Catastrophic shifts in vertical distributions of phytoplankton. The existence in a bifurcation set. (English) Zbl 1091.92057
Summary: A model of phytoplankton dynamics within a water column was analyzed with special consideration of the existence of a bifurcation set in the parameter space. We considered two resources, light and a limiting nutrient, for phytoplankton growth and assumed that the water column is separated into two layers by thermal and/or density stratification. It was shown that there exists a bifurcation set in the parameter space when the growth function meets several conditions that are general for growth functions of two essential resources.
Specifically, these conditions include that a less abundance of the two resources limits the growth while the effect of the other is sufficiently small. A folded structure with two stable states separated by one unstable state appears in the catastrophe manifold when the parameters move to a certain direction with a certain curvature from a point in the bifurcation set. These results suggest that occurrence of discontinuous transitions between two alternative vertical patterns is a possible nature of phytoplankton dynamics within a stratified water column.
Specifically, these conditions include that a less abundance of the two resources limits the growth while the effect of the other is sufficiently small. A folded structure with two stable states separated by one unstable state appears in the catastrophe manifold when the parameters move to a certain direction with a certain curvature from a point in the bifurcation set. These results suggest that occurrence of discontinuous transitions between two alternative vertical patterns is a possible nature of phytoplankton dynamics within a stratified water column.
MSC:
| 92D40 | Ecology |
| 37N25 | Dynamical systems in biology |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 58K35 | Catastrophe theory |
References:
| [1] | Etemad-Shahidi, Oceanogr., 46, 1158 (2001) |
| [2] | Gantmacher, F.R., The Theory of Matrices. Chelsea Publishing, New York, 1974 |
| [3] | Hinch, E.J.: Perturbation Methods. Cambridge University Press, Cambridge, 1991 · Zbl 0746.34001 |
| [4] | Ishii, Math. Biol., 16, 1 (1982) · Zbl 0501.92020 |
| [5] | Klausmeier, Oceanogr., 46, 1998 (2001) |
| [6] | Mann, K.H., Lazier, J.R.N.: Dynamics of Marine Ecosystems: Biological-Physical Interactions in the Oceans. 2nd edn. Blackwell, Oxford, 1996 |
| [7] | Nakanishi, J. Limnol., 60, 125 (1999) |
| [8] | Poston, T., Stewart, I.: Catastrophe Theory and Its Applications. Pitman Publishing, London, 1978 · Zbl 0382.58006 |
| [9] | Scheffer, Evol., 18, 648 (2003) |
| [10] | Scheffer, M.; Carpenter, S. R.; Foley, J. A.; Folke, C., Nature, 413, 591-596 (2001) |
| [11] | Shigesada, Math. Biol., 12, 311 (1981) · Zbl 0477.92018 · doi:10.1007/BF00276919 |
| [12] | Steel, Mar. Biol. Assoc. UK, 39, 217 (1960) · doi:10.1017/S0025315400013266 |
| [13] | Tilman, D.: Resource Competition and Community Structure. Princeton University Press, Princeton, N.J., 1982 |
| [14] | Totaro, S.: Nonlinear Anal. 13, 969-989 (1989) · Zbl 0721.92027 |
| [15] | Yoshiyama, theor. Biol., 216, 397 (2002) · doi:10.1006/jtbi.2002.3007 |
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