[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Stability and Double-Hopf Bifurcations of a Gause–Kolmogorov-Type Predator–Prey System with Indirect Prey-Taxis

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

In this paper, we deal with the Gause–Kolmogorov-type predator–prey system with indirect prey-taxis, which means that directional movement of predators is stimulated by some chemicals emitted by preys. The existence of the positive equilibrium, the effect of the indirect prey-taxis on the stability and the related bifurcations are investigated. The critical values for the occurrence of the Hopf bifurcation, Turing bifurcation, Turing–Hopf bifurcation and double-Hopf bifurcation are explicitly determined. An algorithm for calculating the normal form of the double-Hopf bifurcation for the non-resonance and weak resonance is derived. Moreover, we apply the theoretical results to the system with Holling-II type functional response, the stable region and the bifurcation curves are completely determined in the plane of the indirect prey-taxis and self saturation coefficient. The dynamical classification near the double-Hopf bifurcation point is explicitly determined. In the neighborhood of the double-Hopf bifurcation, there are stable spatially homogeneous/inhomogeneous periodic solutions, stable spatially inhomogeneous quadi-periodic solutions and the pattern transitions from one spatial–temporal patterns to another one with the changes of the indirect taxis and semi saturation coefficients. The results show that spatially inhomogeneous Hopf bifurcations are induced by an indirect prey-taxis parameter \(\chi >0\), which is impossible for the reaction–diffusion predator–prey model with a direct prey-taxis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Ahn, I., Yoon, C.: Global well-posedness and stability analysis of prey–predator model with indirect prey-taxis. J. Differ. Equ. 268(8), 4222–4255 (2020)

    MathSciNet  MATH  Google Scholar 

  2. Ainseba, B., Bendahmane, M., Noussair, A.: A reaction–diffusion system modeling predator–prey with prey-taxis. Nonlinear Anal. Real World Appl. 9(5), 2086–2105 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Allesina, S., Tang, S.: Stability criteria for complex ecosystems. Nature 483(7388), 205–208 (2012)

    Google Scholar 

  4. Arditi, R., Tyutyunov, Y., Morgulis, A., Govorukhin, V., Senina, I.: Directed movement of predators and the emergence of density-dependence in predator–prey models. Theor. Popul. Biol. 59(3), 207–221 (2001)

    MATH  Google Scholar 

  5. Banerjee, M., Ghorai, S., Mukherjee, N.: Study of cross-diffusion induced Turing patterns in a ratio-dependent prey–predator model via amplitude equations. Appl. Math. Model. 55, 383–399 (2018)

    MathSciNet  MATH  Google Scholar 

  6. Berezovskaya, F., Isaev, A., Karev, G.: The role of taxis in dynamics of forest insects. Dokl. Biol. Sci. 365(1–6), 148–151 (1999)

    Google Scholar 

  7. Berezovskaya, F., Karev, G.: Bifurcations of travelling waves in population taxis models. Phys. Uspekhi 42(9), 917–929 (1999)

    Google Scholar 

  8. Cangelosi, R.A., Wollkind, D.J., Kealy-Dichone, B.J., Chaiya, I.: Nonlinear stability analyses of Turing patterns for a mussel–algae model. J. Math. Biol. 70(6), 1249–1294 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Chakraborty, A., Singh, M., Lucy, D., Ridland, P.: Predator–prey model with prey-taxis and diffusion. Math. Comput. Model. 46(3–4), 482–498 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Chow, S., Hale, J.K.: Methods of Bifurcation Theory. Springer, New York (1982)

    MATH  Google Scholar 

  11. Ding, Y., Cao, J., Jiang, W.: Double Hopf bifurcation in active control system with delayed feedback: application to glue dosing processes for particleboard. Nonlinear Dyn. 83(3), 1567–1576 (2016)

    MathSciNet  MATH  Google Scholar 

  12. Ding, Y., Jiang, W., Yu, P.: Double Hopf bifurcation in a container crane model with delayed position feedback. Appl. Math. Comput. 219(17), 9270–9281 (2013)

    MathSciNet  MATH  Google Scholar 

  13. Du, Y., Niu, B., Guo, Y., Wei, J.: Double Hopf bifurcation in delayed reaction–diffusion systems. J. Dyn. Differ. Equat. 32, 313–358 (2020)

    MathSciNet  MATH  Google Scholar 

  14. Duan, D., Niu, B., Wei, J.: Hopf-Hopf bifurcation and chaotic attractors in a delayed diffusive predator–prey model with fear effect. Chaos Solitons Fractals 123, 206–216 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Ei, S.-I., Izuhara, H., Mimura, M.: Spatio-temporal oscillations in the Keller–Segel system with logistic growth. Physica D 277, 1–21 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Faria, T.: Normal forms and Hopf bifurcation for partial differential equations with delays. Trans. Am. Math. Soc. 352(5), 2217–2238 (2000)

    MathSciNet  MATH  Google Scholar 

  17. Faria, T.: Stability and bifurcation for a delayed predator–prey model and the effect of diffusion. J. Math. Anal. Appl. 254(2), 433–463 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Faria, T., Magalhaes, L.: Normal forms for retarded functional differential equations with parameters and applications to Hopf singularity. J. Differ. Equ. 122, 181–200 (1995)

    MATH  Google Scholar 

  19. Govorukhin, V., Morgulis, A., Senina, I., Tyutyunov, Y.: Modelling of active migrations for spatially distributed populations. Surv. Appl. Ind. Math. 6(2), 271–295 (1999)

    MATH  Google Scholar 

  20. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)

    MATH  Google Scholar 

  21. Hassard, B., Kazarinoff, N., Wan, Y.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, New York (1981)

    MATH  Google Scholar 

  22. Jiang, H., Song, Y.: Normal forms of non-resonance and weak resonance double Hopf bifurcation in the retarded functional differential equations and applications. Appl. Math. Comput. 266, 1102–1126 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Jiang, W., Wang, H., Cao, X.: Turing instability and Turing–Hopf bifurcation in diffusive Schnakenberg systems with gene expression time delay. J. Dyn. Differ. Equ. 31(4), 2223–2247 (2019)

    MathSciNet  MATH  Google Scholar 

  24. Jin, H.-Y., Wang, Z.-A.: Global stability of prey-taxis systems. J. Differ. Equ. 262(3, 1), 1257–1290 (2017)

    MathSciNet  MATH  Google Scholar 

  25. Kareiva, P., Odell, G.: Swarms of predators exhibit “preytaxis” if individual predators use area-restricted search. Am. Nat. 130, 233–270 (1987)

    Google Scholar 

  26. Kolmogorov, A.N.: Qualitative analysis of mathematical models of populations. Prob. Cybern. 25, 100–106 (1972)

    Google Scholar 

  27. Kong, L., Lu, F.: Bifurcation branch of stationary solutions in a general predator–prey system with prey-taxis. Comput. Math. Appl. 78(1), 191–203 (2019)

    MathSciNet  MATH  Google Scholar 

  28. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (1998)

    MATH  Google Scholar 

  29. Lee, J., Hillen, T., Lewis, M.: Pattern formation in prey-taxis systems. J. Biol. Dyn. 3(6), 551–573 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Li, C., Wang, X., Shao, Y.: Steady states of a predator–prey model with prey-taxis. Nonlinear Anal. Theory Methods Appl. 97, 155–168 (2014)

    MathSciNet  MATH  Google Scholar 

  31. Losey, J., Denno, R.: The escape response of pea aphids to foliar-foraging predators: factors affecting dropping behaviour. Ecol. Entomol. 23(1), 53–61 (1998)

    Google Scholar 

  32. Lotka, A.: Relation between birth rates and death rates. Science 26, 21–22 (1907)

    Google Scholar 

  33. Murray, J.: Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, Berlin (2003)

    MATH  Google Scholar 

  34. Okubo, A., Levin, S.: Diffusion and Ecological Problems: Modern Perspectives. Springer, New York (2001)

    MATH  Google Scholar 

  35. Painter, K.J., Hillen, T.: Spatio-temporal chaos in a chemotaxis model. Physica D 240, 363–375 (2011)

    MATH  Google Scholar 

  36. Shi, J., Wang, C., Wang, H.: Diffusive spatial movement with memory and maturation delays. Nonlinearity 32(9), 3188–3208 (2019)

    MathSciNet  MATH  Google Scholar 

  37. Shi, J., Wang, C., Wang, H., Yan, X.: Diffusive spatial movement with memory. J. Dyn. Differ. Equ. 32(2), 979–1002 (2020)

    MathSciNet  MATH  Google Scholar 

  38. Song, Y., Jiang, H., Yuan, Y.: Turing–Hopf bifurcation in the reaction–diffusion system with delay and application to a diffusive predator–prey model. J. Appl. Anal. Comput. 9(3), 1132–1164 (2019)

    MathSciNet  MATH  Google Scholar 

  39. Song, Y., Shi, J., Wang, H.: Stability and bifurcation analysis in the resource-consumer model with random and memory-based diffusions. In preparation (2020)

  40. Song, Y., Tang, X.: Stability, steady-state bifurcations, and Turing patterns in a predator–prey model with herd behavior and prey-taxis. Stud. Appl. Math. 139(3), 371–404 (2017)

    MathSciNet  MATH  Google Scholar 

  41. Song, Y., Wu, S., Wang, H.: Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect. J. Differ. Equ. 267(11), 6316–6351 (2019)

    MathSciNet  MATH  Google Scholar 

  42. Song, Y., Zhang, T., Peng, Y.: Turing–Hopf bifurcation in the reaction–diffusion equations and its applications. Commun. Nonlinear Sci. Numer. Simul. 33, 229–258 (2016)

    MathSciNet  MATH  Google Scholar 

  43. Song, Y., Zou, X.: Spatiotemporal dynamics in a diffusive ratio-dependent predator–prey model near a Hopf–Turing bifurcation point. Comput. Math. Appl. 67(10), 1978–1997 (2014)

    MathSciNet  MATH  Google Scholar 

  44. Tao, Y.: Global existence of classical solutions to a predator–prey model with nonlinear prey-taxis. Nonlinear Anal. Real World Appl. 11(3), 2056–2064 (2010)

    MathSciNet  MATH  Google Scholar 

  45. Tello, J.I., Wrzosek, D.: Predator–prey model with diffusion and indirect prey-taxis. Math. Models Methods Appl. Sci. 26(11, SI), 2129–2162 (2016)

    MathSciNet  MATH  Google Scholar 

  46. Tsyganov, M., Biktashev, V.: Half-soliton interaction of population taxis waves in predator–prey systems with pursuit and evasion. Phys. Rev. E 70(3), 031901 (2004)

    MathSciNet  Google Scholar 

  47. Tyutyunov, Y., Titova, L., Arditi, R.: A minimal model of pursuit-evasion in a predator–prey system. Math. Model. Nat. Phenom. 2(4), 122–134 (2007)

    MathSciNet  MATH  Google Scholar 

  48. Tyutyunov, Y., Sen, V.D., Titova, L., Banerjee, I.M.: Predator overcomes the Allee effect due to indirect prey-taxis. Ecol. Complex. 39, 100772 (2019)

  49. Tyutyunov, Y.V., Titova, L.I., Senina, I.N.: Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system. Ecol. Complex. 31, 170–180 (2017)

    Google Scholar 

  50. Volterra, I.: Sui tentativi di applicazione della matematiche alle scienze biologiche esociali. G. Econ. 23(12), 436–458 (1901)

    Google Scholar 

  51. Wang, J., Guo, X.: Dynamics and pattern formations in a three-species predator–prey model with two prey-taxis. J. Math. Anal. Appl. 475(2), 1054–1072 (2019)

    MathSciNet  MATH  Google Scholar 

  52. Wang, X., Wang, W., Zhang, G.: Global bifurcation of solutions for a predator–prey model with prey-taxis. Math. Methods Appl. Sci. 38(3), 431–443 (2015)

    MathSciNet  MATH  Google Scholar 

  53. Wu, S., Shi, J., Wu, B.: Global existence of solutions and uniform persistence of a diffusive predator–prey model with prey-taxis. J. Differ. Equ. 260(7), 5847–5874 (2016)

    MathSciNet  MATH  Google Scholar 

  54. Yousefnezhad, M., Mohammadi, S.A.: Stability of a predator–prey system with prey taxis in a general class of functional responses. Acta Math. Sci. 36(1), 62–72 (2016)

    MathSciNet  MATH  Google Scholar 

  55. Zhang, L., Fu, S.: Global bifurcation for a Holling–Tanner predator–prey model with prey-taxis. Nonlinear Anal. Real World Appl. 47, 460–472 (2019)

    MathSciNet  MATH  Google Scholar 

  56. Zhang, T., Liu, X., Meng, X., Zhang, T.: Spatio-temporal dynamics near the steady state of a planktonic system. Comput. Math. Appl. 75(12), 4490–4504 (2018)

    MathSciNet  MATH  Google Scholar 

  57. Zuo, W., Wei, J.: Stability and Hopf bifurcation in a diffusive predator–prey system with delay effect. Nonlinear Anal. Real World Appl. 12(4), 1998–2011 (2011)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous reviewer for his/her carefully reading the manuscript and many valuable and professional comments and suggestions, which greatly improve the initial manuscript.

Author information

Authors and Affiliations

  1. College of Science, China University of Petroleum (East China), Qingdao, 266580, China

    Wenjie Zuo

  2. Department of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China

    Yongli Song

Authors
  1. Wenjie Zuo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongli Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongli Song.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Partially supported by the NSFC of China (Nos. 11971143, 11671236), Shandong Provincial Natural Science Foundation (No. ZR2019MA006), the Fundamental Research Funds for the Central Universities (No. 19CX02055A) and Natural Science Foundation of Zhejiang Province of China (No. LY19A010010).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zuo, W., Song, Y. Stability and Double-Hopf Bifurcations of a Gause–Kolmogorov-Type Predator–Prey System with Indirect Prey-Taxis. J Dyn Diff Equat 33, 1917–1957 (2021). https://doi.org/10.1007/s10884-020-09878-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-020-09878-9

Keywords

Search

Navigation