The DOI matches its Crossref record; 16 of the 17 references it cites were found in a registry or an index.
Zigzagging: Theoretical insights on climbing strategies
Abstract
Human and animal trails on steep hillsides often exhibit dramatic switchbacks and shortcuts. Helbing et al. have recently examined the emergence of human trail systems on flat terrains while Minetti and Margaria established the effect of gradients on human metabolic efficiency. In this paper we use these ideas to develop a semi-quantitative theoretical model of the behaviour of humans moving on a terrain with relief. The model determines the direction of movement by minimising metabolic cost per unit of distance in a desired direction. The structure of the theory resembles the Landau Theory of Phase Transitions, much used in theoretical physics. We find that both hairpin bends (switchbacks) and shortcuts appear as efficient strategies for downhill walkers, while uphill walkers retain switchbacks. For weakly inclined slopes, the best strategy involves walking directly uphill or downhill. For sufficiently steep slopes, however, we find that the best strategy should undergo a transition to a broken symmetry solution corresponding to the switchback trail patterns typical of rugged environments. The critical slope at which this transition takes place should be less steep for uphill and downhill walkers. The theory should be amenable to empirical investigation. Amongst other applications, this model will enable us to generalize the work of previous authors to real landscapes, eventually permitting the reconstruction of ancient patterns of movement in archaeological landscapes.
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In the citing paper
“Of the two energy-based cost functions tested, that is Herzog (2014c) and Llobera and Sluckin (2007), it is only however Herzog (2014c) that can produce the model outcomes that follow the Roman road (Fig. 10A).”
From Explaining known past routes, underdetermination, and the use of multiple cost functions (2024) · cites this paper as Llobera and Sluckin (2007)
In the citing paper
“Although the hypothesis could be formulated as 'humans minimise steeper slope gradients when traveling by wheeled vehicles', the critical gradient parameter within the cost function is not given a predefined value defined by the cost function but instead to be estimated from the past route being explained.”
From Explaining known past routes, underdetermination, and the use of multiple cost functions (2024) · cites this paper as Llobera and Sluckin (2007)
In the citing paper
“The use of Llobera and Sluckin's (2007) energy expenditure cost function over the more familiar time-based Tobler's Hiking Function (Tobler, 1993) reflects previous success in its ability to model Roman roads (e.g.Fonte et al., 2017;Güimil-Fariña & Parcero-Oubiña, 2015;Parcero-Oubiña et al., 2019).”
From Probabilistic modelling for incorporating uncertainty in least cost path results: a postdictive roman road case study (2021) · cites this paper as Llobera and Sluckin's (2007)
In the citing paper
“The cost of movement has been modelled based again only on the effect of terrain slope and calculated using the metabolic walking costs function proposed in (Llobera & Sluckin, 2007).”
From Exploring the inherent affordability for movement in past landscapes (2027) · cites this paper as Llobera & Sluckin, 2007
In the citing paper
“The cost of movement has been modelled based only on the effect of terrain slope and calculated using the metabolic walking costs function proposed in Llobera & Sluckin, 2007.”
From Exploring the inherent affordability for movement in past landscapes (2027) · cites this paper as Llobera & Sluckin, 2007
In the citing paper
“Furthermore, previous work [11, 29–31] has identified the existence of a critical gradient; the angle at which it is faster to zig-zag up a hill, rather than ascend directly. This occurs at a walking slope of around 15—21 degrees, so models which failed to predict the critical gradient occurring below 21 degrees were removed.”
From Improved prediction of hiking speeds using a data driven approach (2023) · cites this paper as [31]
In the citing paper
“The modified Tobler function has a critical slope for cost-saving zig-zagging of 27%, as defined by Llobera & Sluckin's (2007) equation 20.”
From Budget travel in the Mediterranean: a methodology for reconstructing ancient journeys through least cost networks (2022) · cites this paper as Llobera & Sluckin's (2007)
In the citing paper
“Costs of walking are mostly estimated in terms of energy consumption [Llobera and Sluckin 2007;Minetti et al. 2002] or time [Ericson and Goldstein 1980;Irmischer and Clarke 2017;Langmuir 2004;Tobler 1993].”
From Issues in replication and stability of least-cost path calculations (2022) · cites this paper as Llobera and Sluckin 2007
In the citing paper
“Even though movecost offers a wide range of functions for calculating accumulated slope-dependent anisotropic cost surfaces, we have chosen to use the traditionally most successful function to represent mobility in pre-industrial contexts [55], alongside the function whose results aligned best with fossilised stretches of paths identifiable on satellite imagery (Tobler, see Figure 12), a tried-and”
From Andean landscape legacies: comprehensive remote sensing mapping and GIS analysis of long-term settlement and land use for sustainable futures (NW argentina) (2024) · cites this paper as [55]
Citations
Cites
Social force model for pedestrian dynamics.
Helbing, D., & Molnár, P. (1995) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.51.4282 · Cited by 7,299
Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light
Born, M., Wolf, E., & Hecht, E. (2000) · Physics Today · doi:10.1063/1.1325200 · Cited by 4,185
Ivey, J. E. (1969) · The Musical Quarterly · doi:10.1093/mq/lv.1.87 · Cited by 1,050
Lectures on the Calculus of Variations and Optimal Control Theory
El-Hodiri, M. A., & Young, L. C. (1971) · Econometrica · doi:10.2307/1913279 · Cited by 631
Energy cost of walking and running at extreme uphill and downhill slopes
Minetti, A. E., Moia, C., Roi, G. S., Susta, D., & Ferretti, G. (2002) · Journal of Applied Physiology · doi:10.1152/japplphysiol.01177.2001 · Cited by 624
Biomechanics and Energetics of Muscular Exercise.
(1977) · Annals of Internal Medicine · doi:10.7326/0003-4819-86-3-375_3 · Cited by 470
Modelling the evolution of human trail systems
Helbing, D., Keltsch, J., & Molnár, P. (1997) · Nature · doi:10.1038/40353 · Cited by 302
Elephants avoid costly mountaineering
Wall, J., Douglas‐Hamilton, I., & Vollrath, F. W. (2006) · Current Biology · doi:10.1016/j.cub.2006.06.049 · Cited by 233
A Theory of Metabolic Costs for Bipedal Gaits
Minetti, A. E., & Alexander, R. M. (1997) · Journal of Theoretical Biology · doi:10.1006/jtbi.1997.0407 · Cited by 227
Batty, M., Desyllas, J., & Duxbury, E. (2003) · International Journal of Geographical Information Systems · doi:10.1080/1365881031000135474 · Cited by 197
Research works printed but not linked: 2
- De Smith, M., 2003. In: Longley, P.A. Batty, M. (Eds.), Advanced Spatial Analysis. ESRI Press, Redlands, California (Chapter 15).
- Margaria, 1938, Sulla fisiologia e specialmente sul consume energetico della marcia e della corsa a varia velocita ed inclinazione de1 terreno, Atti Accad. Naz. Lincei Memorie, 7, 299
Cited by
Sorted by citations
Order in movement: a GIS approach to accessibility
Llobera, M., Fábrega-Álvarez, P., & Parcero-Oubiña, C. (2011) · Journal of Archaeological Science · doi:10.1016/j.jas.2010.11.006 · Cited by 105
Self-citationModelling of pathways and movement networks in archaeology: an overview of current approaches
Verhagen, P., Nuninger, L., & Groenhuijzen, M. R. (2019) · Computational Social Sciences · doi:10.1007/978-3-030-04576-0_11 · Cited by 117
Least-cost Paths – Some Methodological Issues
Herzog, I. (2014) · Internet Archaeology · doi:10.11141/ia.36.5 · Cited by 102
The Scaling of Uphill and Downhill Locomotion in Legged Animals
Birn‐Jeffery, A. V., & Higham, T. E. (2014) · Integrative and Comparative Biology · doi:10.1093/icb/icu015 · Cited by 95
Güimil-Fariña, A., & Parcero-Oubiña, C. (2015) · Journal of Archaeological Science · doi:10.1016/j.jas.2014.11.030 · Cited by 95
Metadata
- Authors
- Llobera, M., & Sluckin, T. J.
- Affiliations
- University of Washington (US)
- University of Southampton (GB)
- Corresponding author
- Llobera, M.
- Venue
- Journal of Theoretical Biology
- Publisher
- Elsevier BV
- Published
- 2007-11
- Volume
- 249
- Issue
- 2
- Pages
- 206-217
- ISSN
- 0022-5193 (ISSN-L)
- OpenAlex
- W2049192609
- Cited by
- 103
- References
- 17
- Citations by year
- 2026: 9 · 2025: 15 · 2024: 4 · 2023: 10 · 2022: 9 · 2021: 7 · 2020: 5 · 2019: 8 · 2018: 1 · 2017: 4 · 2016: 2 · 2015: 4 · 2014: 6 · 2013: 6 · 2012: 6
- FWCI
- 9.27
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Theoretical Biology (publishedVersion)
- PubMed (publishedVersion)
- ePrints Soton (University of Southampton) (submittedVersion)
- UCL Discovery (University College London) (submittedVersion)
- Funders
- University of Washington
- Leverhulme Trust
- Arts and Humanities Research Council
- Topics
- Social Sciences › Safety Research › Evacuation and Crowd Dynamics
- Social Sciences › Urban Studies › Urban Design and Spatial Analysis
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
- Keywords
- switchbacks, hiking trails, trail formation, metabolic cost, symmetry breaking, Landau theory, critical slope, archaeological landscape
- MeSH terms
- Altitude, Animals, Biomechanical Phenomena, Energy Metabolism, Humans, Models, Biological, Motor Activity, Walking
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