Skip to main content
SpringerLink
Log in

Dynamics for a Three-Species Predator-Prey Model with Density-Dependent Motilities

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

A Correction to this article was published on 17 September 2021

This article has been updated

Abstract

This work deals with a general cross-diffusion system modeling the dynamics behavior of two predators and one prey with signal-dependent diffusion and sensitivity subject to homogeneous Neumann boundary conditions. Firstly, in light of some \(L^p-\)estimate techniques, we rigorously prove the global existence and uniform boundedness of positive classical solutions in any dimensions with suitable conditions on motility functions and the coefficients of logistic source. Moreover, by constructing some appropriate Lyapunov functionals, we further establish the asymptotic behavior of solutions to a specific model with Lotka-Volterra type functional responses and density-dependent death rates for two predators as well as logistic type for the prey. Our results not only generalize the previously known one, but also present some new conclusions.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

References

  1. Alikakos, N.D.: \(L^p\) bounds of solutions of reaction-diffusion equations. Comm. Partial Diff. Equ. 4, 827–868 (1979)

    Article  MATH  Google Scholar 

  2. Amann, H.: Dynamic theory of quasilinear parabolic equations III. Global existence. Math. Z. 202, 219–250 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amann, H.: Dynamic theory of quasilinear parabolic equations II. Reaction-diffusion systems. Diff. Integr. Equ. 3, 13–75 (1990)

    MathSciNet  MATH  Google Scholar 

  4. Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), volume 133 of Teubner-Texte Math., pages 9-126. Teubner, Stuttgart (1993)

  5. Bai, X., Winkler, M.: Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics. Indiana Univ. Math. J. 65, 553–583 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Lee, J.M., Hilllen, T., Lewis, M.A.: Continuous traveling waves for prey-taxis. Bull. Math. Biol. 70, 654–676 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. He, X., Zheng, S.: Global boundedness of solutions in a reaction-diffusion system of predator-prey model with prey-taxis. Appl. Math. Lett. 49, 73–77 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hirata, M., Kurima, S., Mizukami, M., Yokota, T.: Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics. J. Diff. Equ. 263, 470–490 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Holling, C.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entom. Soc. Can. 45, 1–60 (1965)

    Google Scholar 

  14. Hsiao, L., De Mottoni, P.: Persistence in reacting-diffusing systems: Interaction of two predators and one prey. Nonlinear Anal. 11, 877–891 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jin, H., Wang, Z.: Global stability of prey-taxis systems. J. Diff. Equ. 262, 1257–290 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jin, H., Kim, Y., Wang, Z.A.: Boundedness, stabilization, and pattern formation driven by densitysuppressed motility. SIAM J. Appl. Math. 78, 1632–1657 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jin, H., Wang, Z.: Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion. Eur. J. Appl. Math. (2021). https://doi.org/10.1017/S0956792520000248

  18. Kareiva, P., Odell, G.T.: Swarms of predators exhibit “preytaxis’ if individual predators use area-restricted search. Am. Nat. 130, 233–270 (1987)

  19. Lankeit, J., Wang, Y.: Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discret. Contin. Dyn. Syst. 37, 6099–6121 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lotka, A.J.: Elements of Physical Biology. Williams and Wilkins Co., Baltimore (1925)

    MATH  Google Scholar 

  21. Lin, J., Wang, W., Zhao, C., Yang, T.: Global dynamics and traveling wave solutions of two predators-one prey models. Discret. Contin. Dyn. Syst. Ser. B 20, 1135–1154 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Loladze, I., Kuang, Y., Elser, J.J., Fagan, W.F.: Competition and stoichiometry: coexistence of two predators on one prey. Theo. Popul. Biol. 65, 1–15 (2004)

    Article  MATH  Google Scholar 

  23. Pang, P., Wang, M.: Strategy and stationary pattern in a three-species predator-prey model. J. Diff. Equ. 200, 245–273 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Porzio, M., Vespri, V.: H\(\ddot{o}\)lder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Diff. Equ. 103, 146–178 (1993)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Tello, J.I., Wrzosek, D.: Predator-prey model with diffusion and indirect prey-taxis. Math. Model. Method. Appl. Sci. 26, 2129–62 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tona, T., Hieu, N.: Dynamics of species in a model with two predators and one prey. Nonlinear Anal. 74, 4868–4881 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Volterra, V.: Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560 (1926)

    Article  MATH  Google Scholar 

  29. Wang, J., Wang, M.: Boundeness and global stability of the two-predator and one-prey models with nonlinear prey-taxis. Z. Angew. Math. Phys. 69, 63 (2018)

    Article  MATH  Google Scholar 

  30. Wang, J., Wang, M.: The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary. Math. Method. Appl. Sci. 41, 6741–6762 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, J., Wang, M.: Global solution of a diffusive predator-prey model with prey-taxis. Comput. Math. Appl. 77, 2676–94 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, J., Wang, M.: The dynamics of a predator-prey model with diffusion and indirect prey-taxis. J. Dyn. Differ. Equ. (2019). https://doi.org/10.1007/s10884-019-09778-7

  33. Wang, K., Wang, Q., Yu, F.: Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis. Discret. Contin. Dyn. Syst. 37(1), 505–543 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, Q., Gai, C., Yan, J.: Qualitative analysis of a Lotka-Volterra competition system with advection. Discret. Contin. Dyn. Syst. 35, 1239–1284 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, Q., Song, Y., Shao, L.: Nonconstant positive steady states and pattern formation of 1D prey-taxis systems. J. Nonlinear Sci. 1–27,(2016)

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, Z.A., Xu, J.: On the Lotka-Volterra competition system with dynamical resources and density-dependent diffusion. J. Math. Biol. (2021). https://doi.org/10.1007/s00285-021-01562-w

  38. Winkler, M.: Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation. J. Diff. Equ. 263, 4826–69 (2017)

  39. Wu, S., Shi, J., Wu, B.: Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis. J. Diff. Equ. 260, 5847–74 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Xiang, T.: Global dynamics for a diffusive predator-prey model with prey-taxis and classical Lotka-Volterra kinetics. Nonlinear Anal. Real World Appl. 39, 278–99 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author is supported by Young scholars development fund of SWPU grant 202199010087. The second author is supported by NSFC under grants 11771062 and 11971082, the Fundamental Research Funds for the Central Universities under grants 2019CDJCYJ001, 2020CDJQY-Z001 and XDJK2020C054, Chongqing Key Laboratory of Analytic Mathematics and Applications. The third author is supported by China Postdoctoral Science Foundation under Grant 2020M673102, the Natural Science Foundation of Chongqing, China under grant cstc2020jcyj-bshX0071, Chongqing Post-doctoral Innovative Talent Support program.

Author information

Authors and Affiliations

  1. School of Sciences, Southwest Petroleum University, Chengdu, 610500, People’s Republic of China

    Shuyan Qiu

  2. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, People’s Republic of China

    Chunlai Mu

  3. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Xinyu Tu

Authors
  1. Shuyan Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunlai Mu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xinyu Tu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shuyan Qiu.

Additional information

Publisher's Note

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

The original online version of this article was revised: the error in Acknowledgement section has been corrected.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiu, S., Mu, C. & Tu, X. Dynamics for a Three-Species Predator-Prey Model with Density-Dependent Motilities. J Dyn Diff Equat 35, 709–733 (2023). https://doi.org/10.1007/s10884-021-10020-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-021-10020-6

Keywords

Mathematics Subject Classification

Search

Navigation