Dynamical systems analysis of a two level trophic food web in the Southern Oceans

A theoretical model developed by Stone describing a two level trophic system in the Ocean is analyzed, for the case in which there is unlimited supply of nutrients. We show that spontaneous oscillations in population numbers are possible, but they do not arise from a Hopf bifurcation. Seasonal forcing of the model is also investigated, and it is shown that resonances can occur, in addition to highly non-linear behaviour including high period oscillations, quasi-periodicity and chaos. References L. Edelstein-Keshet. Mathematical Models in Biology . Random House, New York 1988. A. M. Edwards and J. Brindley. Zooplankton mortality and the dynamical behaviour of plankton population models. Bull. Math. Biol. , 61:303--339, 1999. doi:10.1006/bulm.1998.0082 J. A. Freund, S. Mieruch, B. Scholze, K. Wiltshire and U. Feudel. Bloom dynamics in a seasonally forced phytoplankton-zooplankton model: Trigger mechanisms and timing effects. Ecological Complexity , 3:129--139, 2006. doi:10.1016/j.ecocom.2005.11.001 T. Gross, W. Ebenh{oe}h and U. Feudel. Enrichment and foodchain stability: The impact of different forms of predator-prey interaction. J. Theor. Biol. , 227:349--358, 2004. doi:10.1016/j.jtbi.2003.09.020 J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields . Springer-Verlag, New York 1983. A. Huppert, R. Olinky and L. Stone. Bottom-Up excitable models of phytoplankton blooms. Bull. Math. Biol. , 66:865--878, 2004. doi:10.1016/j.bulm.2004.01.003 K. L. Kirk. Enrichment can stabilize population dynamics: Autotoxins and density dependence. Ecology , 79:2456--2462, 1998. doi:10.1890/0012-9658(1998)079[2456:ECSPDA]2.0.CO;2 L. Medio and M. Lines. Nonlinear dynamics a primer . Cambridge University Press, Cambridge 2001. J. D. Murray. Mathematical Biology . Springer--Verlag, New York 1989. J. C. Sprott, J. C. Wildenberg and Y. Azizi. A simple spatiotemporal chaotic Lotka--Volterra model. Chaos Solitons and Fractals , 26:1035--1043, 2005. doi:10.1016/j.chaos.2005.02.015 L. Stone. Phytoplankton-bacteria-protozoa interactions: a qualitative model portraying indirect effects. Mar. Ecol. Prog. Ser. , 64:137--145, 1990. http://www.int-res.com/articles/meps/64/m064p137.pdf J. M. T. Thompson and H. B. Stewart. Nonlinear Dynamics and Chaos . John Wiley and Sons, New York 1989. J. E. Truscott and J. Brindley. Ocean plankton populations as excitable media. Bull. Math. Biol. , 56:981--998, 1994. doi:10.1016/S0092-8240(05)80300-3