\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Singular perturbation approach to a plankton model generating harmful algal bloom

Dedicated to the memory of Professor Masayasu Mimura

This work is supported in part by JSPS KAKENHI, Grant Numbers JP15K04995 and JP19K03618

Abstract Full Text(HTML) Figure(12) Related Papers Cited by
  • Spatially localized blooms of toxic plankton species have negative effects on other organisms owing to the production of toxins, mechanical damage, or other means. A two-prey (toxic and nontoxic phytoplankton) one-predator (zooplankton) Lotka–Volterra system with diffusion has been presented to understand the mechanism underlying the formation of spatial blooms of toxic plankton. In this study, we consider a one-dimensional (1D) system in which the ratio $ D $ of the diffusion rates of the predator and two prey, and the length of the spatial interval $ L $ are both sufficiently large while maintaining $ L^2/D $ as a constant. This system is reduced to a singularly perturbed system. The existence and stability properties of the 1D blooming stationary solutions are demonstrated by improving the previous results.

    Mathematics Subject Classification: Primary: 35B32, 35B35, 35B36, 35K57; Secondary: 35Q92.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  1D blooming stationary solution $ u_1(x;\varepsilon), u_2(x;\varepsilon) $, and $ v(x;\varepsilon) $ of (9) with sharp internal transition layer at $ x = \ell(\varepsilon) $ for $ a = 0.95, b = 1.2, K = 2.9, R = 0.43, d = 2/3, \mu = 0.5, D = 5.0 $, and $ \varepsilon = 0.05 $

    Figure 2.  Outer approximations $ U_1(x;\omega), U_2(x;\omega) $, and $ V(x;\omega) $. $ U_1(x;\omega) $ and $ U_2(x;\omega) $ have a discontinuous point at $ x = \ell(\omega) $

    Figure 3.  Nullclines of two curves

    Figure 4.  The solution $ V(x;\omega^*) \ ( x \in (0, 1)) $ of (13) with $ D = 5.0, \mu = 0.2, d = 1/(1+\mu) = 5/6, R = 0.43 $, and $ K = 2.9 $ (in red) uses both of the monotonically decreasing branches $ y = h_1(V) \ (V \leqq \omega^*) $ and $ y = h_2(V) \ ( V \geqq \omega^*) $, where $ \omega^* = 0.42731 $ and $ \ell^* = 0.324 $

    Figure 5.  The solution $ V(x;\omega^*) \ (x \in (0, 1)) $ of (13) with $ D = 5.0, \mu = 0.5, d = 1/(1+\mu) = 2/3, R = 0.43 $, and $ K = 2.9 $ (in red) uses the monotonically decreasing branch $ y = h_1(V) \ (V \leqq \omega^*) $ but the monotonically increasing branch $ y = h_2(V) \ ( V \geqq \omega^*) $, where $ \omega^* = 0.272147 $ and $ \ell^* = 0.3390 $

    Figure 6.  The solution $ \eta_2(x) $ of (18) on $ [\ell^*, 1] $ with $ D = 5.0, K = 2.9 $, and $ R = 0.43 $, where $ \omega^* = 0.272147 $ and $ \ell^* = 0.3390 $

    Figure 7.  Spatial profiles of four cases of solutions $ z^{(i)}_N(x) $ (in red) and $ z^{(i)}_D(x) $ (in black) $ (i = 1, 2) $

    Figure 8.  Profiles of function $ det^*(x) $ for cases (Ⅰ) and (Ⅱ). In case (Ⅰ), $ det^*(x) < 0 $ on $ I_1 $ and $ I_2 $. In case (Ⅱ), $ det^*(x) < 0 $ on $ I_1 $ and $ det^*(x) > 0 $ on $ I_2 $

    Figure 9.  Profile of solution $ z_N(x) $ for boundary value problem of first three equations of (33) on $ I $ of case (Ⅱ)

    Figure 10.  Solution trajectories of (a) (38) and (b) (39)

    Figure 11.  The unstable manifold of $ (0, 0) $ for (38) meets the stable manifold of $ (1-\frac 1K) $ for (39) at $ V = V^* $

    Figure 12.  Solution trajectories on composite phase plane at $ V = \omega $: (a) $ 0 < \omega \leqq V^* $ and (b) $ V^* < \omega < 1-\frac 1K $

  • [1] I. R. Falconer and A. R. Humpage, Tumor promotion by cyanobacterial toxins, Phycologia, 35 (1996), 74-79. 
    [2] P. C. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations, J. Math. Anal. Appl., 54 (1976), 497-521.  doi: 10.1016/0022-247X(76)90218-3.
    [3] G. E. Hutchinson, The paradox of the plankton, Am. Nat., 95 (1961), 137-145.  doi: 10.1086/282171.
    [4] H. Ikeda, Global bifurcation phenomena of standing pulse solutions for three-component systems with competition and diffusion, Hiroshima Math. J., 32 (2002), 87-124. 
    [5] H. IkedaM. Mimura and T. Scotti, Shadow system approach to a plankton model generating harmful algal bloom, Discrete Contin. Dyn. Syst., 37 (2017), 829-858.  doi: 10.3934/dcds.2017034.
    [6] E. M. Jochimsen, et al., Liver failure and death after exposure to microcystins at at a hemodialysis center in Brazil, N. Engl. J. Med., 338 (1998), 873-878.
    [7] Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556.
    [8] Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221. 
    [9] W. E. A. Kardinaal, et al., Competition for light between toxic and nontoxic strains of the harmful cyanobacterium Microcystis, Appl. Environ. Microbiol., 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] M. Mimura and P. C. Fife, A 3-component system of competition and diffusion, Hiroshima Math. J., 16 (1986), 189-207. 
    [13] M. MimuraM. Tabata and Y. Hosono, Multiple solutions of two-point boundary value problems of Neumann type with a small parameter, SIAM J. Math. Anal., 11 (1980), 613-631.  doi: 10.1137/0511057.
    [14] Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIMA J. Math. Anal., 18 (1987), 1726-1770.  doi: 10.1137/0518124.
    [15] T. ScottiM. Mimura and J. Y. Wakano, Avoiding toxic prey may promote harmful algal blooms, Ecol. Complex., 21 (2015), 157-165.  doi: 10.1016/j.ecocom.2014.07.004.
    [16] J. A. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differ. Equations, 39 (1981), 269-290.  doi: 10.1016/0022-0396(81)90077-2.
  • 加载中
Open Access Under a Creative Commons license

Figures(12)

SHARE

Article Metrics

HTML views(996) PDF downloads(352) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return