Wavetrain solutions of a reaction-diffusion-advection model of mussel-algae interaction

Matt Holzer, Nikola Popovic

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

We consider a system of coupled partial differential equations modeling the interaction of mussels and algae in advective environments. A key parameter in the equations is the ratio of the diffusion rate of the mussel species and the advection rate of the algal concentration. When advection dominates diffusion, one observes large-amplitude solutions representing bands of mussels propagating slowly in the upstream direction. Here, we prove the existence of a family of such periodic wavetrain solutions. Our proof relies on geometric singular perturbation theory to construct these solutions as periodic orbits of the associated traveling wave equations in the large-advection--small-diffusion limit. The construction encounters a number of mathematical obstacles which necessitate a compactification of phase space, geometric desingularization to remedy a loss of normal hyperbolicity, and the application of a generalized exchange lemma at a loss-of-stability turning point. In particular, our analysis uncovers logarithmic (switchback) corrections to the leading-order solution.
Original languageEnglish
Pages (from-to)431-478
Number of pages48
JournalSiam Journal on Applied Dynamical Systems
Volume16
Issue number1
DOIs
Publication statusPublished - 8 Feb 2017

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