- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Collective chemotactic search

Adam Wysocki1,*, Hugues Meyer1,2,†, and Heiko Rieger1,‡

  • *Contact author: a.wysocki@lusi.uni-sb.de
  • †Contact author: hugues.meyer@nottingham.ac.uk
  • ‡Contact author: heiko.rieger@uni-saarland.de
PDF

Phys. Rev. Research 8, 033201 – Published 18 August, 2026

DOI: https://doi.org/10.1103/qdlc-249z

Abstract

We investigate collective search by self-propelled agents that are repelled by their own chemically produced trails, a minimal mechanism that simultaneously generates indirect interactions and memory. Using lattice and off-lattice models, we show that this mechanism enhances search efficiency through two distinct regimes. In a weak-memory regime, chemical cues are short-lived and interactions primarily promote spatial separation between agents. This reduces redundant exploration while preserving mobility, leading to an optimal trade-off between spatial order and persistence. In a strong-memory regime, long-lived chemical trails induce effective self-avoidance, strongly suppressing revisits and long search times. Here, optimal search occurs at finite memory strength: Permanently persistent trails lead to self-caging, while moderate memory enables efficient exploration. At higher densities, overlapping chemical trails give rise to a collective self-avoidance mechanism that yields substantial cooperative speedup without global spatial order. Together, these results show how chemically mediated memory and interactions can optimize collective search across distinct dynamical regimes.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (81)

  1. D. Grebenkov, R. Metzler, and G. Oshanin, Target Search Problems (Springer, Cham, 2024).
  2. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1992).
  3. S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, 2001).
  4. P. C. Bressloff, Stochastic Processes in Cell Biology (Springer, Heidelberg, 1992).
  5. M. Smoluchowski, Versuch einer mathematischen theorie der koagulationskinetik kolloider lösungen, Z. Phys. Chem. 92, 129 (1917).
  6. F. C. Collins and G. E. Kimball, Diffusion-controlled reaction rates, J. Colloid Sci. 4, 425 (1949).
  7. S. Rice, Diffusion-Limited Reactions (Elsevier, Amsterdam, 1985).
  8. O. G. Berg, R. B. Winter, and P. H. Von Hippel, Diffusion-driven mechanisms of protein translocation on nucleic acids. 1. Models and theory, Biochemistry 20, 6929 (1981).
  9. J. Gorman and E. C. Greene, Visualizing one-dimensional diffusion of proteins along DNA, Nat. Struct. Mol. Biol. 15, 768 (2008).
  10. H. C. Berg, E. Coli in Motion (Springer, Heidelberg, 2005).
  11. M. F. Krummel, F. Bartumeus, and A. Gérard, T cell migration, search strategies and mechanisms, Nat. Rev. Immunol. 16, 193 (2016).
  12. M. Hassell and T. Southwood, Foraging strategies of insects, Annu. Rev. Ecol. Syst. 9, 75 (1978).
  13. J. F. Traniello, Foraging strategies of ants, Annu. Rev. Entomol. 34, 191 (1989).
  14. A. M. Edwards, R. A. Phillips, N. W. Watkins, M. P. Freeman, E. J. Murphy, V. Afanasyev, S. V. Buldyrev, M. G. da Luz, E. P. Raposo, H. E. Stanley, et al., Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer, Nature (London) 449, 1044 (2007).
  15. T. T. Hills, C. Kalff, and J. M. Wiener, Adaptive Lévy processes and area-restricted search in human foraging, PLoS One 8, e60488 (2013).
  16. M. Senanayake, I. Senthooran, J. C. Barca, H. Chung, J. Kamruzzaman, and M. Murshed, Search and tracking algorithms for swarms of robots: A survey, Robot. Auton. Syst. 75, 422 (2016).
  17. D. S. Drew, Multi-agent systems for search and rescue applications, Curr. Robot. Rep. 2, 189 (2021).
  18. M. A. Lomholt, K. Tal, R. Metzler, and K. Joseph, Lévy strategies in intermittent search processes are advantageous, Proc. Natl. Acad. Sci. USA 105, 11055 (2008).
  19. V. Tejedor, R. Voituriez, and O. Bénichou, Optimizing persistent random searches, Phys. Rev. Lett. 108, 088103 (2012).
  20. O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, Intermittent search strategies, Rev. Mod. Phys. 83, 81 (2011).
  21. H. Meyer and H. Rieger, Optimal non-Markovian search strategies with n-step memory, Phys. Rev. Lett. 127, 070601 (2021).
  22. A. Klimek and R. R. Netz, Optimal non-Markovian composite search algorithms for spatially correlated targets, Europhys. Lett. 139, 32003 (2022).
  23. L. Kusmierz, S. N. Majumdar, S. Sabhapandit, and G. Schehr, First order transition for the optimal search time of Lévy flights with resetting, Phys. Rev. Lett. 113, 220602 (2014).
  24. A. Chechkin and I. M. Sokolov, Random search with resetting: A unified renewal approach, Phys. Rev. Lett. 121, 050601 (2018).
  25. P. C. Bressloff, Search processes with stochastic resetting and multiple targets, Phys. Rev. E 102, 022115 (2020).
  26. A. Stanislavsky and A. Weron, Optimal non-Gaussian search with stochastic resetting, Phys. Rev. E 104, 014125 (2021).
  27. O. Tal-Friedman, A. Pal, A. Sekhon, S. Reuveni, and Y. Roichman, Experimental realization of diffusion with stochastic resetting, J. Phys. Chem. Lett. 11, 7350 (2020).
  28. A. Altshuler, O. L. Bonomo, N. Gorohovsky, S. Marchini, E. Rosen, O. Tal-Friedman, S. Reuveni, and Y. Roichman, Environmental memory facilitates search with home returns, Phys. Rev. Res. 6, 023255 (2024).
  29. S. Redner and P. L. Krapivsky, Capture of the lamb: Diffusing predators seeking a diffusing prey, Am. J. Phys. 67, 1277 (1999).
  30. G. Oshanin, O. Vasilyev, P. L. Krapivsky, and J. Klafter, Survival of an evasive prey, Proc. Natl. Acad. Sci. USA 106, 13696 (2009).
  31. C. Mejía-Monasterio, G. Oshanin, and G. Schehr, First passages for a search by a swarm of independent random searchers, J. Stat. Mech.: Theory Exp. (2011) P06022.
  32. A. Nandi and D. Das, Capture of a diffusive prey by multiple predators in confined space, Phys. Rev. E 102, 062109 (2020).
  33. D. S. Grebenkov, R. Metzler, and G. Oshanin, From single-particle stochastic kinetics to macroscopic reaction rates: Fastest first-passage time of n random walkers, New J. Phys. 22, 103004 (2020).
  34. D. S. Grebenkov, Depletion of resources by a population of diffusing species, Phys. Rev. E 105, 054402 (2022).
  35. N. P. Tani, A. Blatt, D. A. Quint, and A. Gopinathan, Optimal cooperative searching using purely repulsive interactions, J. Theor. Biol. 361, 159 (2014).
  36. S. Ro, J. Yi, and Y. W. Kim, Target searches of interacting Brownian particles in dilute systems, Phys. Rev. E 107, 064143 (2023).
  37. R. Martínez-García, J. M. Calabrese, T. Mueller, K. A. Olson, and C. López, Optimizing the search for resources by sharing information: Mongolian gazelles as a case study, Phys. Rev. Lett. 110, 248106 (2013).
  38. J. d'Alessandro, A. Barbier-Chebbah, V. Cellerin, O. Benichou, R. Mège, R. Voituriez, and B. Ladoux, Cell migration guided by long-lived spatial memory, Nat. Commun. 12, 4118 (2021).
  39. J. Taktikos, V. Zaburdaev, and H. Stark, Modeling a self-propelled autochemotactic walker, Phys. Rev. E 84, 041924 (2011).
  40. B. Liebchen, D. Marenduzzo, and M. E. Cates, Phoretic interactions generically induce dynamic clusters and wave patterns in active colloids, Phys. Rev. Lett. 118, 268001 (2017).
  41. W. T. Kranz and R. Golestanian, Trail-mediated self-interaction, J. Chem. Phys. 150, 214111 (2019).
  42. Z. Mokhtari, R. I. A. Patterson, and F. Höfling, Spontaneous trail formation in populations of auto-chemotactic walkers, New J. Phys. 24, 013012 (2022).
  43. A. Barbier-Chebbah, O. Bénichou, and R. Voituriez, Self-interacting random walks: Aging, exploration, and first-passage times, Phys. Rev. X 12, 011052 (2022).
  44. H. Meyer and H. Rieger, Alignment interaction and band formation in assemblies of autochemorepulsive walkers, Phys. Rev. E 108, 034604 (2023).
  45. X. Fu, S. Kato, J. Long, H. H. Mattingly, C. He, D. C. Vural, S. W. Zucker, and T. Emonet, Spatial self-organization resolves conflicts between individuality and collective migration, Nat. Commun. 9, 2177 (2018).
  46. B. A. David and P. Kubes, Exploring the complex role of chemokines and chemoattractants in vivo on leukocyte dynamics, Immunol. Rev. 289, 9 (2019).
  47. R. H. Insall, P. Paschke, and L. Tweedy, Steering yourself by the bootstraps: How cells create their own gradients for chemotaxis, Trends Cell Biol. 32, 585 (2022).
  48. E. Donà, J. D. Barry, G. Valentin, C. Quirin, A. Khmelinskii, A. Kunze, S. Durdu, L. R. Newton, A. Fernandez-Minan, W. Huber, M. Knop, and D. Gilmour, Directional tissue migration through a self-generated chemokine gradient, Nature (London) 503, 285 (2013).
  49. B. V. Hokmabad, J. Agudo-Canalejo, S. Saha, R. Golestanian, and C. C. Maass, Chemotactic self-caging in active emulsions, Proc. Natl. Acad. Sci. USA 119, e2122269119 (2022).
  50. B. Nakayama, H. Nagase, H. Takahashi, Y. Saito, S. Hatayama, K. Makino, E. Yamamoto, and T. Saiki, Tunable pheromone interactions among microswimmers, Proc. Natl. Acad. Sci. USA 120, e2213713120 (2023).
  51. B. Liebchen and H. Löwen, Synthetic chemotaxis and collective behavior in active matter, Acc. Chem. Res. 51, 2982 (2018).
  52. J. Grauer, H. Löwen, A. Be'er, and B. Liebchen, Swarm hunting and cluster ejections in chemically communicating active mixtures, Sci. Rep. 10, 5594 (2020).
  53. E. Moreno, R. Großmann, C. Beta, and S. Alonso, From single to collective motion of social amoebae: A computational study of interacting cells, Front. Phys. 9, 750187 (2022).
  54. A. V. Ivlev, J. Bartnick, M. Heinen, C.-R. Du, V. Nosenko, and H. Löwen, Statistical mechanics where Newton's third law is broken, Phys. Rev. X 5, 011035 (2015).
  55. Z. You, A. Baskaran, and M. C. Marchetti, Nonreciprocity as a generic route to traveling states, Proc. Natl. Acad. Sci. USA 117, 19767 (2020).
  56. S. Saha, J. Agudo-Canalejo, and R. Golestanian, Scalar active mixtures: The nonreciprocal Cahn-Hilliard model, Phys. Rev. X 10, 041009 (2020).
  57. M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature (London) 592, 363 (2021).
  58. J. P. Banerjee, R. Mandal, D. S. Banerjee, S. Thutupalli, and M. Rao, Unjamming and emergent nonreciprocity in active ploughing through a compressible viscoelastic fluid, Nat. Commun. 13, 4533 (2022).
  59. K. L. Kreienkamp and S. H. L. Klapp, Nonreciprocal alignment induces asymmetric clustering in active mixtures, Phys. Rev. Lett. 133, 258303 (2024).
  60. M. Mangeat, S. Chatterjee, J. D. Noh, and H. Rieger, Emergent complex phases in a discrete flocking model with reciprocal and non-reciprocal interactions, Commun. Phys. 8, 186 (2025).
  61. J. Choi, J. D. Noh, and H. Rieger, Flocking with random nonreciprocal interactions, Phys. Rev. Res. 7, L042059 (2025).
  62. A. T. Mohite and H. Rieger, Stochastic thermodynamics of nonreciprocally interacting particles and fields, Phys. Rev. E 113, 064136 (2026).
  63. C.-U. Woo, H. Rieger, and J. D. Noh, Collective behavior in the nonreciprocal multi-species Vicsek model, arXiv:2512.18756.
  64. C.-U. Woo, J. D. Noh, and H. Rieger, Extensive spatio-temporal chaos in non-reciprocal flocking, arXiv:2604.06462.
  65. J. Taktikos, V. Zaburdaev, and H. Stark, Collective dynamics of model microorganisms with chemotactic signaling, Phys. Rev. E 85, 051901 (2012).
  66. D. J. Amit, G. Parisi, and L. Peliti, Asymptotic behavior of the “true” self-avoiding walk, Phys. Rev. B 27, 1635 (1983).
  67. P. Grassberger, Self-trapping self-repelling random walks, Phys. Rev. Lett. 119, 140601 (2017).
  68. S. Ro and Y. W. Kim, Parallel random target searches in a confined space, Phys. Rev. E 96, 012143 (2017).
  69. S. Ro and Y. W. Kim, Optimal searcher distribution for parallel random target searches, Phys. Rev. E 106, 024101 (2022).
  70. R. Monchaux, M. Bourgoin, and A. Cartellier, Preferential concentration of heavy particles: A Voronoï analysis, Phys. Fluids 22, 103304 (2010).
  71. J.-S. Ferenc and Z. Néda, On the size distribution of Poisson Voronoi cells, Physica A 385, 518 (2007).
  72. G. A. Arteca and S. Zhang, Quantitative measure of folding in two-dimensional polymers, Phys. Rev. E 59, 4209 (1999).
  73. L. Tweedy, D. A. Knecht, G. M. Mackay, and R. H. Insall, Self-generated chemoattractant gradients: Attractant depletion extends the range and robustness of chemotaxis, PLoS Biol. 14, e1002404 (2016).
  74. L. Tweedy, P. A. Thomason, P. I. Paschke, K. Martin, L. M. Machesky, M. Zagnoni, and R. H. Insall, Seeing around corners: Cells solve mazes and respond at a distance using attractant breakdown, Science 369, eaay9792 (2020).
  75. D. R. Zusman, A. E. Scott, Z. Yang, and J. R. Kirby, Chemosensory pathways, motility and development in Myxococcus xanthus, Nat. Rev. Microbiol. 5, 862 (2007).
  76. M. R. D'Orsogna, M. A. Suchard, and T. Chou, Interplay of chemotaxis and chemokinesis mechanisms in bacterial dynamics, Phys. Rev. E 68, 021925 (2003).
  77. H. Meyer and H. Rieger, Optimal number of agents in a collective search and when to launch them, Phys. Rev. E 111, 064112 (2025).
  78. F. Cichos, K. Gustavsson, B. Mehlig, and G. Volpe, Machine learning for active matter, Nat. Mach. Intell. 2, 94 (2020).
  79. C. Kaspar, B. J. Ravoo, W. G. van der Wiel, S. V. Wegner, and W. H. P. Pernice, The rise of intelligent matter, Nature (London) 594, 345 (2021).
  80. L. Piro, B. Mahault, and R. Golestanian, Optimal navigation of microswimmers in complex and noisy environments, New J. Phys. 24, 093037 (2022).
  81. M. Nasiri, H. Löwen, and B. Liebchen, Optimal active particle navigation meets machine learning, Europhys. Lett. 142, 17001 (2023).

Outline

Information

Phys. Rev. Research 8, 033201– Published 18 August, 2026
  • Received 6 September 2024
  • Accepted 26 June 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation