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

Previous Article
Dynamically consistent discrete-time SI and SIS epidemic models
PROC Home
This Issue
Next Article
Stochastic heat equations with cubic nonlinearity and additive space-time noise in 2D

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

Abstract
Related Papers

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

[1]	Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735
[2]	Pierre Magal. Global stability for differential equations with homogeneous nonlinearity and application to population dynamics. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 541-560. doi: 10.3934/dcdsb.2002.2.541
[3]	Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367
[4]	Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451
[5]	Wei Feng, Jody Hinson. Stability and pattern in two-patch predator-prey population dynamics. Conference Publications, 2005, 2005 (Special) : 268-279. doi: 10.3934/proc.2005.2005.268
[6]	Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics. Conference Publications, 2013, 2013 (special) : 515-524. doi: 10.3934/proc.2013.2013.515
[7]	Salvatore Rionero. A nonlinear $L^2$-stability analysis for two-species population dynamics with dispersal. Mathematical Biosciences & Engineering, 2006, 3 (1) : 189-204. doi: 10.3934/mbe.2006.3.189
[8]	Wei Feng, Xin Lu, Richard John Donovan Jr.. Population dynamics in a model for territory acquisition. Conference Publications, 2001, 2001 (Special) : 156-165. doi: 10.3934/proc.2001.2001.156
[9]	Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233
[10]	Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[11]	Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657
[12]	Liming Cai, Jicai Huang, Xinyu Song, Yuyue Zhang. Bifurcation analysis of a mosquito population model for proportional releasing sterile mosquitoes. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-17. doi: 10.3934/dcdsb.2019139
[13]	Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53
[14]	Masahiro Yamaguchi, Yasuhiro Takeuchi, Wanbiao Ma. Population dynamics of sea bass and young sea bass. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 833-840. doi: 10.3934/dcdsb.2004.4.833
[15]	MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777
[16]	Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633
[17]	Dianmo Li, Zengxiang Gao, Zufei Ma, Baoyu Xie, Zhengjun Wang. Two general models for the simulation of insect population dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 623-628. doi: 10.3934/dcdsb.2004.4.623
[18]	Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643
[19]	Hui Wan, Huaiping Zhu. A new model with delay for mosquito population dynamics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1395-1410. doi: 10.3934/mbe.2014.11.1395
[20]	B. E. Ainseba, W. E. Fitzgibbon, M. Langlais, J. J. Morgan. An application of homogenization techniques to population dynamics models. Communications on Pure & Applied Analysis, 2002, 1 (1) : 19-33. doi: 10.3934/cpaa.2002.1.19

Impact Factor:

American Institute of Mathematical Sciences