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 $μ$.
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Figure 1.
Bifurcation curves of
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
Figure 4.
Dependency of
Figure 10.
The functional forms of
Figure 11.
The functional forms of
Figure 13.
Relations between
Figure 14.
Bifurcation diagram of the shadow system (9) with (10) when
Figure 15.
The functional forms of
Figure 16.
Bifurcation diagram of the shadow system (9) with (10) when
Figure 17.
The graphs of
Figure 19.
Dependency of
[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, et al., 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, et al., 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. |