American Institute of Mathematical Sciences

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2013, 2013(special): 663-672. doi: 10.3934/proc.2013.2013.663

Analysis of a mathematical model for jellyfish blooms and the cambric fish invasion

Florian Rupp 1, and  Jürgen Scheurle 1,

1. 

Lehrstuhl für Höhere Mathematik und Analytische Mechanik, Technische Universität München, Fakultät für Mathematik, D-85747 Garching, Germany

Received  August 2012 Revised  May 2013 Published  November 2013

Dramatic increases in jellyfish populations which lead to the collapse of formerly healthy ecosystems are repeatedly reported from many different sites, cf. [6,8,14]. Due to their devastating effects on fishery the understanding of the causes for such a blooming are of major ecological as well as economical importance. Assuming fish as the dominant predator species we model a combined two species system subject to constant environmental conditions. By totally analytic means we completely classify all biologically relevant equilibria in terms of existence and Lyapunov stability, and give a complete description of this system's non-linear global dynamics supported by numerical simulations. This approach complements, from a systematic point of view, the studies given in the literature to better understand jellyfish blooms.
Keywords: Population dynamics, stability, bifurcation..
Mathematics Subject Classification: Primary: 37N25, 37G10, 37G15, 37C75; Secondary: 34Cx.
Citation: Florian Rupp, Jürgen Scheurle. Analysis of a mathematical model for jellyfish blooms and the cambric fish invasion. Conference Publications, 2013, 2013 (special) : 663-672. doi: 10.3934/proc.2013.2013.663
References:
[1]

A. Bakun and S. J. Weeks, Adverse feedback sequences in exploited marine systems: are deliberate interruptive actions warranted?,, Fish and Fisheries, 7 (2006), 316. Google Scholar

[2]

M.J. Gibbons and A.J. Richardson, Patterns of jellyfish abundance in the north atlantic,, Hydrobiologia, 616 (2009), 51. Google Scholar

[3]

M. Haraldsson, K. Tönnesson, P. Tiselius, T.F. Thingstad and D.L. Aksnes, Relationship between fish and jellyfish as a function of eutrophication and water clarity,, Marine Ecology Progress Series, 471 (2012), 73. Google Scholar

[4]

T. Legovic, A recent increase in jellyfish populations: a predator-prey model and its implications,, Ecological Modelling, 38 (1987), 243. Google Scholar

[5]

C.H. Lucas, Population dynamics of aurelia aurita (scyphozoa) from an isolated brackish lake, with particular reference to sexual reproduction,, J. Plankton Research, 18 (1996), 987. Google Scholar

[6]

M. Kawahara, S. Uye, K. Ohtsu, and H. Iizumi, Unusual population explosion of the giant jellyfish nemopilema nomurai (scyphozoa: rhizostomeae) in east asian waters,, Marine Ecology Progress Series, 307 (2006), 161. Google Scholar

[7]

A. Malej and M. Malej, Population dynamics of the jellyfish pelagia noctiluca,, in, (1992), 215. Google Scholar

[8]

C.E. Mills, Jellyfish blooms: are populations increasing globally in response to changing ocean conditions?,, Hydrobiologia, 451 (2001), 55. Google Scholar

[9]

C. Möllmann, B. Müller-Karulis, G. Kornilovs, and M.A. St John, Effects of climate and overfishing on zooplankton dynamics and ecosystem structure: regime shifts, trophic cascade, and feedback loops in a simple ecosystem,, ICES J. Marine Sciences, 65 (2008), 302. Google Scholar

[10]

A.S. Nielsen, A.W. Pedersen, and H.U. Riisgard, Implications of density driven currents for interaction between jellyfish (aurelia aurita) and zooplankton in a danish fjord,, Sarsia, 82 (1997), 297. Google Scholar

[11]

N.J. Olesen, K. Frandsen, and H.U. Riisgard, Population dynamics, growth and energetics of jellyfish aurelia aurita in a shallow fjord,, Marine Ecology Progress Series, 105 (1994), 9. Google Scholar

[12]

M.L.D. Palomares and D. Pauly, The growth of jellyfishes,, Hydrobiologica, 616 (2009), 11. Google Scholar

[13]

K.A. Pitt, K. Koop, D. Rissik, and M.J. Kingsford, The ecology of scyphozoan jellyfish in lake illawara,, Wetlands (Australia), 21 (2004), 118. Google Scholar

[14]

A.J. Richardson, A. Bakun, G.C. Hays, and M.J. Gibbons, The jellyfish joyride: causes, consequences and management responses to a more gelatinous future,, Trends in Ecology and Evolution, 24 (2009), 312. Google Scholar

[15]

H.U. Riisgard, C.V. Madsen, C. Barth-Jensen, and J.E. Purcell, Population dynamics and zooplankton-predation impact of the indigenous scyphozoan aurelia aurita and the invasive ctenophore mnemiopsis leidyi in limfjorden (Denmark),, Aquatic Invasions, 7 (2011), 147. Google Scholar

[16]

S. Uye, Human forcing of the copepod-fish-jellyfish triangular trophic relationship,, Hydrobiologia, 666 (2011), 71. Google Scholar

[17]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems & Chaos,, Springer (1990)., (1990). Google Scholar

show all references

References:
[1]

A. Bakun and S. J. Weeks, Adverse feedback sequences in exploited marine systems: are deliberate interruptive actions warranted?,, Fish and Fisheries, 7 (2006), 316. Google Scholar

[2]

M.J. Gibbons and A.J. Richardson, Patterns of jellyfish abundance in the north atlantic,, Hydrobiologia, 616 (2009), 51. Google Scholar

[3]

M. Haraldsson, K. Tönnesson, P. Tiselius, T.F. Thingstad and D.L. Aksnes, Relationship between fish and jellyfish as a function of eutrophication and water clarity,, Marine Ecology Progress Series, 471 (2012), 73. Google Scholar

[4]

T. Legovic, A recent increase in jellyfish populations: a predator-prey model and its implications,, Ecological Modelling, 38 (1987), 243. Google Scholar

[5]

C.H. Lucas, Population dynamics of aurelia aurita (scyphozoa) from an isolated brackish lake, with particular reference to sexual reproduction,, J. Plankton Research, 18 (1996), 987. Google Scholar

[6]

M. Kawahara, S. Uye, K. Ohtsu, and H. Iizumi, Unusual population explosion of the giant jellyfish nemopilema nomurai (scyphozoa: rhizostomeae) in east asian waters,, Marine Ecology Progress Series, 307 (2006), 161. Google Scholar

[7]

A. Malej and M. Malej, Population dynamics of the jellyfish pelagia noctiluca,, in, (1992), 215. Google Scholar

[8]

C.E. Mills, Jellyfish blooms: are populations increasing globally in response to changing ocean conditions?,, Hydrobiologia, 451 (2001), 55. Google Scholar

[9]

C. Möllmann, B. Müller-Karulis, G. Kornilovs, and M.A. St John, Effects of climate and overfishing on zooplankton dynamics and ecosystem structure: regime shifts, trophic cascade, and feedback loops in a simple ecosystem,, ICES J. Marine Sciences, 65 (2008), 302. Google Scholar

[10]

A.S. Nielsen, A.W. Pedersen, and H.U. Riisgard, Implications of density driven currents for interaction between jellyfish (aurelia aurita) and zooplankton in a danish fjord,, Sarsia, 82 (1997), 297. Google Scholar

[11]

N.J. Olesen, K. Frandsen, and H.U. Riisgard, Population dynamics, growth and energetics of jellyfish aurelia aurita in a shallow fjord,, Marine Ecology Progress Series, 105 (1994), 9. Google Scholar

[12]

M.L.D. Palomares and D. Pauly, The growth of jellyfishes,, Hydrobiologica, 616 (2009), 11. Google Scholar

[13]

K.A. Pitt, K. Koop, D. Rissik, and M.J. Kingsford, The ecology of scyphozoan jellyfish in lake illawara,, Wetlands (Australia), 21 (2004), 118. Google Scholar

[14]

A.J. Richardson, A. Bakun, G.C. Hays, and M.J. Gibbons, The jellyfish joyride: causes, consequences and management responses to a more gelatinous future,, Trends in Ecology and Evolution, 24 (2009), 312. Google Scholar

[15]

H.U. Riisgard, C.V. Madsen, C. Barth-Jensen, and J.E. Purcell, Population dynamics and zooplankton-predation impact of the indigenous scyphozoan aurelia aurita and the invasive ctenophore mnemiopsis leidyi in limfjorden (Denmark),, Aquatic Invasions, 7 (2011), 147. Google Scholar

[16]

S. Uye, Human forcing of the copepod-fish-jellyfish triangular trophic relationship,, Hydrobiologia, 666 (2011), 71. Google Scholar

[17]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems & Chaos,, Springer (1990)., (1990). Google Scholar

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