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Complete Global and Bifurcation Analysis of a Stoichiometric Predator–Prey Model

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Abstract

Loladze et al. (Bull Math Biol 62:1137–1162, 2000) proposed a highly cited stoichiometric predator–prey system, which is nonsmooth, and thus it is extremely difficult to analyze its global dynamics. The main challenge comes from the phase plane fragmentation and parameter space partitioning in order to perform a detailed and complete global stability and bifurcation analysis. Li et al. (J Math Biol 63:901–932, 2011) firstly discussed its global dynamical behavior with Holling type I functional response and found that the system has no limit cycles, and the internal equilibrium is globally asymptotically stable if it exists. Secondly, for the system with Holling type II functional response, Li et al. (2011) fixed all parameters (with realistic values) except K to perform the bifurcation analysis and obtained some interesting phenomena, for instance, the appearance of bistability and many bifurcation types. The aim of this paper is to provide a complete global analysis for the system with Holling type II functional response without fixing any parameter. Our analysis shows that the model has far richer dynamics than those found in the previous paper (Li et al. 2011), for example, four types of bistability appear: besides the bistability between an internal equilibrium and a limit cycle as shown in Li et al. (2011), the other three bistabilities occur between an internal equilibrium and a boundary equilibrium, between two internal equilibria, or between a boundary equilibrium and a limit cycle. In addition, this paper rigorously provides all possible bifurcation passways of this stoichiometric model with Holling type II functional response.

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References

  1. Andersen, T.: Pelagic Nutrient Cycles: Herbivores as Sourced and Sinks for Nutrients. Springer-Verlag, Berlin (1997)

    Book  Google Scholar 

  2. Cheng, K.S.: Uniqueness of a limit cycle for a predator-prey system. SIAM. J. Math. Anal. 12, 541–548 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  3. Everett, R.A., Nagy, J.D., Kuang, Y.: Dynamics of a data based ovarian cancer growth and treatment model with time delay. J. Dyn. Diff. Equ. 28, 1393–1414 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Grover, J.: Stoichiometry, herbivory and competition for nutrients: simple models based on planktonic ecosystems. J. Theor. Biol. 214, 599–618 (2002)

    Article  Google Scholar 

  5. Hessen, D.O., Bjerkeng, B.: A model approach to planktonic stoichiometry and consumer-resource stability. Freshw. Biol. 38, 447–471 (1997)

    Article  Google Scholar 

  6. Hsu, S.B., Hubbell, S.P., Waltman, P.: Competing predators. SIAM J. Appl. Math. 35, 617–625 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kuang, Y., Huisman, J., Elser, J.J.: Stoichiometric plant-herbivore models and their interpretation. Math. Biosci. Eng. 1, 215–222 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kuang, Y., Nagy, J.D., Elser, J.J.: Biological stoichiometry of tumor dynamics: mathematical models and analysis. Disc. Cont. Dyn. Sys. Ser. B 4, 221–240 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Kuang, Y., Nagy, J.D., Eikenberry, S.E.: Introduction to Mathematical Oncology. Chapman and Hall/CRC, Boca Raton (2016)

    MATH  Google Scholar 

  10. Li, X., Wang, H., Kuang, Y.: Global analysis of a stoichiometric producer-grazer model with Holling type functional responses. J. Math. Biol. 63, 901–932 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Loladze, I., Kuang, Y., Elser, J.J.: Stoichiometry in producer-grazer systems: linking energy flow with element cycling. Bull. Math. Biol. 62, 1137–1162 (2000)

    Article  MATH  Google Scholar 

  12. Lotka, A.: Elements of Physical Biology. Williams & Wilkins, Baltimore, MD (1925)

    MATH  Google Scholar 

  13. Peace, A.: Effects of light, nutrients, and food chain length on trophic efficiencies in simple stoichiometric aquatic food chain models. Ecol. Modell. 312, 125–135 (2015)

    Article  Google Scholar 

  14. Peace, A., Zhao, Y., Loladze, I., Elser, J.J., Kuang, Y.: A stoichiometric producer-grazer model incorporating the effects of excess food-nutrient content on consumer dynamics. Math. Biosci. 244, 107–115 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Peace, A., Wang, H., Kuang, Y.: Dynamics of a producer-grazer model incorporating the effects of excess food-nutrient content on grazer’s growth. Bull. Math. Biol. 76, 2175–2197 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Volterra, V.: Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–600 (1926)

    Article  MATH  Google Scholar 

  17. Wang, H., Smith, H.L., Kuang, Y., Elser, J.J.: Dynamics of stoichiometric bacteria-algae interactions in the epilimnion. SIAM J. Appl. Math. 68, 503–522 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, H., Kuang, Y., Loladze, I.: Dynamics of a mechanistically derived stoichiometric producer-grazer model. J. Biol. Dyn. 2, 286–296 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, H., Sterner, R.W., Elser, J.J.: On the “strict homeostasis” assumption in ecological stoichiometry. Ecol. Modell. 243, 81–88 (2012)

    Article  Google Scholar 

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Acknowledgments

The third author is partially supported by the NSFC (11571041) and the Fundamental Research Funds for the Central Universities. The fourth author is partially supported by the NSERC.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Tian Xie & Xiong Li

  2. School of Information and Mathematics, Yangtze University, Jingzhou, Hubei, 434023, People’s Republic of China

    Xianshan Yang

  3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Hao Wang

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  1. Tian Xie
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  2. Xianshan Yang
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  3. Xiong Li
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  4. Hao Wang
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Correspondence to Hao Wang.

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Xie, T., Yang, X., Li, X. et al. Complete Global and Bifurcation Analysis of a Stoichiometric Predator–Prey Model. J Dyn Diff Equat 30, 447–472 (2018). https://doi.org/10.1007/s10884-016-9551-5

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  • DOI: https://doi.org/10.1007/s10884-016-9551-5

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