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**1 - 3**of**3**### Wavetrain solutions of a reaction-diffusion-advection model of mussel-algae interaction

, 2015

"... We consider a system of coupled partial differential equations modeling the interaction of mussels and algae in advective environments. A key parameter is the relative rate of advection of the algae concentration and diffusion of the mussel species. When advection dominates diffusion, one observes l ..."

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We consider a system of coupled partial differential equations modeling the interaction of mussels and algae in advective environments. A key parameter is the relative rate of advection of the algae concentration and diffusion of the mussel species. 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 pe-riodic 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 deal with a loss of normal hy-perbolicity, and the application of a generalized Exchange Lemma to handle loss-of-stability turning points. In particular, our analysis uncovers logarithmic (switchback) corrections to the leading-order solution.

### A Remark on Geometric Desingularization of a Non-Hyperbolic Point using Hyperbolic Space

, 2014

"... ar ..."

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### From random Poincare ́ maps to stochastic mixed-mode-oscillation patterns

, 2013

"... We quantify the effect of Gaussian white noise on fast–slow dynamical systems with one fast and two slow variables, which display mixed-mode oscillations owing to the presence of a folded-node singularity. The stochastic system can be described by a continuous-space, discrete-time Markov chain, reco ..."

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We quantify the effect of Gaussian white noise on fast–slow dynamical systems with one fast and two slow variables, which display mixed-mode oscillations owing to the presence of a folded-node singularity. The stochastic system can be described by a continuous-space, discrete-time Markov chain, recording the returns of sample paths to a Poincare ́ section. We provide estimates on the kernel of this Markov chain, depending on the system parameters and the noise intensity. These results yield predictions on the observed random mixed-mode oscillation patterns. An unexpected result of the analysis is that in certain cases, noise may increase the number of small-amplitude oscillations between consecutive large-amplitude oscillations. Mathematical Subject Classification. 37H20, 34E17 (primary), 60H10 (secondary)