February  2017, 37(2): 829-858. doi: 10.3934/dcds.2017034

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

Received  April 2015 Revised  December 2015 Published  November 2016

Fund Project: The first author is supported in part by JSPS KAKENHI Grant No. 22244010 and 15K04995. The second author thanks for partial support by JSPS KAKENHI Grant No. 15K13462.

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 $μ$.

Citation: Hideo Ikeda, Masayasu Mimura, Tommaso Scotti. Shadow system approach to a plankton model generating harmful algal bloom. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 829-858. doi: 10.3934/dcds.2017034
References:
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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.  Google Scholar

[2]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.   Google Scholar

[3]

I. R. Falconer and A. R. Humpage, Tumor promotion by cyanobacterial toxins, Phycologia, 35 (1996), 74-79.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[6]

G. E. Hutchinson, The paradox of the plankton, Am. Nat., 95 (1961), 137-145.  doi: 10.1086/282171.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[10]

W. Lampert, Inhibitory and toxic effects of blue-green algae on Daphnia, Int. Revue ges. Hydrobiol., 66 (1981), 285-298.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

T. ScottiM. 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.  Google Scholar

[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.  Google Scholar

[16]

Q. WangC. 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.  Google Scholar

[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.  Google Scholar

Figure 1.  Bifurcation curves of $E_4$ in the $(D, \mu)$-plane ($L=30$) (a) and the $(L, \mu)$-plane ($D=2500$) (b), where $a=0.95$, $b=1.2$, $K=2.9$ and $R=0.43$. The curve $n$ corresponds to the $n$-mode bifurcations, where the zero solution of the linearized problem of (3) with (4) around $E_4$ destabilizes under the $n$th eigenmode $\cos(n\pi x/L)$ perturbation
Figure 2.  Global structure of equilibrium solutions of (3) with (4) when µ is varied, where L = 30, D = 2500. The other parameters are the same as the ones in Figure 1. Solid (resp. dashed) lines represent stable (resp. unstable) equilibrium solutions of (3) with (4). The right figure is a magnification of the left one where µ is close to µc1
Figure 3.  1-mode equilibrium solutions $(\bar{P}_1^+(x), \bar{X}_1^+(x), \bar{Z}_1^+(x))$ of (3) with (4) for (a) $\mu=0.15$, (b) $\mu=0.5$ and (c) $\mu=3.1$. The other parameters are the same as the ones in Figure 1 and $D=2500$. Here $\bar{P}_1^+, \bar{X}_1^+$ and $\bar{Z}_1^+$ are drawn in blue, green and red colors, respectively
Figure 4.  Dependency of $D$ on the global structures of equilibrium solutions of the system (3) with (4) when $L=30$. (a) $D=800$, (b) $D=1500$, (c) $D=2500$, (d) $D=5000$, (e) $D=10000$ and (e$'$) is a magnification of (e) around $\mu=\mu_{c1}$. The other parameters are the same as the ones in Figure 1. Solid (resp. dashed) lines represent stable (resp. unstable) equilibrium solutions of (3) with (4)
Figure 5.  Three different structures of the nullclines of (14) with $d=d(\mu)=1/(1+\mu)$. (a-1) $P$-monostability, (a-2) $X$-monostability, (b) bistability and (c) coexistence. The red and white circles stand for stable and unstable equilibrium solutions of (14), respectively
Figure 6.  Schematic global structure of the constant and non-constant equilibrium solutions $(\bar{P}(\xi), \bar{X}(\xi))$ and $ (\bar{P}^{\pm}_n(x;\xi, \varepsilon), \bar{X}^{\pm}_n(x;\xi, \varepsilon)) \ (n=1, 2, \cdots)$ of (14) when $\varepsilon$ is varied
Figure 7.  Bifurcation curves of $\varepsilon = Q_0(\xi)/(n \pi)^2$ for $\xi \in I$. Here $\xi_n^l(\varepsilon)$ and $\xi_n^r(\varepsilon)$ are bifurcation values of $\xi$ for suitably given $\varepsilon$ $(n=1, 2, \cdots)$
Figure 8.  Schematic global structure of $1$-mode and $2$-mode equilibrium solutions of (14) which bifurcate from $(\bar{P}(\xi), \bar{X}(\xi))$ when $\xi$ is varied. Here the vertical and horizontal axes are $\bar{X}(0;\xi, \varepsilon)$ and $\xi \in I$, respectively
Figure 9.  Dependency of $H^{+}_n(\xi;\varepsilon)$ on $\varepsilon$. (a) $\xi_n^r(\varepsilon) \leqq \bar{\xi}^*$ and (b) $\xi_n^r(\varepsilon) > \bar{\xi}^*$
Figure 10.  The functional forms of $H_1^+(\xi;1/L^2, \mu)$ for (a) $\mu=0.1, \ (\alpha, \beta) = (0.404069, 0.668669)$, (b) $\mu=0.2, \ (\alpha, \beta) = (0.264145, 0.531191)$, (c) $\mu=1, \ (\alpha, \beta) = (0.104722, 0.2785)$, (d) $\mu=6, \ (\alpha, \beta) = (0.0636, 0.184446)$, (e) $\mu=12, \ (\alpha, \beta) = (0.0593018, 0.173621)$ and (f) $\mu=13, \ (\alpha, \beta) = (0.0589696, 0.172776)$, where other parameters are fixed as $a=0.95$, $b=1.2$, $K=2.9$, $R=0.43$ and $L=30$. The horizontal (resp. vertical) axes are $\xi$ (resp. $H^+_1(\xi;1/L^2, \mu)$)
Figure 11.  The functional forms of $H_1^+(\xi;1/L^2, \mu)$ for (a) $\mu=0.12, \ (\alpha, \beta) =(0.353533, 0.638713)$, (b) $\mu=0.1275, \ (\alpha, \beta) =(0.340841, 0.626635)$, (c) $\mu=0.1281, \ (\alpha, \beta) =(0.339872, 0.625696)$, (d) $\mu=0.13, \ (\alpha, \beta) =(0.336846, 0.622744)$, (e) $\mu=17, \ (\alpha, \beta) =(0.0565952, 0.171585)$ and (f) $\mu=18, \ (\alpha, \beta) =(0.0564297, 0.171148)$, where other parameters are fixed as $a=0.95$, $b=1.2$, $K=2.9$, $R=0.43$ and $L=35$. The horizontal (resp. vertical) axes are $\xi$ (resp. $H^+_1(\xi;1/L^2, \mu)$)
Figure 12.  Bifurcation diagram of the shadow system (9) with (10) when $\mu$ is a free parameter. Other parameters are fixed at $a=0.95$, $b=1.2$, $K=2.9$, $R=0.43$ and $L=30$. Solid (resp. dashed) lines represent stable (resp. unstable) equilibrium solutions of (9) with (10)
Figure 13.  Relations between $\varepsilon$ and $\mu$ with respect to the bifurcation curves of $n$-mode equilibrium solutions $(n=1, 2, 3)$ of (9) and (10) with $\varepsilon= 1/L^2$, where $a=0.95$, $b=1.2$, $K=2.9$, $R=0.43$, and $L=30$. (a) $\mu \in (\mu_c, 20.0)$, (b) $\mu \in (\mu_c, 2.0)$, which is a magnification of (a), where $\mu_c = 0.11\cdots$, $\mu_{c1}=0.13\cdots$ and $\mu_{c2}=12.51\cdots$
Figure 14.  Bifurcation diagram of the shadow system (9) with (10) when $\mu$ is a free parameter. Other parameters are fixed at $a=0.95$, $b=1.2$, $K=2.9$, $R=0.43$ and $L=35$. Solid (resp. dashed) lines represent stable (resp. unstable) equilibrium solutions of (9) with (10). In the right corner, it is shown a magnification around the primary bifurcation point $(n=1)$
Figure 15.  The functional forms of $H_1^+(\xi;1/L^2, \mu)$ for (a) $\mu=0.1, \ (\alpha, \beta) = (0.375538, 0.683272)$, (b) $\mu=0.11, \ (\alpha, \beta) = (0.355264, 0.66457)$, (c) $\mu=0.12, \ (\alpha, \beta) = (0.337375, 0.647172)$, (d) $\mu=0.13, \ (\alpha, \beta) = (0.321474, 0.630946)$, (e) $\mu=50, \ (\alpha, \beta) = (0.0521753, 0.168414)$ and (f) $\mu=58, \ (\alpha, \beta) = (0.0520414, 0.168035)$, where other parameters are fixed at $a=0.95$, $b=1.2$, $K=2.9$, $R=0.43$ and $L=60$. The horizontal (resp. vertical) axes are $\xi$ (resp. $H^+_1(\xi;1/L^2, \mu)$)
Figure 16.  Bifurcation diagram of the shadow system (9) with (10) when $\mu$ is a free parameter. Other parameters are fixed at $a=0.95$, $b=1.2$, $K=2.9$, $R=0.43$ and $L=60$. Solid (resp. dashed) lines represent stable (resp. unstable) equilibrium solutions of (9) with (10). In the right corner, it is shown a magnification around the primary bifurcation point ($n=1$)
Figure 17.  The graphs of $L$ and ${\mathcal K}_2={\mathcal K}_2(1)$ as a function of $\mu$, where $a=0.95$, $b=1.2$, $K=2.9$ and $R=0.43$. (a) $\mu \in (\mu_c, 60.0)$, (b) $\mu \in (\mu_c, 20.0)$, (c) $\mu \in (\mu_c, 0.2)$. (b) and (c) are the magnification of (a), where $\mu_c = 0.11\cdots, \mu_{c1}^*=0.12\cdots, \mu_{c1}=0.13\cdots, \mu_{c2}^*=55.57\cdots$ and $\mu_{c2}=12.51\cdots$
Figure 18.  The bifurcation curve of $\varepsilon = \varepsilon^1_0(\xi) = Q_0(\xi)/\pi^2$ for $\xi \in I$. For given $\bar{\xi}^*$, $\varepsilon^1_0(\bar{\xi}^*)$ is determined
Figure 19.  Dependency of $\varepsilon$ on $H^{+}_1(\xi;\varepsilon)$. (ⅰ) $\varepsilon > \varepsilon^1_0(\bar{\xi}^*)$, (ⅱ) $\varepsilon = \varepsilon^1_0(\bar{\xi}^*)$ and (ⅲ) $\varepsilon < \varepsilon^1_0(\bar{\xi}^*)$. These show that $\varepsilon^1_0(\bar{\xi}^*)$ is the bifurcation point of the 1-mode equilibrium solutions
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