## Journal of Applied Mathematics

### Dynamical Systems Analysis of a Five-Dimensional Trophic Food Web Model in the Southern Oceans

#### Abstract

A theoretical model developed by Stone describing a three-level trophic system in the Ocean is analysed. The system consists of two distinct predator-prey networks, linked by competition for nutrients at the lowest level. There is also an interaction at the level of the two preys, in the sense that the presence of one is advantageous to the other when nutrients are low. It is shown that spontaneous oscillations in population numbers are possible, and that they result from a Hopf bifurcation. The limit cycles are analysed using Floquet theory and are found to change from stable to unstable as a solution branch is traversed.

#### Article information

Source
J. Appl. Math., Volume 2009 (2009), Article ID 575047, 17 pages.

Dates
First available in Project Euclid: 2 March 2010

https://projecteuclid.org/euclid.jam/1267538757

Digital Object Identifier
doi:10.1155/2009/575047

Zentralblatt MATH identifier
1196.37125

#### Citation

Hadley, Scott A.; Forbes, Lawrence K. Dynamical Systems Analysis of a Five-Dimensional Trophic Food Web Model in the Southern Oceans. J. Appl. Math. 2009 (2009), Article ID 575047, 17 pages. doi:10.1155/2009/575047. https://projecteuclid.org/euclid.jam/1267538757

#### References

• L. Stone, Phytoplankton-bacteria-protozoa interactions: a qualitative model portraying indirect effects,'' Marine Ecology Progress Series, vol. 64, pp. 137--145, 1990.
• 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.
• R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA, 2001.
• W. W. Murdoch, C. J. Briggs, and R. M. Nisbet, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA, 2001.
• 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.
• G. E. Hutchison, Paradox of the plankton,'' The American Naturalist, vol. 95, pp. 137--145, 1961.
• 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.
• 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.
• 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.
• 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.
• 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.
• M. L. Rosenzweig, Paradox of enrichment: destabilization of exploitation ecosystems in ecological time,'' Science, vol. 171, no. 3969, pp. 385--387, 1971.
• R. M. May, Limit cycles in predator-prey communities,'' Science, vol. 177, no. 4052, pp. 900--902, 1972.
• M. E. Gilpin and M. L. Rosenzweig, Enriched predator-prey systems: theoretical stability,'' Science, vol. 177, no. 4052, pp. 902--904, 1972.
• J. D. Murray, Mathematical Biology, vol. 19 of Biomathematics, Springer, New York, NY, USA, 1989.
• S. Ruan, Oscillations in plankton models with nutrient recycling,'' Journal of Theoretical Biology, vol. 208, no. 1, pp. 15--26, 2001.
• 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.
• L. Edelstein-Keshet, Mathematical Models in Biology, Random House, New York, NY, USA, 1988.
• 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.
• R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Springer, New York, NY, USA, 2nd edition, 1994.