geometric analysis of suction feeding
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Geometric Analysis of Suction Feeding. Math & Nature. The universe is written in the language of mathematics Galileo Galilei , 1623 Quantitative analysis of natural phenomena is at the heart of scientific inquiry Nature provides a tangible context for mathematics instruction. - PowerPoint PPT PresentationTRANSCRIPT
Geometric Analysis of Suction Feeding
The universe is written in the language of mathematics◦ Galileo Galilei, 1623
Quantitative analysis of natural phenomena is at the heart of scientific inquiry
Nature provides a tangible context for mathematics instruction
Math & Nature
Context
1. The part of a text or statement that surrounds a particular word or passage and determines its meaning.
2. The circumstances in which an event occurs; a setting.
The Importance of Context
Context-Specific Learning
◦ Facilitates experiential and associative learning
Demonstration, activation, application, task-centered, and integration principles (Merrill 2002)
◦ Facilitates generalization of principles to other contexts
The Importance of Context
Geometry & Biology◦ Biological structures vary greatly in geometry and
therefore represent a platform for geometric education
◦ Geometric variability functional variability ecological variability Mechanism for illustrating the consequences of
geometry
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Vertebrate skulls vary greatly in form & function
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www.digimorph.org
Vertebrate skulls vary greatly in form & function
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csi.whoi.edu www.digimorph.org
22 bones1 moving part
~50 bones~7 moving parts
Vertebrate skulls vary greatly in form & function
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Liem et al. (2001)
Vertebrate skulls vary greatly in form & function◦ Moveable parts of the fish skull are responsible for
the diversity of feeding mechanisms in fish Jaw protrusion in the sand tiger shark Carcharias
taurus
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D. Huber
Vertebrate skulls vary greatly in form & function◦ Moveable parts of the fish skull are responsible for
the diversity of feeding mechanisms in fish Jaw protrusion in the sling-jaw wrasse Epibulus
insidiator
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P. Wainwright
Fish feeding mechanisms◦ Filter ◦ Biting◦ Ram ◦ Suction
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www.true-wildlife.blogspot.com
C. Fallows
www.z00n.net
Fish feeding mechanisms◦ Filter feeding
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W. Mischler 2013
Motta et al. (2010)
Fish feeding mechanisms◦ Filter feeding
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Motta et al. (2010)
Fish feeding mechanisms◦ Filter feeding
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P. Motta
Fish feeding mechanisms◦ Biting
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Fish feeding mechanisms◦ Biting
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www.digimorph.org
Fish feeding mechanisms◦ Ram feeding
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D. Huber C. Fallows
Fish feeding mechanisms◦ Ram feeding
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S. Huskey www.tennesseeaquarium.com
Fish feeding mechanisms◦ Ram feeding
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D. Huber
Fish feeding mechanisms◦ Suction feeding
Most common fish feeding mechanism Water cohesion Suction performance
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D. Huber
Fish feeding mechanisms◦ Suction feeding
http://www.youtube.com/user/Wainwrightlab
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Wainwright et al (2006)
Fish feeding mechanisms◦ Suction feeding
http://www.youtube.com/user/Wainwrightlab
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Svanback et al (2002)
Fish feeding mechanisms◦ Suction feeding
http://www.youtube.com/user/Wainwrightlab
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Grubich (2001)
Fish feeding mechanisms◦ Suction feeding
Anterior posterior expansion
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Gibb & Ferry-Graham (2005)Wainwright et al (2006)
Fish feeding mechanisms◦ Suction feeding
Fluid flow
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Holzman et al (2008)
Fish feeding mechanisms◦ Suction feeding
Fluid pressure and movement speed
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Svanback et al (2002)
Fish feeding mechanisms◦ Suction feeding
Feeding ecology
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Motta et al (2008)
Fish feeding mechanisms◦ Suction feeding
Geometric modeling
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Van Wassenbergh et al (2007)Bishop et al (2008)
Fish feeding mechanisms◦ Suction feeding
Goliath grouper Epinephelus itajara
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Fish feeding mechanisms◦ Suction feeding
Goliath grouper Epinephelus itajara Questions
What fluid velocity can the goliath grouper generate during suction feeding?
How does suction feeding by the goliath grouper compare to other fish?
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Geometry & Biology◦ CCSS
MACC.912.G-GMD.1.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
MACC.912.G-GMD.2.4: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
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Geometry & Biology◦ CCSS
MACC.912.G-MG.1.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
MACC.K12.MP.1.1: Make sense of problems and persevere in solving them.
MACC.K12.MP.4.1: Model with mathematics
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Goliath grouper model◦ Objective
Determine the velocity of water flow into the mouth
◦ Procedure Determine the volume of components A and B at rest
(t0) and at maximum expansion (t1) t0 = time at rest
t1 = time at maximum expansion
Determine the volume change during feeding
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BA
BA
Goliath grouper model◦ Objective
Determine the velocity of water flow into the mouth
◦ Procedure Determine the area of the mouth at maximum
expansion (t1) t1 = time at maximum expansion
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BA
BA
Goliath grouper model◦ Objective
Determine the velocity of water flow into the mouth
◦ Procedure
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BA
BA
Suction feeding in the goliath grouper◦ Given
Dimensions of cones A and B at rest (t0)
1) Find the volume of the goliath grouper feeding mechanism at rest (t0).
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b
a
c
d
e
a
Suction feeding in the goliath grouper◦ Given
Dimensions of cones A and B at rest (t0)
1) Find the volume of the goliath grouper feeding mechanism at rest (t0).
b
a
c
d
e
a
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Goliath Grouper Suction Feeding
Time 0
Cone A Length (mm) Area (mm2) Volume (mm3)a 34.9
N/A b 153.6
Cone B Length (mm) Area (mm2) Volume (mm3)a 34.9
N/A c 54.3d 6.4e
Volume of feeding mechanism before expansion (t0)
Time 1
Cone A Length (mm) Area (mm2) Volume (mm3)a
N/A b
Cone B Length (mm) Area (mm2) Volume (mm3)a
c d e
Volume of feeding mechanism at maximum expansion (t1)
Volume change during feeding event (mm3) Duration of feeding event (sec) 0.132
Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec)
Goliath Grouper Suction Feeding
Time 0
Cone A Length (mm) Area (mm2) Volume (mm3)a 34.9
N/A 195,916.4b 153.6
Cone B Length (mm) Area (mm2) Volume (mm3)a 34.9
N/A 84,297.3c 54.3d 6.4e 12.2
Volume of feeding mechanism before expansion (t0) 280,213.7
Time 1
Cone A Length (mm) Area (mm2) Volume (mm3)a
N/A b
Cone B Length (mm) Area (mm2) Volume (mm3)a
c d e
Volume of feeding mechanism at maximum expansion (t1)
Volume change during feeding event (mm3) Duration of feeding event (sec) 0.132
Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec)
Suction feeding in the goliath grouper◦ Given
Dimensions of cones A and B at maximum expansion (t1)
2) Find the volume of the goliath grouper feeding mechanism at maximum expansion (t1).
Math & Nature
Goliath Grouper Suction Feeding
Time 0
Cone A Length (mm) Area (mm2) Volume (mm3)a 34.9
N/A 195,916.4 b 153.6
Cone B Length (mm) Area (mm2) Volume (mm3)a 34.9
N/A 84,297.3c 54.3d 6.4e 12.2
Volume of feeding mechanism before expansion (t0) 280,213.7
Time 1
Cone A Length (mm) Area (mm2) Volume (mm3)a 39.5
N/A b 161.3
Cone B Length (mm) Area (mm2) Volume (mm3)a 39.5
c 56.4d 32.6e
Volume of feeding mechanism at maximum expansion (t1)
Volume change during feeding event (mm3) Duration of feeding event (sec) 0.132
Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec)
Goliath Grouper Suction Feeding
Time 0
Cone A Length (mm) Area (mm2) Volume (mm3)a 34.9
N/A 195,916.4b 153.6
Cone B Length (mm) Area (mm2) Volume (mm3)a 34.9
N/A 84,297.3 c 54.3d 6.4e 12.2
Volume of feeding mechanism before expansion (t0) 280,213.7
Time 1
Cone A Length (mm) Area (mm2) Volume (mm3)a 39.5
N/A 263,546.5b 161.3
Cone B Length (mm) Area (mm2) Volume (mm3)a 39.5
230,990.0c 56.4d 32.6e 266.5
Volume of feeding mechanism at maximum expansion (t1)
494,536.5
Volume change during feeding event (mm3) 214,322.8Duration of feeding event (sec) 0.132
Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec)
Suction feeding in the goliath grouper◦ Given
Dimensions of cone B at maximum expansion (t1)
3) Find the area of the goliath grouper mouth at maximum expansion (t1).
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A. Collins
mouth
Goliath Grouper Suction Feeding
Time 0
Cone A Length (mm) Area (mm2) Volume (mm3)a 34.9
N/A 195,916.4b 153.6
Cone B Length (mm) Area (mm2) Volume (mm3)a 34.9
N/A 84,297.3 c 54.3d 6.4e 12.2
Volume of feeding mechanism before expansion (t0) 280,213.7
Time 1
Cone A Length (mm) Area (mm2) Volume (mm3)a 39.5
N/A 263,546.5b 161.3
Cone B Length (mm) Area (mm2) Volume (mm3)a 39.5
230,990.0c 56.4d 32.6e 266.5
Volume of feeding mechanism at maximum expansion (t1)
494,536.5
Volume change during feeding event (mm3) 214,322.8Duration of feeding event (sec) 0.132
Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec)
Goliath Grouper Suction Feeding
Time 0
Cone A Length (mm) Area (mm2) Volume (mm3)a 34.9
N/A 195,916.4b 153.6
Cone B Length (mm) Area (mm2) Volume (mm3)a 34.9
N/A 84,297.3 c 54.3d 6.4e 12.2
Volume of feeding mechanism before expansion (t0) 280,213.7
Time 1
Cone A Length (mm) Area (mm2) Volume (mm3)a 39.5
N/A 263,546.5b 161.3
Cone B Length (mm) Area (mm2) Volume (mm3)a 39.5
3338.8 230,990.0c 56.4d 32.6e 266.5
Volume of feeding mechanism at maximum expansion (t1)
494,536.5
Volume change during feeding event (mm3) 214,322.8Duration of feeding event (sec) 0.132
Area of mouth at maximum expansion (t1) (mm2) 3338.8 Velocity of water flow into mouth (mm/sec)
Suction feeding in the goliath grouper◦ Given
Volume of the goliath grouper feeding mechanism at rest (t0) and at maximum expansion (t1)
Duration of the feeding event (t1 - t0)
Area of the mouth opening at maximum expansion (t1)
4) Find the velocity of water flow into the mouth of the goliath grouper during suction feeding.
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Goliath Grouper Suction Feeding
Time 0
Cone A Length (mm) Area (mm2) Volume (mm3)a 34.9
N/A 195,916.4b 153.6
Cone B Length (mm) Area (mm2) Volume (mm3)a 34.9
N/A 84,297.3 c 54.3d 6.4e 12.2
Volume of feeding mechanism before expansion (t0) 280,213.7
Time 1
Cone A Length (mm) Area (mm2) Volume (mm3)a 39.5
N/A 263,546.5b 161.3
Cone B Length (mm) Area (mm2) Volume (mm3)a 39.5
3338.8 230,990.0c 56.4d 32.6e 266.5
Volume of feeding mechanism at maximum expansion (t1)
494,536.5
Volume change during feeding event (mm3) 214,322.8Duration of feeding event (sec) 0.132
Area of mouth at maximum expansion (t1) (mm2) 3338.8 Velocity of water flow into mouth (mm/sec) 486.3
Wainwright et al (2006)
Suction feeding in the snook Centropomus undecimalis
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Korhnak JJ Photo
Suction feeding in the snook Centropomus undecimalis◦ Given
Dimensions of cones A and B at rest (t0) and at maximum expansion of the feeding mechanism (t1)
Duration of the feeding event (t1 - t0)
5) Find the velocity of water flow into the mouth of the snook during suction feeding.
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Snook Suction Feeding
Time 0
Cone A Length (mm) Area (mm2) Volume (mm3)a 2.1
N/A b 27.6
Cone B Length (mm) Area (mm2) Volume (mm3)a 2.1
N/A c 12.3d 1.8e
Volume of feeding mechanism before expansion (t0)
Time 1
Cone A Length (mm) Area (mm2) Volume (mm3)a 7.0
N/A b 28.9
Cone B Length (mm) Area (mm2) Volume (mm3)a 7.0
c 12.3d 5.9e
Volume of feeding mechanism at maximum expansion (t1)
Volume change during feeding event (mm3) Duration of feeding event (sec) 0.036
Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec)
Snook Suction Feeding
Time 0
Cone A Length (mm) Area (mm2) Volume (mm3)a 2.1
N/A 127.5 b 27.6
Cone B Length (mm) Area (mm2) Volume (mm3)a 2.1
N/A 147.2c 12.3d 1.8e 73.8
Volume of feeding mechanism before expansion (t0) 274.7
Time 1
Cone A Length (mm) Area (mm2) Volume (mm3)a 7.0
N/A 1482.9 b 28.9
Cone B Length (mm) Area (mm2) Volume (mm3)a 7.0
109.4 1611.5c 12.3d 5.9e 66.0
Volume of feeding mechanism at maximum expansion (t1)
3094.4
Volume change during feeding event (mm3) 2819.7Duration of feeding event (sec) 0.036
Area of mouth at maximum expansion (t1) (mm2) 109.4Velocity of water flow into mouth (mm/sec) 716.2
Suction feeding in the longjaw butterfly fish Forcipiger longirostris
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Ferry-Graham et al (2001)
S. Huskey
Suction feeding in the longjaw butterfly fish Forcipiger longirostris◦ Given
Dimensions of cones A and B at rest (t0) and at maximum expansion of the feeding mechanism (t1)
Duration of the feeding event (t1 - t0)
6) Find the velocity of water flow into the mouth of the longjaw butterfly fish during suction feeding.
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Longjaw Butterfly Fish Suction Feeding
Time 0
Cone A Length (mm) Area (mm2) Volume (mm3)a 5.0
N/A b 14.9
Cone B Length (mm) Area (mm2) Volume (mm3)a 5.0
N/A c 31.2d 1.1e
Volume of feeding mechanism before expansion (t0)
Time 1
Cone A Length (mm) Area (mm2) Volume (mm3)a 5.0
N/A b 14.9
Cone B Length (mm) Area (mm2) Volume (mm3)a 5.0
c 31.6d 1.1e
Volume of feeding mechanism at maximum expansion (t1)
Volume change during feeding event (mm3) Duration of feeding event (sec) 0.022
Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec)
Longjaw Butterfly Fish Suction Feeding
Time 0
Cone A Length (mm) Area (mm2) Volume (mm3)a 5.0
N/A 390.1 b 14.9
Cone B Length (mm) Area (mm2) Volume (mm3)a 5.0
N/A 1036.0c 31.2d 1.1e 8.8
Volume of feeding mechanism before expansion (t0) 1426.1
Time 1
Cone A Length (mm) Area (mm2) Volume (mm3)a 5.0
N/A 390.1b 14.9
Cone B Length (mm) Area (mm2) Volume (mm3)a 5.0
3.8 1049.3c 31.6d 1.1e 8.9
Volume of feeding mechanism at maximum expansion (t1)
1439.4
Volume change during feeding event (mm3) 13.3Duration of feeding event (sec) 0.022
Area of mouth at maximum expansion (t1) (mm2) 3.8Velocity of water flow into mouth (mm/sec) 158.8
Suction feeding ◦ Given
Velocities of water flow into the mouths of all three fish
7) Determine which fish is the best suction feeder.
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Korhnak
v(t) = 158.8 mm/secv(t) = 716.2 mm/secv(t) = 486.3 mm/sec
References◦ Bishop, K.L., Wainwright, P.C., and Holzman, R. (2008). Anterior to posterior wave of buccal
expansion in suction feeding fish is critical for optimizing fluid flow velocity profile. Journal of the Royal Society, Interface. 5:1309-1316.
◦ Ferry-Graham, L.A., Wainwright, P.C., and Bellwood, D.R. (2001). Prey capture in long-jawed butterflyfishes (Chaetodontidae): the functional basis of novel feeding habits. Journal of Experimental Marine Biology and Ecology. 256:167-184.
◦ Galileo Galilei, The Assayer, as translated by Stillman Drake (1957), Discoveries and Opinions of Galileo pp. 237 - 238. New York: Doubleday & Company.
◦ Gibb, A.C. and Ferry-Graham, L.A. (2005). Cranial movements during suction feeding in teleost fishes: Are they modified to enhance suction production? Zoology. 108(2): 141-153.
◦ Grubich, J.R. (2001). Prey Capture in Actinopterygian Fishes: A Review of Suction Feeding Motor Patterns with New Evidence from an Elopomorph Fish, Megalops atlanticus. Integrative and Comparative Biology. 41(6): 1258-1265.
◦ Holzman, R., Day, S.W., and Wainwright, P.C. (2007). Timing is everything: coordination of strike kinematics affects the force exerted by suction feeding fish on attached prey. Journal of Experimental Biology. 210: 3328-3336.
◦ Holzman, R., Day, S.W., Mehta, R.S., and Wainwright, P.C. (2008). Jaw protrusion enhances forces exerted on prey by suction feeding fishes. Journal of the Royal Society, Interface. 5(29): 1445-1457.
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References◦ Liem, K., Bemis, W., Walker, W.F., and Grande, L. (2001). Functional Anatomy of the
Vertebrates: An Evolutionary Perspective. New York. Cengage Learning. ◦ Merrill, M.D. (2002). First principles of instruction. Educational Technology Research and
Development. 50 (3): 43 – 59.◦ Motta, P.J., Hueter, R.E., Tricas, T.C., Summers, A.P., Huber, D.R., Lowry, D., Mara, K.R.,
Matott, M.P., Whitenack, L.B., and Wintzer, A.P. (2008). Functional morphology of the feeding apparatus, feeding constraints, and suction performance in the nurse shark Ginglymostoma cirratum. Journal of Morphology. 269(9): 1041-1055.
◦ Motta, P.J., Maslanka, M., Hueter, R.E., Davis, R.L., de la Parra, R., Mulvany, S.L., Habegger, M.L., Strother, J.A., Mara, K.R., Gardiner, J.M., Tyminski, J.P., and Zeigler, L.D. (2010). Feeding anatomy, filter-feeding rate, and diet of whale sharks Rhincodon typus during surface ram filter feeding off the Yucatan Peninsula, Mexico. Zoology. 113: 199-212.
◦ Sanford, C.P.J. and Wainwright, P.C. (2002). Use of sonomicrometry demonstrates the link between prey capture kinematics and suction pressure in largemouth bass. Journal of Experimental Biology. 205: 3445-3457.
◦ Svanback, R., Wainwright, P.C., and Ferry-Graham, L.A. (2002). Linking cranial kinematics, buccal pressure, and suction feeding performance in largemouth bass. Physiological and Biochemical Zoology. 75(6): 532-543.
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References◦ Van Wassenbergh, S., Herrel, A., Adriaens, D., and Aerts, P. (2007). No trade-off between
biting and suction feeding performance in clariid catfishes. Journal of Experimental Biology. 210: 27-36.
◦ Wainwright, P.C., Huskey, S.H., Turingan, R.G., and Carroll, A.M. (2006). Ontogeny of suction feeding capacity in snook, Centropomis undecimalis. Journal of Experimental Zoology. 305A: 246-252.
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Calculating the volume of a truncated cone
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