefore must generate lift either byretaining large amounts of low-density lipids (hydrostatic lift) orby generating flow of water over their fins (hydrodynamic lift). Inspite of the lipids reserves, the majority of sharks are negativelybuoyant and sink if they stop swimming (table 1).2016 The Authors. Published by the Royal Society under the terms of the Creative CommonsAttribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricteduse, provided the original author and source are credited.2rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................Table 1. Nomenclature.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .b span of the pectoral fins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .c0 root chord of a pectoral fin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .C, C+, C∗ cost of transport in general, cost of transport at v = v+ and v = v∗. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Cf friction coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .CD, CL drag and lift coefficients based on reference area S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .CD0 parasite (no lift) drag coefficient based on reference area S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .CD,max drag coefficient at the maximal lift coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .CL,max maximal lift coefficient based on reference area S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .D hydrodynamic drag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .d maximal effective body diameter,√4Sb/π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .g acceleration of gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .K induced drag coefficient based on reference area S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .kK phenomenological constant in (3.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .km prismatic coefficient of the shark’s body defined in (3.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .kP, kτ phenomenological constants in (3.14). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .L hydrodynamic lift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .l pre-caudal (or fork) length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .lt total length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .m mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .P active metabolic rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .P0 basic metabolic rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .P+, P∗, Pmin active metabolic rates at v = v+, v = v∗ and v = vmin(0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .S general reference area, either Sb or Sp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Sb, Sp maximal cross-section area of the body; gross projected are of the pectoral fins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .SD0, S(b)D0 parasite drag area (SD0 = SCD0), parasite drag area of the body. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .s distal margin of a fin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .T thrust. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .u, u¯ characteristic speed defined in (3.17); u¯ = u/w. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v, v¯ swim speed; v¯ = v/w. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v¯c , v¯2 dimensionless quantities defined in (4.14) and (4.21). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vmin(γ ) minimal swim speed at angle γ relative to horizon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v+,v¯+ swim speed yielding the minimal active metabolic rate; v¯+ = v+/w. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v∗, v¯∗ swim speed yielding the minimal cost of transport; v¯∗ = v∗/w. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .W weight of the shark in the water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .w characteristic speed defined in (3.18). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .x generic sensitivity parameter: b, c0, d, l,β , or τ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .α phenomenological parameter in (3.14); angle of attack in figure 1 (only). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Bl reduced excess buoyancy of the liver oils Bl = (ρ − ρl)/ρ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .β reduced excess densityβ = (ρb − ρ) /ρ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .γ swim angle relative to horizon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .η, ηm propulsion and muscle efficiencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(Continued.)3rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................Table 1. (Continued.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ρ ,ρ l density of water, effective density of the liver oils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ρb effective density of the shark. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .τ body temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . a reduced quantity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Being forced to swim continuously to generate hydrodynamic lift, sharks are faced with choicesregarding their swim speed. As the swim speed increases, so does the metabolic cost, and the probabilityof a successful encounter with prey. In all cases, sharks—as other predators—probably select the swimspeed that maximizes the difference between the energy obtained from prey and the energy spentsearching for it. This speed depends on morphology and buoyancy, each affecting the hydrodynamicresistance, as well as on body temperature, which affects the basic metabolic rate [2,3]. Most species ofsharks are ectothermic, so variations in body temperature reflect variations in the water temperature theshark resides in.With a few exceptions, sharks evolved having similar (fusiform) basic body shape, but withconsiderable differences (some of which are ontogenetic) in the relative size of fins, relative bodydiameter and the amount and composition of lipids retained in the body [4–9]. In this study, we suggest aunified theory (theoretical framework) that can relate some of these differences with particular lifestylesand habitats, and can explain some of the ontogenetic differences as direct consequences of allometricscaling laws of swimming performance. It is based on general predictions of energetic costs of activity insharks and swimming speeds that minimize these costs, and specific predictions of the influences of themost conspicuous morphological parameters, buoyancy and temperature on the energetic costs and onthe respective optimal speeds.The theory is presented in §§3 and 4; its few immediate conclusions ensue the developments of §3.6,3.7, 4.7 and 5; overviewing discussion concludes the paper in §6. The data used in the analysis arepresented in §2.2. Underlying dataThe ideal dataset for this study would have included tracking data (speed, depth, body temperature,water temperature and salinity), along with the respective morphological data (length, girth, finsdimensions), and in and out of water weights, for many individuals of different species. At present,no such dataset exists. The set compiled for this study (electronic supplementary material, S1,table S2) included 58 individuals from nine species of morphologically similar requiem sharks:Carcharhinus obscurus, C. leucas, C. plumbeus, C. brevipinna, C. limbatus, C. falciformis, Negaprion brevirostris,Galeocerdo cuvier and Prionace glauca, for which in and out of water weights were reported in [7,9].Morphological data for these individuals were estimated based on relative dimensions reported in[4,5,10]. Hydrodynamic data were estimated from morphological data, using aircraft preliminary designtools [11] (electronic supplementary material, S1). We could evaluate the accuracy of these estimates,using wind tunnel measurements at relevant Reynolds numbers (electronic supplementary material, S2);they were accurate to within a few per cent.3. Fundamentals3.1. Lift and dragConsider a negatively buoyant fish swimming at constant speed along a straight path, inclined at angleγ relative to horizon (positive when ascending). ρ, v, g and m are density of water, the swimming speed,the acceleration of gravity and the displaced mass of water, respectively. The latter can be expressed asm= ρSblkm, (3.1)where l and Sb are the (pre-caudal or fork) length of the fish and its maximal cross-section area andkm is the prismatic coefficient—the ratio between the volume of a body and the volume of the minimalcylinder enclosing it; km ranges between 0.5 and 0.6 for most fish.4rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................0 2 4 6 8 10 12 14 1600.20.40.60.81.01.21.41.61.82.0a (deg)C L00.20.40.60.81.01.21.41.61.82.000.20.40.60.81.000.20.40.60.81.00.1 0.2 0.3 0.4 0.5 0.6CD0.10 0.15 0.20 0.25 0.30(a) (b)Figure 1. Lift and drag coefficients, CL and CD, of a fictitious shark as measured in the wind tunnel at length-based Reynolds numberof 2× 106. This particular shark has the same morphology as the great hammerhead (Sphyrna mokarran), except for the head whichhas been rounded to appear as a typical requiem shark. Details of the experiment can be found in electronic supplementary material,S2. In (a), α is the angle between the shark centreline and the swimming direction when pectoral fins are aligned with the centreline.Reference area is the maximal cross-section area of the body. In (b), the dotted line marks a curve-fitting parabola (3.6). In grey lettersto the right of both figures and on the top of (b) are the corresponding values of lift and drag coefficients when the reference area is thegross projected area of the pectoral fins (which was twice the cross-section area of the body). Separation starts above α = 10°, wherethe curve-fitting parabola on (b) starts to deviate from the data, and develops into a full stall atα = 14°, where the lift coefficient drops.When swimming at constant speed, the hydrodynamic lift L and thrust T counterbalance drag1 D andweight WT=D+W sin γ (3.2)andL=W cos γ . (3.3)Hydrodynamic lift and drag are commonly expressed asL= 12ρv2SCL (3.4)andD= 12ρv2SCD, (3.5)where S is an arbitrary reference area, and CL and CD are the lift and drag coefficients. We assume thatthe lift is contributed mainly by the pectoral fins,2 and therefore, the lift coefficient depends mainly onthe angle between the lifting surfaces (pectoral fins) and the flow (figure 1a). We also assume that the1Lift is universally defined as the component of hydrodynamic force in the direction perpendicular to the direction of swimming.Thrust and drag are both defined as the components of hydrodynamic force in the direction of swimming, the former along it, and thelatter opposing it. For a self-propelling body—as a swimming shark is—separation between the two is essentially impossible [12]. Inthis study, we define drag as the respective component of the hydrodynamic force that would have acted on the shark if it were glidingstretched at the same speed and the same body angle. Concurrently, we define thrust as the sum of the respective component of thehydrodynamic force acting on the (actively swimming) shark and the straight-body drag.2Part of the lift is undoubtedly generated by the caudal fin [13–17]. This part depends on size and aspect ratio of the pectoral fins, theirlocation relative to the centre of mass, and on the location of the center of buoyancy relative to the center of mass. When these twocenters coincide, the lift of the caudal fin counterbalances the pitching moment generated by the shark’s body and pectoral fins. Basedon the pitching moment measured in the wind tunnel on a model shark that was morphologically similar to requiem sharks addressedhere, the lift of the caudal fin will probably not exceed 20% of the total lift generated by the entire shark (electronic supplementarymaterial, S2). An indirect verification that the lift is generated mainly by the pectoral fins can be found in [17]. Effective generation oflift by pectoral fins is stipulated by the Reynolds number being sufficiently high. How high it should be depends on the geometry ofthe fins, and it cannot be specified a priori. As a rough reference, we note that the relation between the pitching moment and lift of themodel shark remained unchanged when the Reynolds number (based on the total length) was decreased from 2.1 down to 0.8 × 106.The latter reflects a 1.5 m shark swimming at 0.5 m s−1.5rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................0.5 1.0 1.5 2.0 2.5 3.0 3.500.010.020.030.040.050.060.07bG. cuvierC. obscurusC. leucasN. brevirostrisC. plumbeusC. brevipinnaC. limbatusC. falciformisP. glaucal (m)Figure 2. Excess density parameterβ of 58 individuals from nine species of requiem sharks. Horizontal barsmark the uncertainty range.Data are based on [7,9,10]. Numerical values underlying this figure can be found in electronic supplementary material, S1, table S2a.drag coefficient depends mainly on the lift coefficient withCD =CD0 + KC2L, (3.6)where CD0 is the parasite (zero lift) drag coefficient, andK= kKπSb2(3.7)is the induced drag coefficient. Here, b is the span of the pectoral fins, kK is a numerical factor accountingfor increased flow separation from the surface of the fin with increasing angle of attack, for non-ellipticaldistribution of lift along the span and, to some extent, for the lift generated by other fins. CD0 dependson the geometry of the shark and (weakly) on the Reynolds number.3 When the reference area is chosenas the cross-section area of the body, typical value of CD0 for a 3 m shark swimming at 0.7 m s−1 is 0.17(figure 1b); typical value of kK is 1.5. The productSD = SCD, (3.8)is referred to as the ‘drag area’. It has the advantage of being independent of the choice of the referencearea. A particular case of (3.8) is SD0 = SCD0.Submerged weight of the shark, W, can be expressed in terms of the excess density parameter, βW =mgβ. (3.9)For most sharks, β varies between 0% and 6% (figure 2).In combination with (3.4), the balance of forces in the direction normal to the direction of swimming(3.3) can be used to define either the lift coefficientCL = 2W cos γρSv2, (3.10)needed to counteract weight at a given swimming speed, or the swimming speedv2 = 2W cos γρSCL, (3.11)needed to counteract weight at a given lift coefficient. Similarly, in combination with (3.5) and (3.10), thebalance of forces in the direction of swimming (3.2) can be used to define either the thrust needed to3Referring to equation (4) in electronic supplementary material, S1, logarithmic derivative of the drag coefficient with respect toReynolds number (Re) can be approximated by ReCD0∂CD0∂Re = − 2.58ln Re ≈ − 1.1logRe . With a typical Reynolds number (based on the bodylength) considered in this study being at least a few hundred thousand (see footnote 2), this estimate suggests that a 10% variationin speed or in characteristic length (each yielding 10% variation in the Reynolds number) should yield 2% variation in parasite dragcoefficient. The concurrent change in parasite drag itself (which changes with velocity and length squared) is estimated at 20%.6rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................sustain speed or the sustained speed for a given thrust. Note that when descending idle (T= 0),tan γ = −CDCLand v2 = 2WρS√C2L + C2D, (3.12)by (3.2) and (3.5).3.2. Active metabolic rateActive metabolic rate is defined here as the total amount of ATP used by the fish per unit timeP= P0 + Tvηηm. (3.13)It comprises the standard metabolic rate, P0, and the cost of activity, Tv/ηmη. ηm is the chemomechanicalefficiency of the muscles (the mechanical work done per mole ATP) and η is the hydrodynamicpropulsion efficiency. Both efficiencies are assumed independent of the shark’s morphology andswimming conditions; their typical values are 24 J per mmol ATP [18] and 0.7 [19], respectively.4,5,6 P0 isapproximated withP0 = kPmαe−kτ /τ , (3.14)where τ is the absolute body temperature, and kP, α and kτ are certain phenomenological parameters.Their typical values are 127 mol ATP per s·kgα, 0.8, and 5020°K, respectively [23], but there can beinterspecific differences in these parameters [24].In a glide, T= 0, and the active metabolic rate equals the standard metabolic rate, P0. In what follows,however, we assume that the shark swims at constant depth and speed; consequently T=D by (3.2), andP= P0 + ρS2ηηm(2WρS)3/2 ( CD0C3/2L+ KC1/2L)= P0 + ρS2ηηm(CD0v3 + Kv(2WρS)2)(3.15)by (3.6), (3.5), (3.10) and (3.11). Equation (3.15) can be rewritten asP= P0(1 + 12v3w3+ 12u4w3v), (3.16)whereu=(KCD0)1/4(2WρS)1/2=(kKπb2SD0)1/4(2mgβρ)1/2(3.17)andw=(ηηmP0ρSD0)1/3(3.18)are a pair of characteristic velocity scales; their physical meaning becomes clear in §3.3. The ratio u/w isa variable parameter but, in general, can be considered an order 1 quantity (figure 3).4Propulsion efficiency can be defined in quite a few ways, depending on the complementary definition of thrust [12]. In this study,we have defined thrust as the sum of the hydrodynamic force acting on the fish in the direction of swimming and the straight-bodydrag at the same speed (see footnote 1). In this way, any possible variations in friction between the body and the water are accountedfor by the propulsion efficiency. When swimming at high Reynolds numbers, these variations are expected to be small [12], and thepropulsion efficiency is expected to be practically the same as the ideal efficiency [20,21].5Requiem sharks can be characterized by reaching maximal depth (the distance between dorsal and ventral edges of the swimmer’soutline in the sagittal plane) at the caudal end. They are near-anguilliform swimmers, and propel themselves by lateral deformationwaves that propagate caudally faster than they swim. The major parameter that determines the (ideal) propulsion efficiency of this typeof anguilliform swimmers, is the ratio between the wave speed and the swimming speed [20,21]. Because this ratio can be adjusted bya particular shark, one can plausibly assume that the propulsion efficiency of all requiem sharks is similar, and practically independentof speed. In numerical examples appearing in this study, we have used 0.7, but the results remained essentially the same when it waschanged to 0.65 or 0.75.6Rendering v as the average speed along the course, and η as locomotion (rather than propulsion) efficiency [22], equation (3.15) andall its descendants, including (3.22), apply for variable depth and speed (but periodic) swimming.7rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................0 0.5 1.0 1.5 2.0 2.5 3.00.40.60.81.01.2l (m) u/w0 0.01 0.02 0.03 0.04 0.05b0.06 0.070.40.60.81.01.2G. cuvierC. obscurusC. leucasN. brevirostrisC. plumbeusC. brevipinnaC. limbatusC. falciformisP. glauca(a) (b)Figure 3. Estimated values of the velocities ratio u/w for the individual sharks from electronic supplementary material, S1, table S2b.The ratio is presented against the pre-caudal length (a) and against the relative excess density (b). The lowest point belongs to a pup ofC. plumbeus. Crosses mark the uncertainty range. Note that asβ increases, the buoyancy decreases.If there were no constraints, then the minimal active metabolic rate,P+ = P0(1 + 233/4u3w3), (3.19)would have been obtained atv+ = u31/4 , (3.20)the solution of the equation∂P∂v= 0. (3.21)For a neutrally buoyant fish (β = 0), v+ = u= 0 and P+ =P0. For a non-neutrally buoyant fish (β = 0),the minimal speed is typically limited by stall of the pectoral fins, and minimal active metabolic rate isobtained at the lowest swimming speed, vmin rather than at v+ (see section 3.6).3.3. Cost of transportThe cost of transport C is defined as the energy used per distance travelledC= Pv. (3.22)Substituting (3.16) it takes on the formC= P0w(wv+ 12u2w2(v2u2+ u2v2))= P0w(wv+ 12v2w2+ 12u4w2v2). (3.23)Minimal cost is obtained at the swimming speed, say v∗, at which∂C∂v= 0. (3.24)This condition leads to the equationw3v∗ − v4∗ + u4 = 0. (3.25)Its solution isv∗ =w(1 + 13u4w4+ · · ·)(3.26)when u→ 0, and v∗ = u(1 + (1/4)(w/u)3 + · · · ) when w→ 0. It lacks a closed-form analytical solutionfor v∗ in between these two extremes, but interpolating formulae in the last two rows of table 2 offervery good fits (figure 4). For future reference, we note that v∗ > max(u,w). This conjecture is apparentin figure 4a; it can be obtained formally by rearranging (3.25) as v∗ =w+ u4(v∗(v2∗ + v∗w+ w2))−1 or as8rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................(b)(a)0.5 1.0 1.5 2.000.51.01.52.00.60.811.41.82.30 0.5 1.0 1.5 2.01.52.02.53.03.54.04.55.0u/wu/wCw/P0v/wC+w/P0C*w/P0v*/wv+/w(c)u/w0 0.5 1.0 1.5 2.01.01.52.02.53.03.54.04.55.0P/P 0P*/P0 P+/P0Figure 4. Minimal cost of transport (b), minimal active metabolic rate (c) and the swimming speed at which they are achieved (a) asfunctions ofu/w. Exact solution ismarkedblue; approximations of the fourth andfifth rowsof table 2 aremarkeddashedblack anddashedred. Grey area marks the range between the minimal active metabolic rate and the minimal cost of transport. The slope of the straightdash-dotted lines in (a) is indicated to the right of each line. The range above steepest line iswhere having large fins is detrimental; belowit is where having large fins is incremental. Crosses mark the estimated minimal swim speed for the individual sharks from electronicsupplementary material, S1, table S2b.Table 2. Sustained performance parameters. In all expressions, an overbar denotes a reduced speed: u¯ = u/w,v¯min(γ )= vmin(γ )/w,v¯+ = v+/w and v¯∗ = v∗/w. Reference equations for the first row are (3.34), (3.20), (3.25), (3.19), (3.28) and (3.27), respectively.type v¯min(0) v¯+ P+/P0 v¯∗ P∗/P0 C∗w/P0solution ofexact u¯(CD0KC2L,max)1/4u¯31/41 + 2u¯333/4v¯∗ − v¯4∗ + u¯4 = 0 1 +v¯3∗2+ u¯42v¯∗1v¯∗+ v¯2∗2+ u¯42v¯2∗series, u¯ 1 1 + 13u¯4 + · · · 32+ u¯4 + · · · 32+ 12u¯4 + · · ·series, u¯ 1 u¯ + 14u¯2+ · · · u¯3 + 54+ · · · u¯2 + 1u¯+ · · ·best fit, u¯ ∈ (0,∞) 6 + 3u¯11/3 + 8u¯76 + 8u¯66 + 10u¯11/3 + 5u¯74 + 5u¯49 + 5u¯11/3 + 3u¯76 + 3u¯5best fit, u¯ ∈ (0, 1.3) 1 + 314u¯332+ 45u¯7/232+ 25u¯3v∗ = u+ w3v∗(v3∗ + v2∗w+ v∗w2 + w3)−1. Thus, w and u are the speeds that minimize the cost of transportin the limits when the buoyancy (and hence the energetic cost of generating hydrodynamic lift) is verysmall and when the basic metabolic rate (the energetic cost of living) is very small, respectively. Typicalvalues of v∗ can be found in electronic supplementary material, S1, table S2b.9rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................The minimal cost of transport and the respective metabolic rate areC∗ = P0w(wv∗+ 12v2∗w2+ 12u4w2v2∗)(3.27)andP∗ =C∗v∗ (3.28)by (3.22) and (3.23). Because v∗ lacks a closed-form expression relating it with u and w, so does C∗ andP∗. They can be approximated withC∗ = 32P0w(1 + 13u4w4+ · · ·)(3.29)andP∗ = 32P0(1 + 23u4w4+ · · ·)(3.30)when say, u/w< 0.6; more elaborate approximations can be found in table 2.The terms in the parentheses on the right-hand side of (3.26), (3.29) and (3.30) manifest the differencebetween negatively and neutrally buoyant fishes (for which u= 0). Negatively buoyant fish have to swimfaster than similarly shaped neutrally buoyant ones, and their cost of transport and active metabolicrate is higher. In fact, estimated optimal swimming speeds of C. leucas, C. limbatus, C. brevipinna andN. breviostris are up to 30% higher than what they would have been if these sharks were neutrallybuoyant; respective costs of transport are up to 40% higher (see electronic supplementary material, S1,table S2b).3.4. The speed ratioChoosing w as a unit of speed, and the basic metabolic rate P0 as a unit of power, all reduced performancecharacteristics—the minimal active metabolic rate P+/P0, the minimal cost of transport C∗w/P0 andthe swimming speeds, v+/w and v∗/w, at which they are obtained—become dependent on a singleparameter, u¯= u/w. All four increase with u¯ and hence, in many cases, energy expenditure of a sharkcan be reduced by making u¯ small.Expression for u¯,uw=(4kKπ)1/4(mgβb)1/2 1(ηηmP0)1/3(SD0ρ2)1/12=(4kKπ)1/4(gβb)1/2 m1/2−α/3ekτ /3τ(ηηmkP)1/3(SD0ρ2)1/12, (3.31)follows by (3.17), (3.18) and (3.14). Because m∝ l3, SD0 ∝ l2 (see footnote 2) and b∝ l, we may expectu¯∝ (βl/b)1/2l7/6−αekτ /3τ or, what is equivalent, u¯∝ (βl/b)1/2m7/18−α/3ekτ /3τ ; the powers with l and m arepositive (0.36 and 0.12, respectively). Typical values of u/w can be found in figure 3; they do not exceed1.1 for all individuals on our list, and do not exceed 0.8 for the two pelagic (P. glauca and C. falciformis)and the two ‘cosmopolitan’ (C. obscurus and G. cuvier) species included thereat. u¯ can be reduced mainlyby decreasing the negative buoyancy β, by increasing the span of the pectoral fins b/l, and by decreasingthe mass. It can also be reduced by increasing the body temperature, but the resulting increase in thestandard metabolic rate more than offsets the beneficial effect of reducing the value of u¯ .3.5. Energy balanceIf prey is uniformly distributed along the swimming path, and the energy intake of the shark is directlyproportional to the amount of prey encountered en route, the energy balance of a shark—the differencebetween energy gained Ein and the energy spent Eout—can be expressed (with help of (3.22)) asE= Ein − Eout =∫T0(ev − P) dt=∫T0(e− C)v dt=∫X0(e− C) dx, (3.32)where X is the distance swum at speed v, T is the swimming duration and e is a certain coefficientreflecting the prey density and the probability of its capture. Minimizing the cost of transport, C,maximizes the energy gain, irrespective of e [25].10rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................If, however, the amount of food encountered by the shark is independent of the volume of watersearched during swimming, but depends only on time, then the energy balance becomesE= Ein − Eout =∫T0(e′ − P) dt, (3.33)where e′ reflects the prey encounter rate and the probability of its capture. Minimizing the activemetabolic rate, P, maximizes the energy gain, irrespective of e′.Realistic scenarios are bounded between these two extremes, suggesting that a shark probably swimsbetween v+, the speed at which its active metabolic rate is minimal, and v∗, the speed at which its costof transport is minimal (this conjecture is assessed in §3.7); its active metabolic rate varies between P+and P∗ (figure 4). The prerequisite to this analysis is that v∗ and v+ exceed a certain minimal swimmingspeed.3.6. Minimal swimming speedFrom a hydrodynamic perspective, the minimal swimming speed is the lowest speed at which the forcesacting on the shark can be balanced. It is an immediate consequence of the existence of the upper boundCL,max on the lift coefficient (figure 1a); in factvmin(γ ) =(2W cos γρSCL,max)1/2= u(CD0KC2L,max)1/4(cos γ )1/2, (3.34)at powered ascent or descent, andvmin(−γ0) = u(CD0KC2L,max)1/4( C2L,maxC2L,max + C2D,max)1/4, (3.35)when gliding at angle γ0 = tan−1(CD,max/CL,max) with zero thrust; CD,max is the drag coefficient at CL =CL,max. Equation (3.34) follows from (3.11) and (3.17); equation (3.35) follows from (3.12). Because C2D,maxis invariably small relative to C2L,max (figure 1), the minimal glide speed vmin(−γ0) and the minimal speedat constant depth vmin(0) are hardly different. At the same time, the minimal speed in vertical ascentvmin(π/2) is identically zero. Typical values of (CD0/KC2L,max)1/4 range between 1 and 1.4 for all practicalcombinations of morphological parameters.7To exploit the minimal active metabolic rate when swimming at constant depth, v+ should exceedvmin(0). It implies (3CD0KC2L,max)1/4< 1, (3.36)by (3.34) and (3.20). This condition cannot be satisfied with any admissible set of morphologicalparameters (see the preceding paragraph), and no shark on our list can exploit the minimal activemetabolic rate when swimming at constant depth (figure 4). Consequently, the lowest active metabolicrate when swimming at constant depth, minCL≤CL,maxP, is achieved at the minimal swimming speed, vmin(0).Given that the difference between vmin(0) and v+ is small, the difference between minCL≤CL,maxP and P+,minCL≤CL,maxP− P+P0= 33/4 uw(vmin(0) − v+w)2+ . . . (3.37)is also small (figure 4c). Nonetheless, because minCL≤CL,maxP> P+, minimizing the active metabolic rate wasnot the evolutionary objective with any of these species.7These combinations included body lengths between 0.5 and 3 m, body diameter to length ratio between 0.1 and 0.22, pectoral finsspan to body length ratio between 0.3 and 0.8, and pectoral fins chord to body length between 0.07 and 0.13. Other parameters can befound in electronic supplementary material, S1.11rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................To exploit the minimal cost of transport when swimming at constant depth, v∗ should exceed vmin(0).It impliesv4min(0) − w3vmin(0) < u4, (3.38)by (3.25), which after some rearrangement, can be recast asv4min(0)u4− 1 < w3u3vmin(0)u. (3.39)At the same timev4min(0)u4= CD0KC2L,max, (3.40)by (3.34); whence (3.39) sets un upper bound on the speed ratiouw<(CD0KC2L,max)1/12(CD0KC2L,max− 1)−1/3. (3.41)For the same combinations of morphological parameters as those listed in footnote 7, the right-handside of (3.41) ranges between 0.8 and 2. The left-hand side varies with buoyancy and body temperature,as well as with basic morphological parameters (see above), and is, in general, an order 1 quantity.Consequently, (3.41) is not automatically satisfied, and buoyancy and body temperature have to becoordinated with morphological parameters to allow a shark to exploit its minimal cost of transport.In particular, because u/w∝ (βl/b)1/2l7/6−αekτ /3τ (see the paragraph following (3.31)), inequality (3.41)implies that large sharks (large l) and/or ectothermic sharks residing in cold water (small τ ) mustalso have small negative buoyancy (small β) and/or large pectoral fins. Examples include the baskingshark Cetorhinus maximus [9], the Portuguese dogfish Centroscymnus coelolepis [26], and the six-gill sharkHexanchus griseus [27].8 Some sharks exhibit ontogenetic increase in hepatosomatic index (proportionalmass of the liver), which is inversely correlated with the value of β. Examples include the oceanicwhitetip shark C. longimanus [6], the dusky shark C. obscurus [28] and the tiger shark G. cuvier [7]. Somesharks exhibit an ontogenetic increase in the span of the pectoral fins; examples include the bull shark C.leucas and dusky shark [5].3.7. Optimal swimming speedIt was predicted in §§3.5 and 3.6 that under most circumstances, the optimal swimming speed of theshark is bounded between the larger of v+ and vmin(0), and v∗. Reliable corroboration of this conjectureis complicated by the fact that average speed measurements are commonly cited without the necessarycomplementary data, which includes length, mass (or girth), temperature, span of the pectoral fins andbuoyancy. Moreover, many of these measurements were made immediately after having released theshark, and hence may not reflect its natural behaviour [29]. Notwithstanding these caveats, reference[30] cites voluntary swimming speeds of two bull sharks and one sandbar shark C. plumbeus in a largewater tank. The bull sharks were 2 and 2.3 m long, the sandbar shark was 2.1 m long (total length).They swam in 26°C water with average speeds of 0.72, 0.62 and 0.64 m s−1, respectively, accelerating anddecelerating a few hundredth m s−1 about these values. Referring to electronic supplementary material,S1, tables S2a and S2b, v∗ for comparably sized bull sharks (sharks 5,8,13 and 15) is between 0.67 and0.78 m s−1, depending on buoyancy and morphology; v∗ for comparably sized sandbar sharks (sharks1–4,6 and 7) is between 0.62 and 0.74 m s−1. For both species, v+ is roughly 0.28 m s−1 smaller than v∗.Reference [31] cites average swimming speeds of three blue sharks P. glauca, tracked over the periodof a few days (sharks 16, 22 and 23). With body temperatures of about 18°C, the three sharks, measuring2.2, 2.7 and 2.6 m (fork length) averaged 0.48, 0.4 and 0.44 m s−1. There are no comparably sized sharkson our list, but the optimal speeds can be estimated based on the same formulae that underlay table S2in electronic supplementary material, S1. With β = 0.02, and depending on the length of the pectoral finsand body mass, they yield v∗ between 0.55 and 0.6 m s−1 for the two larger sharks, and between 0.52 and0.56 for the smaller one; v+ is 0.27 m s−1 smaller than v∗.For the two bull sharks and the three blue sharks, we predict vmin(0) between 0.13 and a fewhundredth m s−1 smaller than the respective v∗, whereas for the sandbar shark, we predict it is between8In this reference, the authors have unequivocally demonstrated that a six-gill shark may have positive, rather than negative, buoyancy,but made no estimate of the buoyancy itself. It can be estimated by several methods. The simplest, perhaps, is by estimating themechanical power needed to overcome drag (1/2)ρv3SCD and equating it with the rate of change of potential energy in unpoweredascent, −βmgva (va being the rate of ascent). It turns out to be a negligible few hundredths percent.12rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................xv∗∂v∗∂x= 14v4∗ − w3v∗(4u4xu∂u∂x+ 3w3v∗ xw∂w∂x)(4.1)andxC∗∂C∗∂x= v∗C∗∂C∗∂v∗xv∗∂v∗∂x+ P0C∗∂C∗∂P0xP0∂P0∂x+ uC∗∂C∗∂uxu∂u∂x+ wC∗∂C∗∂wxw∂w∂x, (4.2)and, given v,xP∂P∂x= xC∂C∂x= P0P∂P∂P0xP0∂P0∂x+ uP∂P∂uxu∂u∂x+ wP∂P∂wxw∂w∂x. (4.3)They follow from (3.25), (3.22) and (3.16). Being inherently dimensionless, logarithmic derivatives offerboth simplicity of the final expressions and a convenient interpretation of the result. For example,the relative change in the cost of transport, C∗/C∗, owing to a (small) relative change x/x in anindependent parameter, isC∗C∗=(xC∗∂C∗∂x)xx, (4.4)The logarithmic derivative (x/C∗)(∂C∗/∂x) serves as an amplification factor between x/x and C∗/C∗;with (x/C∗)(∂C∗/∂x) = 0.1, say, a 10% change in x yields 1% change in C∗.The first term in (4.2) vanishes by (3.24). Substituting (3.23) for C∗ in (4.3), and (3.16) for P in (4.3)yieldsxC∗∂C∗∂x= xP0∂P0∂x+ 12w3v∗ + v4∗ + u4(4u4xu∂u∂x− 3(v4∗ + u4)xw∂w∂x)(4.5)andxP∂P∂x= xC∂C∂x= xP0∂P0∂x+ 12w3v + v4 + u4(4u4xu∂u∂x− 3(v4 + u4) xw∂w∂x). (4.6)Note that because ∂C∗/∂v∗ = 0 by (3.24),xC∗∂C∗∂x=(xC∂C∂x)v=v∗; (4.7)this equivalence is manifested in (4.5) and (4.6).0.2 and 0.08 m s−1 smaller. In other words, there are possible combinations of morphological parameters and buoyancy for which vmin(0) exceeds the observed swimming speed. vmin(0) is extremely sensitive to buoyancy (it vanishes with β), and hence obtaining unrealistic vmin(0) demonstrates the importance of having the ideal dataset mentioned in §2, as well as the importance of coordination between morphological parameters and buoyancy.4. Derivatives4.1. PreliminariesSustained performance of a shark is characterized mainly by the active metabolic rate, the cost of transport and the speed at which the minimal cost of transport is achieved. Essentially, there are six major morphological parameters affecting the sustained performance: length, l; span and chord of the pectoral fins, b and c0; body diameter, d; buoyancy, β and body temperature, τ . The first three can be considered an evolutionary adaptation; the next two also depend on an individual’s body condition; the last two also depend on the habitat the animal uses. Sensitivity of the sustained performance to variations in these parameters is manifested in the partial derivatives computed below.Quite generally, if x denotes one of the independent parameters, namely β, b, c0, d, l and τ , we can write a series of logarithmic derivatives13rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................Table 3. Sensitivity derivatives. In all expressions, an overbar denotes a reduced quantity; in particular, u¯ = u/w, v¯ = v/w and v¯∗ =v∗/w. v¯c and v¯2 have been defined in (4.14) and (4.21); S¯(b)D0 has been defined in (4.28). Expressions in the second and the fourth columnshave been modified with the help of (3.25). If increasing the diameter does not increase the basic metabolic rate, α in the fifth rowshould be set to zero. Expressions in the third and fifth columns are approximations of the respective expressions to their lefts under theassumption that v¯4c 1 and u3 w3.xv∗∂v∗∂xxC∗∂C∗∂xxC∂C∂xorxP∂P∂xx exact approximation exact approximation exactβ2u¯43v¯∗ + 4u¯423u¯42u¯43v¯∗ + 2u¯423u¯42u¯42v¯ + v¯4 + u¯4τkττv¯∗3v¯∗ + 4u¯4kτ3τkττ2v¯∗3v¯∗ + 2u¯42kτ3τkττ2v¯2v¯ + v¯4 + u¯4b − 2v¯4cv¯∗ + u¯4(v¯4c + 1)3v¯∗ + 4u¯4 −231 + u¯4v¯4cv¯4c2v¯4cv¯4∗ − u¯4v¯4c3v¯∗ + 2u¯4231 − u¯4v¯4cv¯4c2v¯4cv¯4 − u¯4v¯4c2v¯ + v¯4 + u¯4c0 − 2v¯42v¯∗ + u¯43v¯∗ + 4u¯4 −23v¯422v¯42v¯∗ + u¯43v¯∗ + 2u¯423v¯422v¯42v¯42v¯ + v¯4 + u¯4d(2α − S¯(b)D0 )v¯∗3v¯∗ + 4u¯4 +(4− S¯(b)D0 )u¯43v¯∗ + 4u¯42α − S¯(b)D03(4α + S¯(b)D0 )v¯∗3v¯∗ + 2u¯4 +(4+ S¯(b)D0 )u¯43v¯∗ + 2u¯44α + S¯(b)D034αv¯ + 4u¯4 + v¯4S¯(b)D02v¯ + v¯4 + u¯4l(3α − 2)v¯∗ + 2u¯43v¯∗ + 4u¯4 α −23(6α + 2)v¯∗ + 6u¯43v¯∗ + 2u¯4 2α +236αv¯ + 4u¯4 + 2v¯42v¯ + v¯4 + u¯4Essentially, there are three primitive derivatives that are required to find all the others: xP0∂P0∂x ,xu∂u∂xand xw∂w∂x . The last two,xu∂u∂x= 12xm∂m∂x+ 12xβ∂β∂x− 12xb∂b∂x− 14xSD0∂SD0∂x(4.8)andxw∂w∂x= 13xP0∂P0∂x− 13xSD0∂SD0∂x, (4.9)follow from (3.17) and (3.18); we assume that the remaining parameters, ρ, kK, η and ηm are, essentially,constants. In turn, P0 is commonly considered a function of mass and body temperature τ only, andhencexP0∂P0∂x= 2α xm∂m∂x+ kττxτ∂τ∂x. (4.10)Explicit expressions for all pertinent derivatives are summarized in table 3; the underlying derivationsand a few comments follow in §§4.2–4.8.4.2. BuoyancyFirst, consider the effect of negative buoyancy, β. Because SD0, b, m and P0 are independent of ββu∂u∂β= 12(4.11)andβw∂w∂β= βP0∂P0∂β= 0 (4.12)by (4.8), (4.9) and (4.10). The particular derivatives in the first row of table 3 follow these by (4.1),(4.5) and (4.6). All these derivatives are non-negative (figure 5a,e); that is, negative buoyancy increasesthe active metabolic rate, the cost of transport and the speeds at which the minimal values of theseparameters are obtained. At the same time, all the derivatives diminish with the ratio u/w. Indeed,when this ratio becomes smaller than, say 0.7, the cost of transport and the speed at which it is obtainedbecome insensitive to changes in buoyancy (figure 5a). As mentioned already, the ratio u/w can be made on January 4, 2018http://rsos.royalsocietypublishing.org/Downloaded from 14rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................0.5 1.0 1.5 2.000.20.40.60.81.0u/w0 0.5 1.0 1.5 2.0−1.0−0.8−0.6−0.4−0.200.21.21.41.82.310u/w0.5 1.0 1.5 2.0u/w0 0.5 1.0 1.5 2.0u/w0.5 1.0 1.5 2.0u/w0 0.5 1.0 1.5 2.0u/w024681012−2−10120123456−1.6−1.2−0.8−0.400.40.8 dblbC *∂C*∂bbC *∂C*∂ttu *∂u*∂ttC *∂C*∂bbC *∂C*∂ddu *∂u*∂xx(a) (b)(c) (d)(e) ( f )Figure 5. Relative changes in the minimal cost of transport with negative buoyancy (a), pectorals span (b), body temperature (c) anddiameter (d). Relative changes in the minimal cost of transport speed with body temperature (e), pectoral fin span (grey range in (f )),negative buoyancy (dashed-dotted line in (f )), diameter (broken and dotted lines in (f )) and length (solid line in (f )). Dotted lines in (d)and (f ) reflect the case when increase in diameter does not yield an increase in the basic metabolic rate; light grey ranges in the samefigures mark the composite case where the increase in diameter also increases buoyancy. The borders of these ranges are Bl/β = 0.1(bottom) and Bl/β = 0.5 (top). The borders of the grey ranges in (c) and (e) are τ = 16°C (top) and τ = 28°C (bottom). Relevant rangeof u/w values for requiem sharks is indicated by dark grey horizontal bars in (e) and (f ). Crossesmark the individual sharks from electronicsupplementary material, S1, tables S2c and S2d.small by increasing the buoyancy or/and the body temperature, and, to a lesser extent, the span of thepectoral fins.4.3. Span of the pectoral finsWe assume that an infinitesimal increase b in the span of the pectoral fins is accompanied by an increaseSp = c0b in their projected area, c0 being the root (proximal, base) chord of the fins. These changesaffect induced and parasite drag coefficients alike. The change in the induced drag coefficient is implicitlyincluded in u; the change in the parasite drag coefficient is modelled by setting SD0 = 2kf CfSp, whereCf is the effective friction coefficient, kf is the form factor,9 and ‘2’ comes from the fact that surface area of9This is an empirical correction accounting for fin/body interference; its typical value is 1.4. In electronic supplementary material, S1,it appears as Fn.15rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................the fins is twice their projected area. Thus,bSD0∂SD0∂b= 2v¯4c, (4.13)wherev¯c =(SD0kf c0bCf)1/4(4.14)is a dimensionless parameter that is interpreted below. For a typical requiem shark, v¯c varies in a narrowrange between 1.8 and 2.2 (see electronic supplementary material, S1, table S2b). Thus,bP0∂P0∂b= 0, (4.15)bu∂u∂b= −12(1 + 1v¯4c)(4.16)andbw∂w∂b= − 23v¯4c(4.17)by (4.8)–(4.10) and (4.13). The partial derivatives in the third row of table 3 follow these by (4.1), (4.5) and(4.6). Referring to the last column in this row, increasing the span of the pectoral fins decreases the activemetabolic rate and the cost of transport only ifv < uv¯c. (4.18)Large fins are detrimental for high-speed swimming. For a neutrally buoyant fish, for which u= 0,increasing the span of the pectoral fins increases the active metabolic rate at any speed other than zero.Increasing the span of the pectoral fins can reduce the minimal cost of transport only ifv∗ < uv¯c. (4.19)Condition (4.19) can be elucidated in figure 4a. The straight lines in figure 4 can be envisioned as thelines representing v/w= v¯cu/w for different values of v¯c. With v¯c > 1 (see the paragraph immediatelyfollowing (4.14)), there is an intersection between the respective line and the line v∗/w. The region tothe right of the intersection point is the region where increasing the span of the fins reduces the cost oftransport, the region to the left of it is where it increases it. In general, however, if u/w is small (say,smaller than 0.7), all performance parameters become insensitive to the span of the pectoral fins.4.4. Area of the pectoral finsWe assume an infinitesimal increase c0 of the chord length of the pectoral fins that leaves their spanunchanged. It yields an increase SD0 = 4kf Cf sc0 in the drag area of the shark, s being the length (distalmargin) of a single fin and the multiplier 4 comes from having two fins and each fin having two sides.Thus,c0SD0∂SD0∂c0= 2v¯42, (4.20)wherev¯2 =(SD02kf c0sCf)1/4= v¯c(b2s)1/4. (4.21)Because b≈ 2s+ d, v¯2 > v¯c; and hence v¯42 is a large quantity indeed. Thus,c0P0∂P0∂c0= 0, (4.22)c0u∂u∂c0= −14c0SD0∂SD0∂c0= − 12v¯42(4.23)andc0w∂w∂c0= −13c0SD0∂SD0∂c0= − 23v¯42(4.24)by (4.8)–(4.10) and (4.13). The partial derivatives listed in the fourth row of table 3 follow these threeby (4.1), (4.5) and (4.6). All derivatives are of the order of 1/v¯42 and hence small. Thus, although minimalactive metabolic rate and minimal cost of transport increase with increasing chord of the fins, this increase16rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................τu∂u∂τ= 0 (4.25)andτw∂w∂τ= kτ3τ. (4.26)The partial derivatives listed in the second row of table 3 follow these two by (4.1), (4.5) and (4.6).All derivatives are large and positive, suggesting that lowered body temperatures decrease the activemetabolic rate, the cost of transport and the speeds at which the minimal values of these parametersare obtained. For small values of u/w, a 3°C decrease in the body temperature yields a 12% decrease inthe minimal cost of transport (and the active metabolic rate) and a 6% decrease in speed at which theminimal cost of transport is obtained (figure 5c,e).4.6. DiameterIn this section, we assess the effect of increasing body diameter on the minimal cost of transport in twoprimitive cases: when the basic metabolic rate increases with mass, and when it remains invariant of it(a composite case is addressed in §4.7). In both cases, it is explicitly assumed that the change in diameteryields no change in buoyancy and no change in span of the pectoral fins, but it does change the drag areaand the mass. The change in mass,dm∂m∂d= 2, (4.27)follows from (3.1); the change in the drag area isdSD0∂SD0∂d= S(b)D0SD0, (4.28)where S(b)D0 is the drag area of the body. Tacit assumptions underlying (4.28) are that S(b)D0 is linear in d,and that the body diameter has no effect on the drag area of the fins. The ratio S¯(b)D0 = S(b)D0/SD0 on theright-rand side of (4.28) is of the order of 2/3.In the first case—where an increase in diameter yields an increase in the basic metabolic ratedP0∂P0∂d= 2α (4.29)by (4.27) and (3.14), whereasdu∂u∂d= 1 − 14S¯(b)D0 (4.30)anddw∂w∂d= 13(2α − S¯(b)D0) (4.31)by (4.8), (4.9), (4.27), (4.29) and (4.28). The partial derivatives listed in the fifth row of table 3 follow thesetwo by (4.1), (4.5) and (4.6). All derivatives are positive—increasing body diameter increases the activemetabolic rate, the cost of transport and the speeds at which the minimal values of these parameters areobtained (figure 5d,f ). The second case—where an increase in body diameter does not increase the basicmetabolic rate—is obtained from the first by setting α = 0; this case is exploited below.4.7. Body conditioningBased on the previous results, we can address now the composite case where an increase in the bodydiameter is a consequence of storing low-density lipids, and therefore is accompanied by an increase in10Decreasing water temperature increases viscosity. A change of 3°C reduces the associated Reynolds number by 7% and henceincreases the drag coefficient by 1.5% (see footnote 3). We render this effect small when compared with the effect of decreasingtemperature on the basic metabolic rate.is ignorable. That being said, increasing the chord decreases the minimal swim speed (see (3.34)), and hence may reduce the minimal active metabolic rate.4.5. TemperatureAgain, to start with, a few primitive derivatives are needed. It is assumed that the change in temperature affects only the basic metabolic rate and does not affect any other parameter. Specifically, it does not affect the parasite drag coefficient.10 Thus, using (3.14),17rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................0 0.5 1.0 1.5 2.0 2.5 3.0−2.4−2.0−1.6−1.2−0.8−0.40l (m)G. cuvierC. obscurusC. leucasN. brevirostrisC. plumbeusC. brevipinnaC. limbatusC. falciformisP. glaucaddC*ddC *Figure 6. Relative change in the minimal cost of transport with body diameter under the assumption that the addition to the diametercomes from stored lipids. Crosses mark the uncertainty range; relevant numerical values can be found in electronic supplementarymaterial, S1, table S2c. The highest point belongs to a pup of C. plumbeus. Assumptions: S¯(b)D0 ≈ 2/3, Bl ∈ (0.1, 0.12).buoyancy. Adding a small volume V of oil of density ρl = (1 − Bl)ρ to a body of volume V and densityρb = (1 + β)ρ, yields a changedd= V2V, (4.32)in the body diameter, and a changeββ= −VV(1 + Blβ), (4.33)in the relative excess density.The combined effect of diameter and buoyancy on the minimal cost of transport and the speed atwhich it is obtained areC∗C∗=(dC∗dC∗dd)dd(4.34)andv∗v∗=(dv∗dv∗dd)dd, (4.35)wheredC∗dC∗dd= dC∗∂C∗∂d+ βC∗∂C∗∂βdβ∂β∂d(4.36)anddv∗dv∗dd= dv∗∂v∗∂d+ βv∗∂v∗∂βdβ∂β∂d. (4.37)The derivatives dC∗∂C∗∂d ,βC∗∂C∗∂β, dv∗∂v∗∂d andβv∗∂v∗∂βare found in table 3. Stored lipids do not require energy tobe maintained, and therefore, the increase in diameter does not yield an increase in the basic metabolicrate. Consequently, the first derivative has to be accessed with α = 0 (see above). The last derivative onthe right-hand side of (4.36) and (4.37),dβ∂β∂d= −2β + Blβ, (4.38)follows from (4.33) and (4.32). The derivatives dC∗dC∗dd anddv∗dv∗dd are shown in figures 5d,f and 6. In general,dC∗dC∗dd is negative, manifesting the energetic advantage of becoming fatter—the advantage that increasesas the animal becomes larger (figure 6). The exception is the pup of C. plumbeus, for which this derivativepractically vanishes. In spite of the diminutive size of the pup, which actually makes it too small to beadequately described by the present theory, this result seems to accord observations of Hussey et al. [28]:having the cost of transport insensitive to conditioning, pups can turn stored lipids into growth withno energetic penalty. In general, dv∗dv∗dd is negative and large, manifesting that the optimal swim speed ishighly sensitive to conditioning, with fat individuals swimming considerably slower than their skinny,but otherwise similar, counterparts.18rsos.royalsocietypublishing.orgR.Soc.opensci.3:160406................................................lm∂m∂l= 3, lSD0∂SD0∂l= 2, lb∂b∂l= 1, (4.39)and, consequently,lP0∂P0∂l= 3α, (4.40)lu∂u∂l= 12(4.41)andlw∂w∂l= α − 23(4.42)by (4.8)–(4.10). The partial derivatives listed in the last row of table 3 follow these three by (4.1), (4.5) and(4.6). All derivatives are positive—increased body length increases the active metabolic rate, the cost oftransport and the speeds at which the minimal values of these parameters are obtained. The derivativesincrease with increasing value of u/w. For small values of this parameter, a 10% increase in length yields25% increase in the minimal cost of transport, but less than 3% increase in speed at which the minimalcost is obtained.5. Pelagic sharksAmong the six major parameters affecting performance of a shark, probably the most intriguing is thespan of the pectoral fins. While length, diameter, temperature and negative buoyancy unconditionallyincrease the cost of transport, and pectoral fins chord hardly affects it, pectoral fins span either increasesor decreases it, depending on the ratio of u/w. If minimizing the cost of transport was the only selectivepressure, and if (4.19) would have been unconditionally satisfied, then we would have seen sharksincreasing the span of their fins over the generations. This is not the case, and there is large variability inrelative size of the pectoral fins within the same species [5].A partial explanation can be from the hypothesis that selection favours maximizing the differencebetween energy gained and energy spent, as opposed to just minimizing the energy spent. For reefsharks, which hunt on reefs, large pectoral fins are detrimental to prey capture as the animals haveto manoeuvre within the complex three-dimensional structure of the reef itself. At the same time, noessential constraints are placed on the size of pectoral fins of pelagic sharks. In those species, in orderto avoid one-directional evolutionary change in the span of the pectoral fins, the cost of transport mustbe insensitive to it. To this end, the morphology of the shark and its buoyancy should be coordinated insuch a way that ∣∣∣∣ bC∗∂C∗∂b∣∣∣∣< ε, (5.1)where ε is a certain small number that is specified later. For example, ε = 0.1 implies that a 10% changein the pectorals span yields less than 1% change in the minimal cost of transport. Using table 3 andexploiting (3.25), (5.1) yields ∣∣∣∣∣ 2v¯4cw3v∗ − u4(v¯4c − 1)3w3v∗ + 2u4∣∣∣∣∣< ε. (5.2)Subject to an a posteriori verification, this condition is satisfied when u/w is small (figure 5b).Consequently, it can be recast as1w4 + u4∣∣∣∣w4 − u4(v¯4c −43)∣∣∣∣< 32εv¯4c (5.3)by (3.26), or, explicitly, as(1 − 32εv¯4c)(v¯4c(1 + 32ε)− 43)−1

How Sharks Float

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How Sharks Float

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