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A dynamical approach to phytoplankton blooms
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Rotationally symmetric solutions to the Cahn-Hilliard equation
Shadow system approach to a plankton model generating harmful algal bloom
1. | Department of Mathematics, University of Toyama, Gofuku 3190, Toyama, 930-8555, Japan |
2. | Graduate School of Advanced Mathematical Sciences, Meiji University, Nakano 4-21-1, Tokyo, 164-8525, Japan |
Spatially localized blooms of toxic plankton species have negative impacts on other organisms via the production of toxins, mechanical damage, or by other means. Such blooms are nowadays a worldwide spread environmental issue. To understand the mechanism behind this phenomenon, a two-prey (toxic and nontoxic phytoplankton)-one-predator (zooplankton) Lotka-Volterra system with diffusion has been considered in a previous paper. Numerical results suggest the occurrence of stable non-constant equilibrium solutions, that is, spatially localized blooms of the toxic prey. Such blooms appear for intermediate values of the rate of toxicity $μ$ when the ratio $D$ of the diffusion rates of the predator and the two prey is rather large. In this paper, we consider a one-dimensional limiting system (we call it a shadow system) in $(0,L)$ as $D \to \infty $ and discuss the existence and stability of non-constant equilibrium solutions with large amplitude when $μ$ is globally varied. We also show that the structure of non-constant equilibrium solutions sensitively depends on $L$ as well as $μ$.
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Y. Kan-on,
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Competition for light between toxic and nontoxic strains of the farmful cyanobacterium Microcystis, Applied and Environmental Microbiology, 73 (2007), 2939-2946.
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[12] |
Y. Nishiura,
Global structure of bifurcating solutions of some reaction-diffusion systems, SIMA J. Math. Anal., 13 (1982), 555-593.
doi: 10.1137/0513037. |
[13] |
K. G. Porter and J. D. Orcutt Jr, Nutritional adequacy, manageability, and toxicity as factors that determine food quality of green and blue-green algae for Daphnia, in Evolution and Ecology of Zooplankton Communities (ed. W. C. Kerfoot), University Press of New England, Hanover, NH, USA, (1980), 268-281. |
[14] |
T. Scotti, M. Mimura and J. Y. Wakano,
Avoiding toxic prey may promote harmful algal blooms, Ecological Complexity, 21 (2015), 157-165.
doi: 10.1016/j.ecocom.2014.07.004. |
[15] |
A. M. Turing,
The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. Lond. Ser. B, 237 (1952), 37-72.
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[16] |
Q. Wang, C. Gai and J. Yan,
Qualitative analysis of a Lotka-Volterra competition system with advection, Discreat and Contin. Dyn. Syst. -Series A, 35 (2015), 1239-1284.
doi: 10.3934/dcds.2015.35.1239. |
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L. Zhou and C. V. Pao,
Asymptotic behavior of a competition-diffusion system in population dynamics, Nonlinear Analysis, 6 (1982), 1163-1184.
doi: 10.1016/0362-546X(82)90028-1. |
show all references
References:
[1] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[2] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.
|
[3] |
I. R. Falconer and A. R. Humpage,
Tumor promotion by cyanobacterial toxins, Phycologia, 35 (1996), 74-79.
|
[4] |
R. S. Fulton Ⅲ and H. W. Paerl,
Toxic and inhibitory effects of the blue-green alga Microcystis aeruginosa on herbivorous zooplankton, J. of Plankton Research, 9 (1987), 837-855.
|
[5] |
R. S. Fulton Ⅲ and H. W. Paerl,
Zooplankton feeding selectivity for unicellular and colonial Microcystis aeruginosa, Bull. of Marine Science, 43 (1988), 500-508.
|
[6] |
G. E. Hutchinson,
The paradox of the plankton, Am. Nat., 95 (1961), 137-145.
doi: 10.1086/282171. |
[7] |
E. M. Jochimsen,
Liver failure and death after exposure to microcystins at Hemo-dialysis Center in Brazil, N. Engl. J. Med., 338 (1998), 873-878.
|
[8] |
Y. Kan-on,
Global bifurcation structure of positive stationary solutions for a classical Lotka-Volterra competition model with diffusion, Japan Journal of Industrial and Applied Mathematics, 20 (2003), 285-310.
doi: 10.1007/BF03167424. |
[9] |
W. E. A. Kardinaal,
Competition for light between toxic and nontoxic strains of the farmful cyanobacterium Microcystis, Applied and Environmental Microbiology, 73 (2007), 2939-2946.
|
[10] |
W. Lampert,
Inhibitory and toxic effects of blue-green algae on Daphnia, Int. Revue ges. Hydrobiol., 66 (1981), 285-298.
|
[11] |
W. Lampert,
Further studies on the inhibitory effect of the toxic blue-green Microcystis aeruginosa on the filtering rate of zooplankton, Arch. Hydrobiol., 95 (1982), 207-220.
|
[12] |
Y. Nishiura,
Global structure of bifurcating solutions of some reaction-diffusion systems, SIMA J. Math. Anal., 13 (1982), 555-593.
doi: 10.1137/0513037. |
[13] |
K. G. Porter and J. D. Orcutt Jr, Nutritional adequacy, manageability, and toxicity as factors that determine food quality of green and blue-green algae for Daphnia, in Evolution and Ecology of Zooplankton Communities (ed. W. C. Kerfoot), University Press of New England, Hanover, NH, USA, (1980), 268-281. |
[14] |
T. Scotti, M. Mimura and J. Y. Wakano,
Avoiding toxic prey may promote harmful algal blooms, Ecological Complexity, 21 (2015), 157-165.
doi: 10.1016/j.ecocom.2014.07.004. |
[15] |
A. M. Turing,
The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. Lond. Ser. B, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[16] |
Q. Wang, C. Gai and J. Yan,
Qualitative analysis of a Lotka-Volterra competition system with advection, Discreat and Contin. Dyn. Syst. -Series A, 35 (2015), 1239-1284.
doi: 10.3934/dcds.2015.35.1239. |
[17] |
L. Zhou and C. V. Pao,
Asymptotic behavior of a competition-diffusion system in population dynamics, Nonlinear Analysis, 6 (1982), 1163-1184.
doi: 10.1016/0362-546X(82)90028-1. |



















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