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Slow-Fast Dynamics and Its Application to a Biological Model

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Mathematical Sciences with Multidisciplinary Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 157))

Abstract

In this article we introduce some basic concepts about slow-fast dynamics and its application to a biological model, that is a predator–prey system with response functions of Holling type. The relevant studies were collaborated with Kening Lu in Li and Lu (J Differ Equ 257:4437–4469, 2014) and with Huiping Zhu in Li and Zhu (J Differ Equ 254:879–910, 2013). Another application to a medical model, especially a SIS epidemic model with nonlinear incidence, was published in Li et al. (J Math Anal Appl 420:987–1004, 2014), collaborated with Jiaquan Li, Zhien Ma, and Huiping Zhu. The studies are based on singular perturbation theory developed by F. Dumortier, R. Roussarie, and P. De Maesschalck, see, for example, Dumortier and Roussarie (Mem Am Math Soc 121(577):1–100, 1996), Dumortier and Roussarie (J Differ Equ 174:1–29, 2001), Dumortier and Roussarie (Discrete Continuous Dyn Syst Ser S 2:723–781, 2009), De Maesschalck and Dumortier (Trans Am Math Soc 358(5):2291–2334, 2006), De Maesschalck and Dumortier (Proc R Soc Edinb A 138(2):265–299, 2008), De Maesschalck et al. (Indag Math 22:165–206, 2011), and De Maesschalck et al. (C R Math Acad Sci Paris 352(4):317–320, 2014).

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References

  1. Albrecht, F., Gatzke, H., Wax, N.: Stable limit cycles in prey-predator populations. Science 181, 1073–1074 (1973)

    Article  Google Scholar 

  2. Andrews, J.F.: A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng. 10, 707–723 (1968)

    Article  Google Scholar 

  3. Bazykin, A.D.: Nonlinear Dynamics of Interacting Populations. World Scientific Series on Nonlinear Science Series A, vol. 11. World Scientific, Singapore/New Jersey/London/Hong Kong (2000)

    Google Scholar 

  4. Broer, H.W., Naudot, V., Roussarie, R., Saleh, K.: A predator-prey model with non-monotonic response function. Regul. Chaotic Dyn. 11, 155–165 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Broer, H.W., Saleh, K., Naudot, V., Roussarie, R.: Dynamics of a predator-prey model with non-monotonic response function. Discrete Continuous Dyn. Syst. 18, 221–251 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, L., Jing, Z.: Existence and uniqueness of limit cycles of differential equations with predator-prey interactions. Kexue Tongbao 29 (9), 521–523 (1984) (in Chinese)

    MathSciNet  Google Scholar 

  7. Chen, J., Zhang, H.: The qualitative analysis of two species predator-prey model with Holling’s type III functional response. Appl. Math. Mech. 7 (1), 73–80 (1986) (in Chinese)

    MathSciNet  MATH  Google Scholar 

  8. Cheng, K.S.: Uniqueness of a limit cycle for a predator-prey system. SIAM J. Math. Anal. 12 (4), 541–548 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  9. Collings, J.B.: The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model. J. Math. Biol. 36, 149–168 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. De Maesschalck, P., Dumortier, F.: Canard solutions at non-generic turning points. Trans. Am. Math. Soc. 358 (5), 2291–2334 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. De Maesschalck, P., Dumortier, F.: Canard cycles in the presence of slow dynamics with singularities. Proc. R. Soc. Edinb. A 138 (2), 265–299 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. De Maesschalck, P., Dumortier, F., Roussarie, R.: Cyclicity of common slow-fast cycles. Indag. Math. 22, 165–206 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. De Maesschalck, P., Dumortier, F., Roussarie, R.: Canard cycle transition at a slow-fast passage through a jump point. C. R. Math. Acad. Sci. Paris 352 (4), 317–320 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Mem. Am. Math. Soc. 121 (577), 1–100 (1996)

    MathSciNet  MATH  Google Scholar 

  15. Dumortier, F., Roussarie, R.: Multiple canard cycles in generalized Liénard equations. J. Differ. Equ. 174, 1–29 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dumortier, F., Roussarie, R.: Birth of canard cycles. Discrete Continuous Dyn. Syst. Ser. S 2, 723–781 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fenichel, N.: Geometric singular perturbation theory. J. Differ. Equ. 31, 53–98 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  18. Freedman, H.I., Wolkowicz, G.S.K.: Predator-prey systems with group defence: the paradox of Bull. Math. Biol. 48 (5–6), 493–508 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  19. Holling, C.S.: The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Can. Entomol. 91, 293–320 (1959)

    Article  Google Scholar 

  20. Holling, C.S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. 45, 3–60 (1965)

    Google Scholar 

  21. Krupa, M., Szmolyan, P.: Extending singular perturbation theory to non-hyperbolic points–fold and canard points in two dimensions. SIAM J. Math. Anal. 33, 286–314 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174, 312–368 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lamontagne, Y., Coutu, C., Rousseau, C.: Bifurcation analysis of a predator-prey system with generalized Holling type III function response. J. Dyn. Differ. Equ. 20, 535–571 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, C., Lu, K.: Slow divergence integral and its application to classical Liénard equations of degree 5. J. Differ. Equ. 257, 4437–4469 (2014)

    Article  MATH  Google Scholar 

  25. Li, C., Zhu, H.: Canard limit cycles for predator-prey systems with Holling types of functional response. J. Differ. Equ. 254, 879–910 (2013)

    Article  MATH  Google Scholar 

  26. Li, C, Li, J., Ma, Z., Zhu, H.: Canard phenomenon for an SIS epidemic model with nonlinear incidence. J. Math. Anal. Appl. 420, 987–1004 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. May, R.M.: Limit cycles in predator-prey communities. Science 17, 900–902 (1972)

    Article  Google Scholar 

  28. Rosenzweig, M.L.: Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171, 385–387 (1971)

    Article  Google Scholar 

  29. Rothe, F., Shafer, D.S.: Multiple bifurcation in a predator-prey system with nonmonotonic predator response. Proc. R. Soc. Edinb. Sect. A 120 (3–4), 313–347 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ruan, S., Xiao, D.: Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 61 (4), 1445–1472 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wolkowicz, G.S.K.: Bifurcation analysis of a predator-prey system involving group defence. SIAM J. Appl. Math. 48 (3), 592–606 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wrzosek, D.: Limit cycles in predator-prey models. Math. Biosci. 98 (1), 1–12 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  33. Xiao, D., Zhu, H.: Multiple focus and Hopf bifurcations in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 66 (3), 802–819 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhu, H., Campbell, S.A., Wolkowicz, G.S.K.: Bifurcation analysis of a predator-prey system with nonmonotonic function response. SIAM J. Appl. Math. 63 (2), 636–682 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author appreciates the reviewer of this paper for the valuable comments that help him to improve the earlier version of the manuscript. This work is partially supported by grant NSFC-11271027.

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Chengzhi Li

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  1. Chengzhi Li
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Correspondence to Chengzhi Li .

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Editors and Affiliations

  1. Dept of Mathematics and Economics, Virginia State University, Petersburg, Virginia, USA

    Bourama Toni

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Li, C. (2016). Slow-Fast Dynamics and Its Application to a Biological Model. In: Toni, B. (eds) Mathematical Sciences with Multidisciplinary Applications. Springer Proceedings in Mathematics & Statistics, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-319-31323-8_14

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