# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2017, 37 (2) : 829-858. doi: 10.3934/dcds.2017034
##### 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.

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.
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
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
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
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)
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
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
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 $\varepsilon$ $(n=1, 2, \cdots)$
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
Dependency of $H^{+}_n(\xi;\varepsilon)$ on $\varepsilon$. (a) $\xi_n^r(\varepsilon) \leqq \bar{\xi}^*$ and (b) $\xi_n^r(\varepsilon) > \bar{\xi}^*$
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. $H^+_1(\xi;1/L^2, \mu)$)
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. $H^+_1(\xi;1/L^2, \mu)$)
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)
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 $\mu_{c2}=12.51\cdots$
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 $(n=1)$
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. $H^+_1(\xi;1/L^2, \mu)$)
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 ($n=1$)
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 $\mu_{c2}=12.51\cdots$
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
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
 [1] Qi Wang. On the steady state of a shadow system to the SKT competition model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2941-2961. doi: 10.3934/dcdsb.2014.19.2941 [2] Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5519-5549. doi: 10.3934/dcdsb.2020357 [3] Tohru Tsujikawa, Kousuke Kuto, Yasuhito Miyamoto, Hirofumi Izuhara. Stationary solutions for some shadow system of the Keller-Segel model with logistic growth. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 1023-1034. doi: 10.3934/dcdss.2015.8.1023 [4] Zhipeng Qiu, Huaiping Zhu. Complex dynamics of a nutrient-plankton system with nonlinear phytoplankton mortality and allelopathy. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2703-2728. doi: 10.3934/dcdsb.2016069 [5] Xiong Li, Hao Wang. A stoichiometrically derived algal growth model and its global analysis. Mathematical Biosciences & Engineering, 2010, 7 (4) : 825-836. doi: 10.3934/mbe.2010.7.825 [6] Rajinder Pal Kaur, Amit Sharma, Anuj Kumar Sharma. Dynamical complexity in a delayed Plankton-Fish model with alternative food for predators. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021036 [7] Shin-Ichiro Ei, Kota Ikeda, Yasuhito Miyamoto. Dynamics of a boundary spike for the shadow Gierer-Meinhardt system. Communications on Pure and Applied Analysis, 2012, 11 (1) : 115-145. doi: 10.3934/cpaa.2012.11.115 [8] Theodore Kolokolnikov, Michael J. Ward. Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1033-1064. doi: 10.3934/dcdsb.2004.4.1033 [9] Xin-You Meng, Yu-Qian Wu, Jie Li. Bifurcation analysis of a Singular Nutrient-plankton-fish model with taxation, protected zone and multiple delays. Numerical Algebra, Control and Optimization, 2020, 10 (3) : 391-423. doi: 10.3934/naco.2020010 [10] R. P. Gupta, Shristi Tiwari, Shivam Saxena. The qualitative behavior of a plankton-fish interaction model with food limited growth rate and non-constant fish harvesting. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2791-2815. doi: 10.3934/dcdsb.2021160 [11] Hideaki Takaichi, Izumi Takagi, Shoji Yotsutani. Global bifurcation structure on a shadow system with a source term - Representation of all solutions-. Conference Publications, 2011, 2011 (Special) : 1344-1350. doi: 10.3934/proc.2011.2011.1344 [12] Patrick Henning, Anders M. N. Niklasson. Shadow Lagrangian dynamics for superfluidity. Kinetic and Related Models, 2021, 14 (2) : 303-321. doi: 10.3934/krm.2021006 [13] Feng-Bin Wang, Sze-Bi Hsu, Wendi Wang. Dynamics of harmful algae with seasonal temperature variations in the cove-main lake. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 313-335. doi: 10.3934/dcdsb.2016.21.313 [14] François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262 [15] Amy Henderson, Emek Kose, Allison Lewis, Ellen R. Swanson. Mathematical modeling of algal blooms due to swine CAFOs in Eastern North Carolina. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 555-572. doi: 10.3934/dcdss.2021151 [16] Tao Jie, Gao Yan. Computing shadow prices with multiple Lagrange multipliers. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2307-2329. doi: 10.3934/jimo.2020070 [17] Lorena Rodríguez-Gallego, Antonella Barletta Carolina Cabrera, Carla Kruk, Mariana Nin, Antonio Mauttone. Establishing limits to agriculture and afforestation: A GIS based multi-objective approach to prevent algal blooms in a coastal lagoon. Journal of Dynamics and Games, 2019, 6 (2) : 159-178. doi: 10.3934/jdg.2019012 [18] Jianhong Wu, Weiguang Yao, Huaiping Zhu. Immune system memory realization in a population model. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 241-259. doi: 10.3934/dcdsb.2007.8.241 [19] Pierre Degond, Simone Goettlich, Axel Klar, Mohammed Seaid, Andreas Unterreiter. Derivation of a kinetic model from a stochastic particle system. Kinetic and Related Models, 2008, 1 (4) : 557-572. doi: 10.3934/krm.2008.1.557 [20] Jesse Berwald, Marian Gidea. Critical transitions in a model of a genetic regulatory system. Mathematical Biosciences & Engineering, 2014, 11 (4) : 723-740. doi: 10.3934/mbe.2014.11.723

2020 Impact Factor: 1.392