Advertisement

Journal of Mathematical Biology

, Volume 78, Issue 4, pp 943–984 | Cite as

An ant navigation model based on Weber’s law

  • Paulo AmorimEmail author
  • Thierry Goudon
  • Fernando Peruani
Article
  • 122 Downloads

Abstract

We analyze an ant navigation model based on Weber’s law, where the ants move across a pheromone landscape sensing the area using two antennae. The key parameter of the model is the angle \(2\beta \) representing the span of the ant’s sensing area. We show that when \(\beta <\pi /2\) ants are able to follow (straight) pheromone trails proving that for initial conditions close to the trail, there exists a Lyapunov function that ensures ant trajectories converge on and follow the pheromone trail, with these solutions being locally asymptotically stable. Furthermore, we indicate that the features of the ant trajectories such as convergence speed or oscillation wave length are controlled by the angle \(\beta \). For \(\beta >\pi /2\), we present numerical evidence that indicates that ants are unable to follow pheromone trails. We also assess our model by comparing it to previous experimental results, showing that the solutions’ behavior falls into biologically meaningful ranges. Our work provides solid mathematical support for experimental studies where it was found that ant perception follows a Weber’s law, by proving that such models lead to the desired robust and stable trail following.

Keywords

Ant navigation Individual-based model Pheromones Foraging Animal movement 

Mathematics Subject Classification

92D50 70K20 34D20 

Notes

Acknowledgements

We acknowledge support form the Brazilian–French Network in Mathematics, which has made possible a visit in Nice where a large part of this work was done. P.A. was partially supported by FAPERJ “Jovem Cientista do Nosso Estado” Grant No. 202.867/2015, and CNPq Grant No. 442960/2014-0.

References

  1. Bandeira de Melo EB, Araújo AFR (2011) Modelling foraging ants in a dynamic and confined environment. Biosystems 104:23–31CrossRefGoogle Scholar
  2. Barberis L, Peruani F (2016) Large-scale patterns in a minimal cognitive flocking model: incidental leaders, nematic patterns, and aggregates. Phys Rev Lett 117:248001–248006CrossRefGoogle Scholar
  3. Beckers R, Deneubourg J-L, Goss S (1992) Trail laying behaviour during food recruitment in the ant Lasius niger (L.). Insectes Soc 39(1):59–72CrossRefGoogle Scholar
  4. Boissard E, Degond P, Motsch S (2013) Trail formation based on directed pheromone deposition. J Math Biol 66(6):1267–1301MathSciNetCrossRefzbMATHGoogle Scholar
  5. Calenbuhr V, Deneubourg J-L (1992a) A model for osmotropotactic orientation (I). J Theor Biol 158(3):359–393CrossRefGoogle Scholar
  6. Calenbuhr V, Deneubourg J-L (1992b) A model for osmotropotactic orientation (II). J Theor Biol 158(3):395–407CrossRefGoogle Scholar
  7. Camazine S, Deneubourg J-L, Franks NR, Sneyd J, Theraulaz G, Bonabeau E (2001) Self-organization in biological systems. Princeton studies in complexity. Princeton University Press, PrincetonzbMATHGoogle Scholar
  8. Carrillo JA, Fornasier M, Toscani G, Vecil F (2010) Particle, kinetic, and hydrodynamic models of swarming. In: Naldi G, Pareschi L, Toscani G (eds) Mathematical modeling of collective behavior in socio-economic and life sciences. Modeling and simulation in science, engineering and technology. Birkhäuser, BostonGoogle Scholar
  9. Cucker F, Smale S (2007a) Emergent behavior in flocks. IEEE Trans Autom Control 52:852–862MathSciNetCrossRefzbMATHGoogle Scholar
  10. Cucker F, Smale S (2007b) On the mathematics of emergence. Jpn J Math 2:197–227MathSciNetCrossRefzbMATHGoogle Scholar
  11. Degond P, Frouvelle A, Liu J-G (2013) Macroscopic limits and phase transition in a system of self-propelled particles. J Nonlinear Sci 23(3):427–456MathSciNetCrossRefzbMATHGoogle Scholar
  12. Deneubourg J-L, Aron S, Goss S, Pasteels JM (1990) The self-organizing exploratory pattern of the argentine ant. J Insect Behav 3(2):150–168Google Scholar
  13. D’Orsogna MR, Chuang YL, Bertozzi AL, Chayes LS (2006) Self-propelled particles with soft-core interactions: patterns, stability, and collapse. Phys Rev Lett 96(10):104302CrossRefGoogle Scholar
  14. Edelstein-Keshet L (1994) Simple models for trail-following behaviour; trunk trails versus individual foragers. J Math Biol 32:303–328CrossRefzbMATHGoogle Scholar
  15. Edelstein-Keshet L, Watmough J, Ermentrout BG (1995) Trail following in ants: individual properties determine population behaviour. Behav Ecol Sociobiol 36(2):119–133CrossRefGoogle Scholar
  16. Ehmer B, Gronenberg W (1997) Antennal muscles and fast antennal movements in ants. J Comp Physiol B 167(4):287–296CrossRefGoogle Scholar
  17. Fontelos MA, Friedman A (2015) A PDE model for the dynamics of trail formation by ants. J Math Anal Appl 425(1):1–19MathSciNetCrossRefzbMATHGoogle Scholar
  18. Fontelos MA, Garnier A, Vela-Perez M (2015) From individual to collective dynamics in argentine ants (Linepithema humile). Math Biosci 262:56–64MathSciNetCrossRefzbMATHGoogle Scholar
  19. Ha S-Y, Tadmor E (2008) From particle to kinetic and hydrodynamic descriptions of flocking. Kinet Relat Models 1(3):415–435MathSciNetCrossRefzbMATHGoogle Scholar
  20. Hangartner W (1967) Spezifität und inaktivierung des spurpheromons von lasius fuliginosus latr. und orientierung der arbeiterinnen im duftfeld. Zeitschrift für vergleichende Physiologie 57(2):103–136CrossRefGoogle Scholar
  21. Hölldobler B, Wilson EO (1990) The ants. Harvard University Press, CambridgeCrossRefGoogle Scholar
  22. Jackson D, Holcombe M, Ratnieks F (2004) Coupled computational simulation and empirical research into the foraging system of pharaoh’s ant (Monomorium pharaonis). Biosystems 76(1–3):101–112CrossRefGoogle Scholar
  23. John A, Schadschneider A, Chowdhury D, Nishinari K (2009) Traffic-like collective movement of ants on trails: absence of jammed phase. Phys Rev Lett 102:108001CrossRefGoogle Scholar
  24. Johnson K, Rossi LF (2006) A mathematical and experimental study of ant foraging trail dynamics. J Theor Biol 241:360–369MathSciNetCrossRefGoogle Scholar
  25. Perna A, Granovskiy B, Garnier S, Nicolis SC, Labédan M, Theraulaz G, Fourcassié V, Sumpter DJT (2012) Individual rules for trail pattern formation in argentine ants (Linepithema humile). PLoS Comput Biol 8(7):e1002592MathSciNetCrossRefGoogle Scholar
  26. Ramsch K, Reid CR, Beekman M, Middendorf M (2012) A mathematical model of foraging in a dynamic environment by trail-laying argentine ants. J Theor Biol 306:32–45MathSciNetCrossRefzbMATHGoogle Scholar
  27. Reid CR, Lattya T, Beekam M (2012) Making a trail: informed argentine ants lead colony to the best food by U-turning coupled with enhanced pheromone laying. Anim Behav 84(6):1579–1587CrossRefGoogle Scholar
  28. Ryan SD (2015) A model for collective dynamics in ant raids. J Math Biol 72:1579–1606MathSciNetCrossRefzbMATHGoogle Scholar
  29. Schweitzer F, Lao K, Family F (1997) Active random walkers simulate trunk trail formation by ants. Biosystems 41:153–166CrossRefGoogle Scholar
  30. Sharpe T, Webb B (1998) Simulated and situated models of chemical trail following in ants. In: Proceedings of the 5th international conference, simulation of adaptive behavior. MIT Press, pp 195–204Google Scholar
  31. Vicsek T, Zafeiris A (2012) Collective motion. Phys Rep 517(3–4):71–140CrossRefGoogle Scholar
  32. Watmough J, Edelstein-Keshet L (1995) A one-dimensional model of trail propagation by army ants. J Math Biol 33:459–476MathSciNetCrossRefzbMATHGoogle Scholar
  33. Wilson EO (1962a) Chemical communication among workers of the fire ant Solenopsis saevissima (Fr. Smith). 1. The organization of mass-foraging. Anim Behav 10(1–2):134–138CrossRefGoogle Scholar
  34. Wilson EO (1962b) Chemical communication among workers of the fire ant Solenopsis saevissima (Fr. Smith). 2. An information analysis of the odour trail. Anim Behav 10(1–2):148–158CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Paulo Amorim
    • 1
    Email author
  • Thierry Goudon
    • 2
  • Fernando Peruani
    • 3
  1. 1.Instituto de MatemáticaUniversidade Federal do Rio de JaneiroRio de JaneiroBrasil
  2. 2.Inria, CNRS, LJADUniversité Côte d’AzurNiceFrance
  3. 3.CNRS, LJADUniversité Côte d’AzurNiceFrance

Personalised recommendations