TY - JOUR
T1 - Large scale patterns in mussel beds
T2 - stripes or spots?
AU - Bennett, Jamie J.R.
AU - Sherratt, Jonathan A.
N1 - Funding Information:
Jamie J. R. Bennett’s work was supported by The Maxwell Institute Graduate School in Analysis and Its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (Grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University, and the University of Edinburgh.
Funding Information:
We would like to thank Johan van de Koppel for kindly providing the photograph in Fig.?1.
Publisher Copyright:
© 2018, The Author(s).
PY - 2019/2/15
Y1 - 2019/2/15
N2 - An aerial view of an intertidal mussel bed often reveals large scale striped patterns aligned perpendicular to the direction of the tide; dense bands of mussels alternate periodically with near bare sediment. Experimental work led to the formulation of a set of coupled partial differential equations modelling a mussel–algae interaction, which proved pivotal in explaining the phenomenon. The key class of model solutions to consider are one-dimensional periodic travelling waves (wavetrains) that encapsulate the abundance of peak and trough mussel densities observed in practice. These solutions may, or may not, be stable to small perturbations, and previous work has focused on determining the ecologically relevant (stable) wavetrain solutions in terms of model parameters. The aim of this paper is to extend this analysis to two space dimensions by considering the full stripe pattern solution in order to study the effect of transverse two-dimensional perturbations—a more true to life problem. Using numerical continuation techniques, we find that some striped patterns that were previously deemed stable via the consideration of the associated wavetrain solution, are in fact unstable to transverse two-dimensional perturbations; and numerical simulation of the model shows that they break up to form regular spotted patterns. In particular, we show that break up of stripes into spots is a consequence of low tidal flow rates. Our consideration of random algal movement via a dispersal term allows us to show that a higher algal dispersal rate facilitates the formation of stripes at lower flow rates, but also encourages their break up into spots. We identify a novel hysteresis effect in mussel beds that is a consequence of transverse perturbations.
AB - An aerial view of an intertidal mussel bed often reveals large scale striped patterns aligned perpendicular to the direction of the tide; dense bands of mussels alternate periodically with near bare sediment. Experimental work led to the formulation of a set of coupled partial differential equations modelling a mussel–algae interaction, which proved pivotal in explaining the phenomenon. The key class of model solutions to consider are one-dimensional periodic travelling waves (wavetrains) that encapsulate the abundance of peak and trough mussel densities observed in practice. These solutions may, or may not, be stable to small perturbations, and previous work has focused on determining the ecologically relevant (stable) wavetrain solutions in terms of model parameters. The aim of this paper is to extend this analysis to two space dimensions by considering the full stripe pattern solution in order to study the effect of transverse two-dimensional perturbations—a more true to life problem. Using numerical continuation techniques, we find that some striped patterns that were previously deemed stable via the consideration of the associated wavetrain solution, are in fact unstable to transverse two-dimensional perturbations; and numerical simulation of the model shows that they break up to form regular spotted patterns. In particular, we show that break up of stripes into spots is a consequence of low tidal flow rates. Our consideration of random algal movement via a dispersal term allows us to show that a higher algal dispersal rate facilitates the formation of stripes at lower flow rates, but also encourages their break up into spots. We identify a novel hysteresis effect in mussel beds that is a consequence of transverse perturbations.
KW - Activator-inhibitor
KW - Mussels
KW - Mussel–algae interaction
KW - Pattern formation
KW - Self-organisation
KW - Tranverse perturbations
KW - Turing–Hopf bifurcation
KW - Two-dimensional stability
UR - http://www.scopus.com/inward/record.url?scp=85053379792&partnerID=8YFLogxK
U2 - 10.1007/s00285-018-1293-z
DO - 10.1007/s00285-018-1293-z
M3 - Article
C2 - 30187225
AN - SCOPUS:85053379792
SN - 0303-6812
VL - 78
SP - 815
EP - 835
JO - Journal of Mathematical Biology
JF - Journal of Mathematical Biology
IS - 3
ER -