Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Dynamics of a stoichiometric producer-grazer system with seasonal effects on light level

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA

Special Issues: Resource Explicit Population Models

Many population systems are subject to seasonally varying environments. As a result, many species exhibit seasonal changes in their life-history parameters. It is quite natural to try to understand how seasonal forcing affects population dynamics subject to stoichiometric constraints, such as nutrient/light availability and food quality. Here, we use a variation of a stoichiometric Lotka-Volterra type model, known as the LKE model, as a case study, focusing on seasonal variation in the producer’s light-dependent carrying capacity. Positivity and boundedness of model solutions are studied, as well as numerical explorations and bifurcations analyses. In the absence of seasonal effects, the LKE model suggests that the dynamics are either stable equilibrium or limit cycles. However, through bifurcation analysis we observe that seasonal forcing can lead to complicated population dynamics, including periodic and quasi-periodic solutions.
  Article Metrics

Keywords ecological stoichiometry; predator-prey model; seasonal forcing; carrying capacity; quasi-periodic solution

Citation: Lale Asik, Jackson Kulik, Kevin Long, Angela Peace. Dynamics of a stoichiometric producer-grazer system with seasonal effects on light level. Mathematical Biosciences and Engineering, 2019, 16(1): 501-515. doi: 10.3934/mbe.2019023


  • 1. R.W. Sterner, and J.J. Elser, Ecological stoichiometry: the biology of elements from molecules to the biosphere, Princeton University Press, 2002.
  • 2. M. Danger and F. Maunoury-Danger, Ecological Stoichiometry, in Encyclopedia of Aquatic Ecotoxicology (eds. J.F. Frard, and B. Christian), Springer, (2013), 317–326.
  • 3. V.H. Smith, Nutrient dependence of primary productivity in lakes, Limnol. Oceanogr., 24 (1979), 1051-1064.
  • 4. I. Loladze, Y. Kuang and J.J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bull. Math Biol., 62 (2000), 1137-1162.
  • 5. S.E. Jørgensen and B.D. Fath, Modelling Population Dynamics, in Developments in Environmental Modelling, Elsevier, (2011), 129–158.
  • 6. S.E. Jorgenson and G. Bendoricchio, Fundamentals of ecological modelling, Elsevier Science Publishers BV, 2001.
  • 7. T. Andersen, Pelagic Nutrient Cycles: Herbivores as Sources and Sinks, NY: Springer-Verlag, 1997.
  • 8. J. Urabe and R.W. Sterner, Regulation of herbivore growth by the balance of light and nutrients, Proc. Nat. Acad. Sci, 93 (1996), 8465–8469.
  • 9. J. Cebrián and I.I. Valiela, Seasonal patterns in phytoplankton biomass in coastal ecosystems, J. Plankton Res., 21 (1999), 429–444.
  • 10. M.L. Rosenzweig, Paradox of enrichment: destabilization of exploitation ecosystems in ecological time, Science, 171 (1971), 385–387.
  • 11. S. Lynch, Maps and Nonautonomous Systems in the Plane, in Dynamical Systems with Applications using Mathematica®, Birkhüser Boston, (2007), 171--194.
  • 12. M. Scheffer, S. Rinaldi, Y.A. Kuznetsov and E.H. van Nes, Seasonal dynamics of Daphnia and algae explained as a periodically forced predator-prey system, Oikos, (1997), 519–532.
  • 13. S. Ulrich, G.Z. Maciej, L. Winfried and D. Annie, The PEG-model of seasonal succession of planktonic events in fresh waters, Arch. Hydrobiol, 106 (1986), 433–471.
  • 14. H. Wang, Y. Kuang and I. Loladze, Dynamics of a mechanistically derived stoichiometric producer-grazer model, J. Biol. Dynam., 2 (2008), 286–296.
  • 15. I. Loladze, Y. Kuang, J.J. Elser and W.F. Fagan, Competition and stoichiometry: coexistence of two predators on one prey, Theor. Popul. Biol., 65 (2004), 1–15.
  • 16. J. Urabe, J.J. Elser, M. Kyle, T. Yoshida, T. Sekino and Z. Kawabata, Herbivorous animals can mitigate unfavourable ratios of energy and material supplies by enhancing nutrient recycling, Ecol. Lett., 5 (2002), 177–185.


This article has been cited by

  • 1. Lale Asik, Jackson Kulik, Kevin R. Long, Angela Peace, Seasonal Variation of Nutrient Loading in a Stoichiometric Producer–Consumer System, Bulletin of Mathematical Biology, 2019, 10.1007/s11538-019-00629-6
  • 2. Md. Nazmul Hassan, Lale Asik, Jackson Kulik, K.R. Long, Angela Peace, Environmental seasonality on predator–prey systems under nutrient and toxicant constraints, Journal of Theoretical Biology, 2019, 480, 71, 10.1016/j.jtbi.2019.08.001
  • 3. Ming Chen, Lale Asik, Angela Peace, Stoichiometric knife-edge model on discrete time scale, Advances in Difference Equations, 2019, 2019, 1, 10.1186/s13662-019-2468-7

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved