Skip to main content
SpringerLink
Log in

Effects of Protection Zone and Nonlinear Growth on a Predator-Prey Model

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

In this paper, we are concerned about a predator-prey model with a protection zone for the prey and nonlinear growth rate for the predator. Global bifurcation, uniqueness and stability of the positive steady-state solution are given as well as the related dynamical behavior. Our study shows that there exists a critical growth rate of the prey, which depends on the protection zone and its diffusion coefficient. If the protection zone is suitably small, then the prey cannot be saved effectively and it may become extinct as the death rate of the predator is small; whereas, if the protection zone is appropriately large, then the prey persists and may coexist with the predator even though the death rate of the predator is small. In addition, it is shown that the nonlinear growth enlarges the coexistence region of the prey and predator.

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

References

  1. Beverton, R.J.H., Holt, S.J.: On the Dynamics of Exploited Fish Populations. Fish. Invest. (1957)

    Google Scholar 

  2. Cantrell, R.S., Cosner, C.: Spatial Ecology via Reaction-Diffusion Equations. Wiley, New York (2003)

    MATH  Google Scholar 

  3. Cantrell, R.S., Cosner, C., Hutson, V.: Permanence in ecological systems with spatial heterogeneity. Proc. R. Soc. Edinb., Sect. A 123, 533–559 (1993)

    Article  MathSciNet  Google Scholar 

  4. Cantrell, R.S., Cosner, C., Lam, K.-Y.: Ideal free dispersal under general spatial heterogeneity and time periodicity. SIAM J. Appl. Math. 81, 789–813 (2021)

    Article  MathSciNet  Google Scholar 

  5. Chen, S., Yu, J.: Dynamics of a diffusive predator-prey system with a nonlinear growth rate for the predator. J. Differ. Equ. 260, 7923–7939 (2016)

    Article  MathSciNet  Google Scholar 

  6. Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)

    Article  MathSciNet  Google Scholar 

  7. Cui, R., Shi, J., Wu, B.: Strong Allee effect in a diffusive predator-prey system with a protection zone. J. Differ. Equ. 256, 108–129 (2014)

    Article  MathSciNet  Google Scholar 

  8. Dancer, E.N., Du, Y.: Effects of certain degeneracies in the predator-prey model. SIAM J. Math. Anal. 34, 292–314 (2002)

    Article  MathSciNet  Google Scholar 

  9. De la Sen, M.: The generalized Beverton-Holt equation and the control of populations. Appl. Math. Model. 32, 2312–2328 (2008)

    Article  MathSciNet  Google Scholar 

  10. De la Sen, M., Alonso-Quesada, S.: Control issues for the Beverton-Holt equation in ecology by locally monitoring the environment carrying capacity: non-adaptive and adaptive cases. Appl. Math. Comput. 215, 2616–2633 (2009)

    MathSciNet  MATH  Google Scholar 

  11. DeAngelis, D.L., Ni, W.-M., Zhang, B.: Dispersal and spatial heterogeneity: single species. J. Math. Biol. 72, 239–254 (2016)

    Article  MathSciNet  Google Scholar 

  12. Du, Y.: Effects of a degeneracy in the competition model, part I. Classical and generalized steady-state solutions. J. Differ. Equ. 181, 92–132 (2002)

    Article  Google Scholar 

  13. Du, Y., Hsu, S.-B.: A diffusive predator-prey model in heterogeneous environment. J. Differ. Equ. 203, 331–364 (2004)

    Article  MathSciNet  Google Scholar 

  14. Du, Y., Liang, X.: A diffusive competition model with a protection zone. J. Differ. Equ. 244, 61–86 (2008)

    Article  MathSciNet  Google Scholar 

  15. Du, Y., Shi, J.: A diffusive predator-prey model with a protection zone. J. Differ. Equ. 229, 63–91 (2006)

    Article  MathSciNet  Google Scholar 

  16. Du, Y., Shi, J.: Allee effect and bistability in a spatially heterogeneous predator-prey model. Trans. Am. Math. Soc. 359, 4557–4593 (2007)

    Article  MathSciNet  Google Scholar 

  17. Du, Y., Peng, R., Wang, M.: Effect of a protection zone in the diffusive Leslie predator-prey model. J. Differ. Equ. 246, 3932–3956 (2009)

    Article  MathSciNet  Google Scholar 

  18. He, X., Ni, W.-M.: Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity I. Commun. Pure Appl. Math. 69, 981–1014 (2016)

    Article  MathSciNet  Google Scholar 

  19. He, X., Zheng, S.: Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response. J. Math. Biol. 75, 239–257 (2017)

    Article  MathSciNet  Google Scholar 

  20. Hutson, V., Lou, Y., Mischaikow, K., Poláčik, P.: Competing species near a degenerate limit. SIAM J. Math. Anal. 35, 453–491 (2003)

    Article  MathSciNet  Google Scholar 

  21. Kato, T.: Perturbation Theory for Linear Operators. Die Grundlehren der Mathematischen Wissenschaften, vol. 132. Springer, New York (1966)

    Book  Google Scholar 

  22. Kuto, K.: Bifurcation branch of stationary solutions for a Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment. Nonlinear Anal., Real World Appl. 10, 943–965 (2009)

    Article  MathSciNet  Google Scholar 

  23. Li, H., Pang, P.Y.H., Wang, M.: Qualitative analysis of a diffusive prey-predator model with trophic interactions of three levels. Discrete Contin. Dyn. Syst., Ser. B 17, 127–152 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Li, S., Wu, J., Dong, Y.: Uniqueness and non-uniqueness of steady states of a diffusive predator-prey-mutualist model with a protection zone. J. Differ. Equ. 274, 151–187 (2021)

    Article  MathSciNet  Google Scholar 

  25. López-Gómez, J.: Spectral Theory and Nonlinear Functional Analysis. Chapman and Hall/CRC, Boca Raton (2001)

    Book  Google Scholar 

  26. Lou, Y.: On the effects of migration and spatial heterogeneity on single and multiple species. J. Differ. Equ. 223, 400–426 (2006)

    Article  MathSciNet  Google Scholar 

  27. Lou, Y., Ni, W.-M.: Diffusion, self-diffusion and cross-diffusion. J. Differ. Equ. 131, 79–131 (1996)

    Article  MathSciNet  Google Scholar 

  28. Lou, Y., Zhao, X.-Q., Zhou, P.: Global dynamics of a Lotka-Volterra competition-diffusion-advection system in heterogeneous environments. J. Math. Pures Appl. 121, 47–82 (2019)

    Article  MathSciNet  Google Scholar 

  29. Oeda, K.: Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone. J. Differ. Equ. 250, 3988–4009 (2011)

    Article  MathSciNet  Google Scholar 

  30. Pao, C.V.: On Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)

    MATH  Google Scholar 

  31. Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)

    Article  MathSciNet  Google Scholar 

  32. Shi, J., Wang, X.: On global bifurcation for quasilinear elliptic systems on bounded domains. J. Differ. Equ. 246, 2788–2812 (2009)

    Article  MathSciNet  Google Scholar 

  33. Wang, Y.-X., Li, W.-T.: Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone. Nonlinear Anal., Real World Appl. 14, 224–245 (2013)

    Article  MathSciNet  Google Scholar 

  34. Yang, W.-B.: Existence asymptotic behavior of solutions for a predator-prey system with a nonlinear growth rate. Acta Appl. Math. 152, 57–72 (2017)

    Article  MathSciNet  Google Scholar 

  35. Yang, W.-B., Wu, J.-H., Nie, H.: Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate. Commun. Pure Appl. Anal. 14, 1183–1204 (2015)

    Article  MathSciNet  Google Scholar 

  36. Zeng, X., Zeng, W., Liu, L.: Effect of the protection zone on coexistence of the species for a ratio-dependent predator-prey model. J. Math. Anal. Appl. 462, 1605–1626 (2018)

    Article  MathSciNet  Google Scholar 

  37. Zhao, X.-Q., Zhou, P.: On a Lotka-Volterra competition model: the effects of advection and spatial variation. Calc. Var. Partial Differ. Equ. 55, 73 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, People’s Republic of China

    Yu-Xia Wang

Authors
  1. Yu-Xia Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu-Xia Wang.

Additional information

Publisher’s Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, YX. Effects of Protection Zone and Nonlinear Growth on a Predator-Prey Model. Acta Appl Math 176, 15 (2021). https://doi.org/10.1007/s10440-021-00461-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10440-021-00461-y

Keywords

Mathematics Subject Classification (2000)

Search

Navigation