Phytoplankton growth in oligotrophic oceans: Weakly nonlinear theory
Arjen Doelman, Mathematisch Instituut, Leiden University (June 28, 2011)
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In this talk, we will consider the problem of bifurcating DCMs under nutrient-light co-limitation from a weakly nonlinear point of view. In particular, we will work with the plankton-nutrient model in one spatial dimension introduced in A. Zagaris's talk and investigate the weakly nonlinear stability problem for these bifurcating DCMs.
The most intriguing mathematical aspect of this problem concerns the existence of an infinite number of eigenvalues tightly clustered around the origin. Although the corresponding modes are latent (non-bifurcating), they have to be included in the analysis as they interact nonlinearly with active (bifurcating) modes.
We will present explicit asymptotic results valid both close to and far from the bifurcation point, verifying that the bifurcating DCM is stable. Then, we will see that the latent modes have a decisive impact on the dynamics, solely through nonlinear interactions and although a strictly linear point of view dictates that they should be utterly irrelevant. In fact, the bifurcating stable DCM is soon annihilated in a saddle-node bifurcation induced by these latent modes, offering its place to a secondary pattern.