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


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.


Ant navigation Individual-based model Pheromones Foraging Animal movement 

Mathematics Subject Classification

92D50 70K20 34D20 



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.


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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

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