Article:
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F. Rupp & J. Scheurle (2014): The Dynamics of the Jellyfish Joyride: Mathematical Discussion of the Causes Leading to Blooming, Mathematical Methods in the Applied Sciences, accepted. 


Abstract:
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Dramatic increases in jellyfish populations which lead to the collapse of formally healthy ecosystems are repeatedly reported
from many different sites. Due to their devastating effects on fish populations the understanding of the
causes for such bloomings are of major ecological as well as economical importance. Following [a], we set up a twodimensional
predator-prey model for the fish-jellyfish interactions by taking fishery as well as environmental conditions into
account. It assumes fish as the dominant predator species. By totally analytic means, we completely classify all equilibria
in terms of existence and non-linear stability, and give a description of this system's non-linear global dynamics. We
analytically study the non-equilibrium dynamics and detect homoclinic, Andronov-Hopf, and Takens-Bogdanov bifurcations
as well as non-existence of periodic and homoclinic orbits in certain relevant regions of the phase space. A hereby found, and
analytically rigorously stated, funnel phenomenon gives rise to a mathematical explanation of jellysh blooming beyond
the typical one in terms of large-amplitude limit cycles based on the well-known Rosenzweig-MacArthur equations. Also, we
illustrate the plethora of bifurcation scenarios by numerical results.

[a]: F. Rupp & J. Scheurle (2013): Analysis of a Mathematical Model for Jellyfish Blooms
     and the Cambric Fish Invasion, Dynamical Systems and Differential Equations, DCDS
     Supplement 2013 Proceedings of the 9th AIMS International Conference (Orlando, USA),
     pp. 663-672.