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Article Contents

# Shadow system approach to a plankton model generating harmful algal bloom

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 $μ$.

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

 Citation:

• 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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