A reaction-diffusion system and its shadow system describing harmful algal blooms

Main Article Content

Shintaro kondo
Masayasu Mimura

Abstract

The occurrence of harmful algal blooms (HAB) in ecosystems is a worldwide environmental issue that currently needs to be addressed. An attempt to theoretically understand the mechanism behind the formation of HAB has led to the proposal of a reaction-diffusion model of the Lotka--Volterra type. In particular, a shadow system, as a limiting system of the model in which the diffusion rate tends to infinity, has been proposed to study whether or not stable nonconstant equilibrium solutions of the system exist, because these solutions are mathematically associated with HAB. In this paper, we discuss the convergence property between solutions of the full system and its shadow system from the point of view of an evolutional problem.

Article Details

How to Cite
kondo, S., & Mimura, M. (2016). A reaction-diffusion system and its shadow system describing harmful algal blooms. Tamkang Journal of Mathematics, 47(1). https://doi.org/10.5556/j.tkjm.47.2016.1916
Section
Special Issue
Author Biographies

Shintaro kondo

Meiji Institute for Advanced Study ofMathematical Sciences,Meiji University, 4-21-1 Nakano Nakano-ku, Tokyo.

Masayasu Mimura

Graduate School of Advanced Mathematical Sciences, Meiji University, 4-21-1 Nakano Nakano-ku, Tokyo 164- 8525, Japan.

References

R. A. Adams, Sobolev Spaces, Academic Press, 1975.

S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand, Princeton, 1965.

H. Brezis, Analyse Fonctionnelle, Theorie et applications, (2nd ed.) Masson, Paris, 1983.

E. Conway, D. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35(1978), 1--16.

E. J. Doedel and B. E. Oldeman et al. AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations. Concordia University, Montreal, 2010.

A. Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York, 1969.

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg New York, 2001.

G. E. Hutchinson, The Paradox of the Plankton, Am. Nat., 95(1961), 137--145.

H. Ikeda, M. Mimura and T. Scotti, Shadow system approach to a plankton model generation harmful algal bloom, manuscript.

Y. Kan-on, Global bifurcation structure of positive stationary solutions for a classical Lotka-Volterra competition model with diffusion, Japan J. Indust. Appl. Math., 20(2003), 285--310.

W. E. A. Kardinaal, L. Tonk, I. Janse, S. Hol, P. Slot, J. Huisman and P. M. Visser, Competition for Light between Toxic and Nontoxic Strains of the Harmful Cyanobacterium Microcystis, Appl. Environ. Microbiol.,73(2007), 2939--2946.

K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential

Equations, 58(1985), 15-21.

Y. Miyamoto, Upper semicontinuity of the global attractor for the Gierer--Meinhardt model, J. Differential Equations, 223(2006), 185--207.

Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13(1982), 555--593.

M. Pierre, Global Existence in Reaction-Diffusion Systems with Control of Mass: A Survey, Milan J. Math., 78(2010), 417--455.

F. Rothe, Global solutions of reaction-diffusion systems, Lecture Notes in Math. 1072, Springer-Verlag, Berlin, New York, 1984.

T. Scotti, M. Mimura and J. Y. Wakano, Avoiding toxic prey may promote harmful algal blooms, Ecological Complexity, 21(2015), 157--165.

A. M. Turing, The Chemical Basis of Morphogenesis, Phil.Transaction of the Royal Society of London, Series (B): Biological Sciences, 237(1952), 37--72.

N. Tarfulea and A. Minut, Qualitative analysis of a diffusive prey-predator model, Appl. Math. Lett., 25(2012), 803--807.