Dynamical systems analysis of a five-dimensional trophic food web model in the southern oceans. (English) Zbl 1196.37125

The paper investigates the solutions of a dynamical system representing a theoretical model for a three-level trophic system in the ocean. The system consists of two distinct prey-predator networks, linked by competition for nutrients at the lowest level. There is also an interaction at the level of the two preys; the presence of one of the preys gives an advantage to the other one when nutrients are low. The model displays the interaction between phytoplankton, bacteria, zooplankton and nutrients. A particular emphasis in the analysis is on the stability of the steady-state populations and self-sustained solutions. Previous work on this model considered that the nutrient concentration was invariable, showing that the system becomes degenerate, in the sense that the unforced equations gave rise to equilibrium points that are centres. In this paper, the nutrient concentration is considered to be variable and, instead of degeneracy, Hopf bifurcation was observed for a range of parameter values. Using the Floquet theory, it is shown that the limit cycles change the stability as one moves along the solution branches. Some numerical results in MATLAB are presented at the end of the paper.


37N25 Dynamical systems in biology
92D25 Population dynamics (general)


Full Text: DOI EuDML


[1] L. Stone, “Phytoplankton-bacteria-protozoa interactions: a qualitative model portraying indirect effects,” Marine Ecology Progress Series, vol. 64, pp. 137-145, 1990. · doi:10.1046/j.1365-2656.2002.00645.x
[2] S. Hadley and L. Forbes, “Dynamical systems analysis of a two level trophic food web in the Southern Oceans,” The ANZIAM Journal, vol. 50, pp. E24-E55, 2009. · Zbl 1196.37125
[3] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA, 2001.
[4] W. W. Murdoch, C. J. Briggs, and R. M. Nisbet, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA, 2001.
[5] K. W. Shertzer, S. P. Ellner, G. F. Fussmann, and N. G. Hairston Jr., “Predator-prey cycles in an aquatic microcosm: testing hypotheses of mechanism,” Journal of Animal Ecology, vol. 71, no. 5, pp. 802-815, 2002. · doi:10.1046/j.1365-2656.2002.00645.x
[6] G. E. Hutchison, “Paradox of the plankton,” The American Naturalist, vol. 95, pp. 137-145, 1961.
[7] M. Scheffer, S. Rinaldi, J. Huisman, and F. J. Weissing, “Why plankton communities have no equilibrium: solutions to the paradox,” Hydrobiologia, vol. 491, pp. 9-18, 2003. · doi:10.1023/A:1024404804748
[8] A. M. Verschoor, M. Vos, and I. van der Stap, “Inducible defences prevent strong population fluctuations in bi- and tritrophic food chains,” Ecology Letters, vol. 7, no. 12, pp. 1143-1148, 2004. · doi:10.1111/j.1461-0248.2004.00675.x
[9] I. van der Stap, M. Vos, R. Tollrian, and W. M. Mooij, “Inducible defenses, competition and shared predation in planktonic food chains,” Oecologia, vol. 157, no. 4, pp. 697-705, 2008. · doi:10.1007/s00442-008-1111-1
[10] T. Gross, W. Ebenhoh, and U. Feudel, “Enrichment and foodchain stability: the impact of different forms of predator-prey interaction,” Journal of Theoretical Biology, vol. 227, no. 3, pp. 349-358, 2004. · doi:10.1016/j.jtbi.2003.09.020
[11] A. M. Edwards and J. Brindley, “Zooplankton mortality and the dynamical behaviour of plankton population models,” Bulletin of Mathematical Biology, vol. 61, no. 2, pp. 303-339, 1999. · Zbl 1323.92162 · doi:10.1006/bulm.1998.0082
[12] M. L. Rosenzweig, “Paradox of enrichment: destabilization of exploitation ecosystems in ecological time,” Science, vol. 171, no. 3969, pp. 385-387, 1971.
[13] R. M. May, “Limit cycles in predator-prey communities,” Science, vol. 177, no. 4052, pp. 900-902, 1972.
[14] M. E. Gilpin and M. L. Rosenzweig, “Enriched predator-prey systems: theoretical stability,” Science, vol. 177, no. 4052, pp. 902-904, 1972.
[15] J. D. Murray, Mathematical Biology, vol. 19 of Biomathematics, Springer, New York, NY, USA, 1989. · Zbl 0682.92001
[16] S. Ruan, “Oscillations in plankton models with nutrient recycling,” Journal of Theoretical Biology, vol. 208, no. 1, pp. 15-26, 2001. · doi:10.1006/jtbi.2000.2196
[17] H.-L. Wang, J.-F. Feng, F. Shen, and J. Sun, “Stability and bifurcation behaviors analysis in a nonlinear harmful algal dynamical model,” Applied Mathematics and Mechanics, vol. 26, no. 6, pp. 729-734, 2005. · Zbl 1144.37439 · doi:10.1007/BF02465423
[18] L. Edelstein-Keshet, Mathematical Models in Biology, Random House, New York, NY, USA, 1988. · Zbl 0674.92001
[19] L. K. Forbes, “Forced transverse oscillations in a simple spring-mass system,” SIAM Journal on Applied Mathematics, vol. 51, no. 5, pp. 1380-1396, 1991. · Zbl 0743.70022 · doi:10.1137/0151069
[20] R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Springer, New York, NY, USA, 2nd edition, 1994. · Zbl 0806.34028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.