Quantifying chaos for ecological stoichiometry.

The theory of ecological stoichiometry considers ecological interactions among species with different chemical compositions. Both experimental and theoretical investigations have shown the importance of species composition in the outcome of the population dynamics. A recent study of a theoretical three-species food chain model considering stoichiometry [B. Deng and I. Loladze, Chaos 17, 033108 (2007)] shows that coexistence between two consumers predating on the same prey is possible via chaos. In this work we study the topological and dynamical measures of the chaotic attractors found in such a model under ecological relevant parameters. By using the theory of symbolic dynamics, we first compute the topological entropy associated with unimodal Poincaré return maps obtained by Deng and Loladze from a dimension reduction. With this measure we numerically prove chaotic competitive coexistence, which is characterized by positive topological entropy and positive Lyapunov exponents, achieved when the first predator reduces its maximum growth rate, as happens at increasing δ1. However, for higher values of δ1 the dynamics become again stable due to an asymmetric bubble-like bifurcation scenario. We also show that a decrease in the efficiency of the predator sensitive to prey's quality (increasing parameter ζ) stabilizes the dynamics. Finally, we estimate the fractal dimension of the chaotic attractors for the stoichiometric ecological model.

[1]  T. Andersen.  Pelagic Nutrient Cycles: Herbivores as Sources and Sinks , 2011 .

[2]  M. Misiurewicz,et al.  Entropy of piecewise monotone mappings , 1980 .

[3]  B. Weiss,et al.  Sensitive dependence on initial conditions , 1993 .

[4]  Bo Deng,et al.  Food chain chaos with canard explosion. , 2004, Chaos.

[5]  James P. Crutchfield,et al.  Geometry from a Time Series , 1980 .

[6]  Michael E. Gilpin,et al.  Spiral Chaos in a Predator-Prey Model , 1979, The American Naturalist.

[7]  B. Deng.  Equilibriumizing all food chain chaos through reproductive efficiency. , 2006, Chaos.

[8]  K. McCann,et al.  Nonlinear Dynamics and Population Disappearances , 1994, The American Naturalist.

[9]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[10]  B. Hao,et al.  Applied Symbolic Dynamics and Chaos , 1998 .

[11]  J. Urabe,et al.  Regulation of herbivore growth by the balance of light and nutrients. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[12]  M. Malkin,et al.  Computing aspects of the entropic theory of one-dimensional dynamical systems , 1991 .

[13]  Yang Kuang,et al.  Stoichiometry in producer-grazer systems: Linking energy flow with element cycling , 2000, Bulletin of mathematical biology.

[14]  J. Elser,et al.  Stoichiometry and population dynamics , 2004 .

[15]  W. Schaffer.  Order and Chaos in Ecological Systems , 1985 .

[16]  Brian Dennis,et al.  Chaotic Dynamics in an Insect Population , 1997, Science.

[17]  J. Elser,et al.  Stoichiometric impacts of increased carbon dioxide on a planktonic herbivore , 2003 .

[18]  W. Thurston,et al.  On iterated maps of the interval , 1988 .

[19]  Bo Deng,et al.  Food chain chaos due to transcritical point. , 2003, Chaos.

[20]  D. Checkley.  The egg production of a marine planktonic copepod in relation to its food supply: Laboratory studies1 , 1980 .

[21]  J. Elser,et al.  Herbivorous animals can mitigate unfavourable ratios of energy and material supplies by enhancing nutrient recycling , 2002 .

[22]  Marten Scheffer,et al.  Chaos in a long-term experiment with a plankton community , 2008, Nature.

[23]  P. Grassberger,et al.  Measuring the Strangeness of Strange Attractors , 1983 .

[24]  Ming-Chia Li.  Point Bifurcations and Bubbles for a Cubic Family , 2003 .

[25]  Chaotic Feature in the Light Curve of 3C 273 , 2006, astro-ph/0603304.

[26]  Robert W. Sterner,et al.  Daphnia growth on varying quality of Scenedesmus: Mineral limitation of zooplankton , 1993 .

[27]  Bo Deng,et al.  Food chain chaos due to junction-fold point. , 2001, Chaos.

[28]  T. Weisse,et al.  Carbon:phosphorus stoichiometry and food chain production. , 1998 .

[29]  Sebastiaan A.L.M. Kooijman,et al.  Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain , 1999 .

[30]  J. Sprott.  Chaos and time-series analysis , 2001 .

[31]  J. Huisman,et al.  reply: Coexistence and resource competition , 2000, Nature.

[32]  A. Hastings,et al.  Chaos in a Three-Species Food Chain , 1991 .

[33]  Yang Kuang,et al.  Competition and stoichiometry: coexistence of two predators on one prey. , 2004, Theoretical population biology.

[34]  Kevin S. McCann,et al.  Bifurcation Structure of a Three-Species Food-Chain Model , 1995 .

[35]  J. Elser,et al.  Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere , 2002 .

[36]  S. McNaughton,et al.  Mineral nutrition and spatial concentrations of African ungulates , 1988, Nature.

[37]  William M. Schaffer,et al.  Stretching and Folding in Lynx Fur Returns: Evidence for a Strange Attractor in Nature? , 1984, The American Naturalist.

[38]  P. Grassberger,et al.  Characterization of Strange Attractors , 1983 .

[39]  E. Mccauley,et al.  Multiple dynamics in a single predator–prey system: experimental effects of food quality , 2001, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[40]  J. Huisman,et al.  Biodiversity of plankton by species oscillations and chaos , 1999, Nature.

[41]  Bo Deng,et al.  Food chain chaos due to Shilnikov's orbit. , 2002, Chaos.

[42]  F. Blanchard,et al.  Topological chaos: what may this mean? , 2008, 0805.0232.

[43]  R. Gilmore,et al.  The Topology of Chaos: Alice in Stretch and Squeezeland , 2002 .

[44]  Yuri A. Kuznetsov,et al.  Belyakov Homoclinic Bifurcations in a Tritrophic Food Chain Model , 2001, SIAM J. Appl. Math..