Recent results on prey detection in a spider orb web

Alexandre Kawano; Antonino Morassi; Ramón Zaera; Alexandre Kawano; Antonino Morassi; Ramón Zaera

doi:10.3934/nhm.2025014

Networks and Heterogeneous Media

2025, Volume 20, Issue 1: 286-323. doi: 10.3934/nhm.2025014

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Recent results on prey detection in a spider orb web

1.
Escola Politécnica, University of São Paulo, Av. Prof. Luciano Gualberto, 380 - Butantã, São Paulo 05508-010, Brazil
2.
Polytechnic Department of Engineering and Architecture, University of Udine, via Cotonificio 114, Udine 33100, Italy
3.
Department of Continuum Mechanics and Structural Analysis, Universidad Carlos III de Madrid, Av. de la Universidad 30, Leganés 28911, Spain

Received: 18 November 2024 Revised: 19 March 2025 Accepted: 21 March 2025 Published: 02 April 2025

In this paper, we present a review of some results concerning the inverse problem of detecting a prey in a spider orb web. First, an overview of the discrete and numerical models in the literature is provided to understand the mechanics of the orb web and a descriiption of their structural features is presented. Then, a continuous model was introduced in which the orb web was described as a membrane with a specific fibrous structure and subject to tensile prestress in the referential configuration. The prey's impact was modeled as a pressure field acting on the orb web with magnitude of the form $ g(t)f(x) $, where $ g(t) $ is a known function of the time and $ f(x) $ is the unknown spatial term that depends on the position variable $ x $. Next, for axially symmetric orb webs supported at the boundary and undergoing infinitesimal deformations, a uniqueness results for $ f(x) $ was proven in terms of dynamic displacement measurements taken on an arbitrarily small and thin ring centered at the origin of the web, for a sufficiently large interval of time. A reconstruction algorithm suggested by the uniqueness result was implemented for $ f(x) $, and how the results of identification are affected by the key geometric and mechanical parameters were studied. The results, obtained on a realistic family of orb webs, revealed that the dynamic signals propagating through the web immediately after impact contained enough information for the spider to capture the prey. In the final part of the paper, we discuss an alternative method for prey localization and include some extension to account for different boundary conditions. A short list of open problems is proposed in the conclusions section. This paper is dedicated to Professor Emilio Turco for his 60th Birthday.
- spider orb web,
- membrane model,
- inverse problems,
- identification of sources,
- infinitesimal vibration
Citation: Alexandre Kawano, Antonino Morassi, Ramón Zaera. Recent results on prey detection in a spider orb web[J]. Networks and Heterogeneous Media, 2025, 20(1): 286-323. doi: 10.3934/nhm.2025014

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Abstract

In this paper, we present a review of some results concerning the inverse problem of detecting a prey in a spider orb web. First, an overview of the discrete and numerical models in the literature is provided to understand the mechanics of the orb web and a descriiption of their structural features is presented. Then, a continuous model was introduced in which the orb web was described as a membrane with a specific fibrous structure and subject to tensile prestress in the referential configuration. The prey's impact was modeled as a pressure field acting on the orb web with magnitude of the form $ g(t)f(x) $, where $ g(t) $ is a known function of the time and $ f(x) $ is the unknown spatial term that depends on the position variable $ x $. Next, for axially symmetric orb webs supported at the boundary and undergoing infinitesimal deformations, a uniqueness results for $ f(x) $ was proven in terms of dynamic displacement measurements taken on an arbitrarily small and thin ring centered at the origin of the web, for a sufficiently large interval of time. A reconstruction algorithm suggested by the uniqueness result was implemented for $ f(x) $, and how the results of identification are affected by the key geometric and mechanical parameters were studied. The results, obtained on a realistic family of orb webs, revealed that the dynamic signals propagating through the web immediately after impact contained enough information for the spider to capture the prey. In the final part of the paper, we discuss an alternative method for prey localization and include some extension to account for different boundary conditions. A short list of open problems is proposed in the conclusions section. This paper is dedicated to Professor Emilio Turco for his 60th Birthday.

References

[1]	L. H. Lin, W. Sobek, Structural hierarchy in spider webs and spiderweb-type systems, Struct. Engineer, 76 (1998), 59–64.
[2]	P. Jiang, L. H. Wu, T. Y. Lv, S. S. Tang, M. L. Hu, Z. M. Qiu, et al., Memory effect of spider major ampullate silk in loading-unloading cycles and the structural connotations, J. Mech. Behav. Biomed. Mater., 146 (2023), 106031. https://doi.org/10.1016/j.jmbbm.2023.106031 doi: 10.1016/j.jmbbm.2023.106031
[3]	T. Köhler, F. Vollrath, Thread biomechanics in the two orb-weaving spiders araneus diadematus (araneae, araneidae) and uloborus walckenaerius (araneae, uloboridae), J. Exp. Zool., 271 (1995), 1–17. https://doi.org/10.1002/jez.1402710102 doi: 10.1002/jez.1402710102
[4]	C. L. Craig, The ecological and evolutionary interdependence between web architecture and web silk spun by orb web weaving spiders, Biol. J. Linn. Soc., 30 (1987), 135–162. https://doi.org/10.1111/j.1095-8312.1987.tb00294.x doi: 10.1111/j.1095-8312.1987.tb00294.x
[5]	L. H. Lin, D. T. Edmonds, F. Vollrath, Structural engineering of an orb-spider's web, Nature, 373 (1995), 146–148.
[6]	M. S. Alam, C. H. Jenkins, Damage tolerance in naturally compliant structures, Int. J. Damage Mech., 14 (2005), 365–384. https://doi.org/10.1177/1056789505054313 doi: 10.1177/1056789505054313
[7]	M. S. Alam, M. A. Wahab, C. H. Jenkins, Mechanics in naturally compliant structures, Mech. Mater., 39 (2007), 145–160. https://doi.org/10.1016/j.mechmat.2006.04.005 doi: 10.1016/j.mechmat.2006.04.005
[8]	Y. Aoyanagi, K. Okumura, Simple model for the mechanics of spider webs, Phys. Rev. Lett., 104 (2010), 038102. https://doi.org/10.1103/PhysRevLett.104.038102 doi: 10.1103/PhysRevLett.104.038102
[9]	S. W. Cranford, A. Tarakanova, N. M. Pugno, M. J. Buehler, Nonlinear material behaviour of spider silk yields robust webs, Nature, 482 (2012), 72–76.
[10]	F. K. Ko, J. Jovicic, Modeling of mechanical properties and structural design of spider web, Biomacromolecules, 5 (2004), 780–785. https://doi.org/10.1021/bm0345099 doi: 10.1021/bm0345099
[11]	R. Zaera, A. Soler, J. Teus, Uncovering changes in spider orb-web topology owing to aerodynamic effects, J. R. Soc. Interface, 11 (2014), 20140484. https://doi.org/10.1098/rsif.2014.0484 doi: 10.1098/rsif.2014.0484
[12]	A. Soler, R. Zaera, The secondary frame in spider orb webs: The detail that makes the difference, Sci. Rep., 6 (2016), 31265.
[13]	H. Yu, J. Yang, Y. Sun, Energy absorption of spider orb webs during prey capture: A mechanical analysis, J. Bionic Eng., 12 (2015), 453–463. https://doi.org/10.1016/S1672-6529(14)60136-0 doi: 10.1016/S1672-6529(14)60136-0
[14]	L. Zheng, M. Behrooz, F. Gordaninejad, A bioinspired adaptive spider web, Bioinspiration Biomimetics, 12 (2017), 016012. https://doi.org/10.1088/1748-3190/12/1/016012 doi: 10.1088/1748-3190/12/1/016012
[15]	B. Mortimer, A. Soler, C. R. Siviour, R. Zaera, F. Vollrath, Tuning the instrument: Sonic properties in the spider's web, J. R. Soc. Interface, 13 (2016), 20160341. https://doi.org/10.1098/rsif.2016.0341 doi: 10.1098/rsif.2016.0341
[16]	B. Mortimer, A. Soler, L. Wilkins, F. Vollrath, Decoding the locational information in the orb web vibrations of araneus diadematus and zygiella x-notata, J. R. Soc. Interface, 16 (2019), 20190201. https://doi.org/10.1098/rsif.2019.0201 doi: 10.1098/rsif.2019.0201
[17]	A. W. Otto, D. O. Elias, R. L. Hatton, Modeling transverse vibration in spider webs using frequency-based dynamic substructuring, in Dynamics of Coupled Structures, Volume 4: Proceedings of the 36th IMAC, 2018,143–155. https://doi.org/10.1007/978-3-319-74654-8_12
[18]	A. Morassi, A. Soler, R. Zaera, A continuum membrane model for small deformations of a spider orb-web, Mech. Syst. Signal Process., 93 (2017), 610–633. https://doi.org/10.1016/j.ymssp.2017.02.018 doi: 10.1016/j.ymssp.2017.02.018
[19]	A. Kawano, A. Morassi, Closed-form solutions to the static transverse deformation of a spider orb-web, Meccanica, 54 (2019a), 2421–2442. https://doi.org/10.1007/s11012-019-01083-3 doi: 10.1007/s11012-019-01083-3
[20]	A. Kawano, A. Morassi, Detecting a prey in a spider orb web, SIAM J. Appl. Math., 79 (2019b), 2506–2529. https://doi.org/10.1137/19M1262322 doi: 10.1137/19M1262322
[21]	A. Kawano, A. Morassi, R. Zaera, Detecting a prey in a spider orb-web from in-plane vibration, SIAM J. Appl. Math., 81 (2021a), 2297–2322. https://doi.org/10.1137/20M1372792 doi: 10.1137/20M1372792
[22]	A. Kawano, A. Morassi, R. Zaera, Determination of the prey impact region in a spider orb-web from in-plane vibration, Appl. Math. Comput., 424 (2022a), 126947. https://doi.org/10.1016/j.amc.2022.126947 doi: 10.1016/j.amc.2022.126947
[23]	R. Zaera, Ó. Serrano, J. Fernández-Sáez, A. Morassi, Eco-localization of a prey in a spider orb web, J. Vib. Control, 28 (2022), 1229–1238. https://doi.org/10.1177/1077546321993548 doi: 10.1177/1077546321993548
[24]	A. Kawano, A. Morassi, Can the spider hear the position of the prey, Mech. Syst. Signal Process., 143 (2020), 106838. https://doi.org/10.1016/j.ymssp.2020.106838 doi: 10.1016/j.ymssp.2020.106838
[25]	A. Kawano, A. Morassi, R. Zaera, The prey's catching problem in an elastically supported spider orb-web, Mech. Syst. Signal Process., 151 (2021b), 107310. https://doi.org/10.1016/j.ymssp.2020.107310 doi: 10.1016/j.ymssp.2020.107310
[26]	Y. Guo, Z. Chang, B. Li, Z. L. Zhao, H. P. Zhao, X. Q. Feng, et al., Functional gradient effects on the energy absorption of spider orb webs, Appl. Phys. Lett., 113 (2018), 103701. https://doi.org/10.1063/1.5039710 doi: 10.1063/1.5039710
[27]	Y. Jiang, H. Nayeb-Hashemi, Dynamic response of spider orb webs subject to prey impact, Int. J. Mech. Sci., 186 (2020a), 105899. https://doi.org/10.1016/j.ijmecsci.2020.105899 doi: 10.1016/j.ijmecsci.2020.105899
[28]	Y. Jiang, H. Nayeb-Hashemi, Energy dissipation during prey capture process in spider orb webs, J. Appl. Mech., 87 (2020b), 091009. https://doi.org/10.1115/1.4047364 doi: 10.1115/1.4047364
[29]	B. Xie, X. Wu, X. Ji, Investigation on the energy-absorbing properties of bionic spider web structure, Biomimetics, 8 (2023), 537. https://doi.org/10.3390/biomimetics8070537 doi: 10.3390/biomimetics8070537
[30]	S. Kaewunruen, C. Ngamkhanong, T. Yang, Large-amplitude vibrations of spider web structures, Appl. Sci., 10 (2020b), 6032. https://doi.org/10.3390/app10176032 doi: 10.3390/app10176032
[31]	S. Kaewunruen, C. Ngamkhanong, S. Xu, Large amplitude vibrations of imperfect spider web structures, Sci. Rep., 10 (2020a), 19161.
[32]	K. Yavuz, A. M. Soler, R. Zaera, S. Jahangirov, Effect of spider's weight on signal transmittance in vertical orb webs, R. Soc. Open Sci., 11 (2024), 240986. https://doi.org/10.1098/rsos.240986 doi: 10.1098/rsos.240986
[33]	M. Denny, The physical properties of spider's silk and their role in the design of orb-webs, J. Exp. Biol., 65 (1976), 483–506. https://doi.org/10.1242/jeb.65.2.483 doi: 10.1242/jeb.65.2.483
[34]	E. J. Kullmann, The convergent development of orb-webs in cribellate and ecribellate spiders, Am. Zool., 12 (1972), 395–405. https://doi.org/10.1093/icb/12.3.395 doi: 10.1093/icb/12.3.395
[35]	E. Wirth, F. G. Barth, Forces in the spider orb web, J. Comp. Physiol. A, 171 (1992), 359–371. https://doi.org/10.1007/BF00223966
[36]	W. G. Eberhard, Construction behaviour and the distribution of tensions in orb webs, Bull. Br. Arachnological Soc., 5 (1981), 189–204.
[37]	N. Tew, T. Hesselberg, The effect of wind exposure on the web characteristics of a tetragnathid orb spider, J. Insect Behav., 30 (2017), 273–286. https://doi.org/10.1007/s10905-017-9618-0 doi: 10.1007/s10905-017-9618-0
[38]	Y. C. Li, Y. P. Zhao, F. K. Gui, B. Teng, Numerical simulation of the hydrodynamic behaviour of submerged plane nets in current, Ocean Eng., 33 (2006), 2352–2368. https://doi.org/10.1016/j.oceaneng.2005.11.013 doi: 10.1016/j.oceaneng.2005.11.013
[39]	W. G. Eberhard, Function and phylogeny of spider webs, Annu. Rev. Ecol. Syst., 21 (1990), 341–372.
[40]	R. M. Young, An Introduction to Nonharmonic Fourier Series, first edition, Academic Press, 2001.
[41]	A. Olevskii, A. Ulanovskii, On multi-dimensional sampling and interpolation, Anal. Math. Phys., 2 (2012), 149–170. https://doi.org/10.1007/s13324-012-0027-4 doi: 10.1007/s13324-012-0027-4
[42]	A. Kawano, A. Zine, Uniqueness and nonuniqueness results for a certain class of almost periodic distributions, SIAM J. Math. Anal., 43 (2011), 135–152. https://doi.org/10.1137/090763524 doi: 10.1137/090763524
[43]	L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Classics in Mathematics, Springer Berlin Heidelberg, 2003.
[44]	A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, 3rd edition, Springer International Publishing, 2021.
[45]	J. Baumeister, Stable Solution of Inverse Problems, Friedr. Vieweg and Sohn, Braunschweig, 1986.
[46]	R. Hergenröder, F. G. Barth, Vibratory signals and spider behavior: How do the sensory inputs from the eight legs interact in orientation, J. Comp. Physiol., 152 (1983), 361–371. https://doi.org/10.1007/bf00606241 doi: 10.1007/bf00606241
[47]	B. D. Opell, J. E. Bond, Capture thread extensibility of orb-weaving spiders: Testing punctuated and associative explanations of character evolution, Biol. J. Linn. Soc., 70 (2000), 107–120. https://doi.org/10.1111/j.1095-8312.2000.tb00203.x doi: 10.1111/j.1095-8312.2000.tb00203.x
[48]	B. D. Opell, J. E. Bond, Changes in the mechanical properties of capture threads and the evolution of modern orb-weaving spiders, Evol. Ecol. Res., 3 (2001), 507–519.
[49]	B. Mortimer, S. D. Gordon, C. Holland, C. R. Siviour, F. Vollrath, J. F. C. Windmill, The speed of sound in silk: linking material performance to biological function, Adv. Mater., 26 (2014), 5179–5183. https://doi.org/10.1002/adma.201401027 doi: 10.1002/adma.201401027
[50]	T. Mulder, B. Mortimer, F. Vollrath, Functional flexibility in a spider's orb web, J. Exp. Biol., 223 (2020), jeb234070. https://doi.org/10.1242/jeb.234070 doi: 10.1242/jeb.234070
[51]	Y. C. Yang, C. Dorn, T. Mancini, Z. Talken, G. Kenyon, C. Farrar, et al., Blind identification of full-field vibration modes from video measurements with phase-based video motion magnification, Mech. Syst. Signal Process., 85 (2017), 567–590. https://doi.org/10.1016/j.ymssp.2016.08.041 doi: 10.1016/j.ymssp.2016.08.041
[52]	Y. C. Yang, H. K. Jung, C. Dorn, G. Park, C. Farrar, D. Mascarenas, Estimation of full-field dynamic strains from digital video measurements of output-only beam structures by video motion processing and modal superposition, Struct. Control Health Monit., 26 (2019), e2408. https://doi.org/10.1002/stc.2408 doi: 10.1002/stc.2408
[53]	A. Kawano, A. Morassi, R. Zaera, Dynamic identification of pretensile forces in a spider orb-web, Mech. Syst. Signal Process., 169 (2022b), 108703. https://doi.org/10.1016/j.ymssp.2021.108703 doi: 10.1016/j.ymssp.2021.108703
[54]	A. Savitzky, M. J. E. Golay, Smoothing and differentiation of data by simplified least squares procedures, Anal. Chem., 36 (1964), 1627–1639. https://doi.org/10.1021/ac60214a047 doi: 10.1021/ac60214a047
[55]	W. H. Press, S. Teukolsky, B. P. Flannery, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, 2007.
[56]	C. V. Boys, The influence of a tuning-fork on the garden spider, Nature, 23 (1880), 149–150.
[57]	H. C. McCook, American Spiders and Their Spinningwork: A Natural History of the Orbweaving Spiders of the United States with Special Regard to Their Industry and Habits, Academy of Natural Sciences of Philadelphia, 1890.
[58]	M. Asch, G. Lebeau, The spectrum of the damped wave operator for a bounded domain in $R^2$, Exp. Math., 12 (2003), 227–240. https://doi.org/10.1080/10586458.2003.10504494 doi: 10.1080/10586458.2003.10504494
[59]	G. Di Fazio, M. S. Fanciullo, A. Morassi, P. Zamboni, Bounded deformation for a spider orb web, preprint, 2025.
[60]	F. Vollrath, M. Downes, S. Krackow, Design variability in web geometry of an orb-weaving spider, Physiol. Behav., 62 (1997), 735–743. https://doi.org/10.1016/S0031-9384(97)00186-8 doi: 10.1016/S0031-9384(97)00186-8
[61]	A. Kawano, A. Morassi, Asymmetrical spider orb webs, preprint, 2024.

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