Journal of Mathematical Biology

, Volume 78, Issue 3, pp 655–682 | Cite as

Counter-propagating wave patterns in a swarm model with memory

  • Angelika ManhartEmail author


Hyperbolic transport-reaction equations are abundant in the description of movement of motile organisms. Here, we focus on a system of four coupled transport-reaction equations that arises from an age-structuring of a species of turning individuals. By modeling how the behavior depends on the time since the last reversal, we introduce a memory effect. The highlight consists of the explicit construction and characterization of counter-propagating traveling waves, patterns which have been observed in bacterial colonies. Stability analysis reveals conditions for the wave formation as well as pulsating-in-time spatially constant solutions.


Traveling waves Hyperbolic equations Viscous limit Myxobacteria Wave formation Age-structured equations Pattern formation 

Mathematics Subject Classification

35L60 35Q70 35B32 35B35 35B36 35B40 35B65 34D20 92D25 92D50 



The author wants to express gratitude to P. Degond and C. Schmeiser for the insightful discussions and A. Mogilner for his support.


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Copyright information

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

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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