Skip to main content
SpringerLink
Log in

Turing Patterns in a Predator–Prey Reaction–Diffusion Model with Seasonality and Fear Effect

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

In this paper, we study a predator–prey reaction–diffusion model with seasonality and fear effect. In this model, the predator species reproduces only at a certain time of each year. We calculate the steady states of the system and study their stabilities, and then, we derive the conditions for Turing instability to occur. Numerical simulations show that (i) spatial patterns can emerge for Beddington–DeAngelis functional response, (ii) the system can have oscillatory behavior for both linear functional response and Beddington–DeAngelis functional response.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

Availability of data and materials

The authors declare that the manuscript has no associated data.

References

  • Beddington, J.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44, 331–340 (1975)

  • Casewell, H.: Matrix Population Models: Construction, Analysis and Interpretation. Sinauer Association, Sunderland, Masssachusetts (2001)

  • Creel, S., Christianson, D.: Relationships between direct predation and risk effects. Trends Ecol. Evol. 23(4), 194–201 (2008)

    Article  Google Scholar 

  • Cresswell, W.: Predation in bird populations. J. Ornithol. 152, 251–263 (2011)

    Article  Google Scholar 

  • DeAngelis, D., Goldstein, R., O’Neill, R.: A model for trophic interaction. Ecology 56, 881–892 (1975)

    Article  Google Scholar 

  • Fazly, M., Lewis, M., Wang, H.: On impulsive reaction-diffusion models in higher dimensions. SIAM J. Appl. Math. 77(1), 224–246 (2017)

    Article  MathSciNet  Google Scholar 

  • Fazly, M., Lewis, M., Wang, H.: Analysis of propagation for impulsive reaction-diffusion models. SIAM J. Appl. Math. 80(1), 521–542 (2020)

    Article  MathSciNet  Google Scholar 

  • Huang, Q., Wang, H., Lewis, M.: A hybrid continuous/discrete-time model for invasion dynamics of zebra mussels in rivers. SIAM J. Appl. Math. 77(3), 854–880 (2017)

    Article  MathSciNet  Google Scholar 

  • Jiang, W., Cao, X., Wang, C.: Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain. Discrete Contin. Dyn. Syst. Ser. B 27(2), 1163–1178 (2022)

    Article  MathSciNet  Google Scholar 

  • Kelley, W., Peterson, A.: Difference Equations: An Introduction with Applications, 2nd edn. Academic Press, San Diego (2001)

    Google Scholar 

  • Kuznetsov, Y., Meijer, H.: Numerical Bifurcation Analysis of Maps: From Theory To Software. Cambridge University Press, Cambridge (2019)

    Book  Google Scholar 

  • Lewis, M., Li, B.: Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models. Bull. Math. Biol. 74(10), 2383–2402 (2012)

    Article  MathSciNet  Google Scholar 

  • Lima, S.: Nonlethal effects in the ecology of predator–prey interactions. Bioscience 48(1), 25–34 (1998)

    Article  Google Scholar 

  • Lima, S.: Predators and the breeding bird: behavioural and reproductive flexibility under the risk of predation. Biol. Rev. 84(3), 485–513 (2009)

    Article  Google Scholar 

  • Lin, Y., Wang, Q.: Spreading speed and traveling wave solutions in impulsive reactiondiffusion models. Commun. Nonlinear Sci. Numer. Simul. 23(1), 185–191 (2015)

    Article  MathSciNet  Google Scholar 

  • Murray, J.: Mathematical biology II. Spatial Models and Biomedical Applications. Springer, New York (2001)

    Google Scholar 

  • Ni, W., Tang, M.: Turing patterns in the Lengyel–Epstein system for the CIMA reaction. Trans. Am. Math. Soc. 357, 3953–3969 (2005)

    Article  MathSciNet  Google Scholar 

  • Pachepsky, E., Nisbet, R., Murdoch, W.: Between discrete and continuous: consumer-resource dynamics with synchronized reproduction. Ecology 89, 280–288 (2008)

    Article  Google Scholar 

  • Peng, R., Yi, F., Zhao, X.: Spatiotemporal patterns in a reaction–diffusion model with the Degn–Harrison reaction scheme. J. Differ. Equ. 254, 2465–2498 (2013)

    Article  MathSciNet  Google Scholar 

  • Shi, H., Ruan, S.: Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference. IMA J. Appl. Math. 80, 1534–1568 (2015)

    Article  MathSciNet  Google Scholar 

  • Singh, A., Nisbet, R.: Semi-discrete host-parasitoid models. J. Theor. Biol. 247(4), 733–742 (2007)

    Article  MathSciNet  Google Scholar 

  • Turing, A.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. Ser. B 237, 37–72 (1952)

    Article  MathSciNet  Google Scholar 

  • Vasilyeva, O., Lutscher, F., Lewis, M.: Analysis of spread and persistence for stream insects with winged adult stages. J. Math. Biol. 72, 851–875 (2016)

    Article  MathSciNet  Google Scholar 

  • Wang, X., Lutscher, F.: Turing patterns in a predator-prey model with seasonality. J. Math. Biol. 78, 711–737 (2019)

    Article  MathSciNet  Google Scholar 

  • Wang, Z., Wang, H.: Persistence and propagation of a PDE and discrete-time map hybrid animal movement model with habitat shift driven by climate change. SIAM J. Appl. Math. 80(6), 2608–2630 (2020)

    Article  MathSciNet  Google Scholar 

  • Wang, Z., Wang, H.: Bistable traveling waves in impulsive reaction-advection-diffusion models. J. Differ. Equ. 285, 17–39 (2021)

    Article  MathSciNet  Google Scholar 

  • Wang, X., Zou, X.: Modeling the fear effect in predator-prey interactions with adaptive avoidance of predators. Bull. Math. Biol. 79, 1325–1359 (2017)

    Article  MathSciNet  Google Scholar 

  • Wang, Y., Zou, X.: On a predator-prey system with digestion delay and anti-predation strategy. J. Nonlinear Sci. 30, 1579–1605 (2020)

    Article  MathSciNet  Google Scholar 

  • Wang, J., Shi, J., Wei, J.: Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey. J. Differ. Equ. 251, 1276–1304 (2011)

    Article  MathSciNet  Google Scholar 

  • Wang, X., Zanette, L., Zou, X.: Modelling the fear effect in predator-prey interactions. J. Math. Biol. 73, 1179–1204 (2016)

    Article  MathSciNet  Google Scholar 

  • Wang, Z., Salmaniw, Y., Wang, H.: Persistence and propagation of a discrete-time map and PDE hybrid model with strong Allee effct. Nonlinear Anal. Real World Appl. 61, 103336 (2021)

    Article  Google Scholar 

  • Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Springer-Verlag, New York (2003)

    Google Scholar 

  • Wu, R., Zhao, X.: Spatial invasion of a birth pulse population with nonlocal dispersal. SIAM J. Appl. Math. 79(3), 1075–1097 (2019)

    Article  MathSciNet  Google Scholar 

  • Yi, F., Wei, J., Shi, J.: Diffusion-driven instability and bifurcation in the Lengyel–Epstein system. Nonlinear Anal. Real World Appl. 9, 1038–1051 (2008)

    Article  MathSciNet  Google Scholar 

  • Yi, F., Gaffney, E., Seirin-Lee, S.: The bifurcation analysis of Turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete Contin. Dyn. Syst. Ser. B 22(2), 647–668 (2017)

    MathSciNet  Google Scholar 

  • Yi, F., Liu, S., Tuncer, N.: Spatiotemporal patterns of a reaction-diffusion substrate-inhibition Seelig model. J. Dyn. Differ. Equ. 29, 219–241 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Dr. Xiaoying Wang for sharing the numerical simulation data of paper (Wang and Lutscher 2019), which helped a lot in improving our numerical simulation codes. We thank the editors and the anonymous reviewers for their valuable comments, which helped to improve the presentation of this paper.

Funding

This research was supported by the National Natural Science Foundation of China (No. 12071491).

Author information

Authors and Affiliations

  1. School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, Guangdong, People’s Republic of China

    Tianyang Li & Qiru Wang

Authors
  1. Tianyang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiru Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

TL contributed to writing—original draft. QW contributed to writing—review and editing.

Corresponding author

Correspondence to Qiru Wang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interests.

Ethical Approval

Not applicable.

Additional information

Communicated by Anthony Bloch.

Publisher's Note

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

This research was supported by the National Natural Science Foundation of China (No. 12071491).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, T., Wang, Q. Turing Patterns in a Predator–Prey Reaction–Diffusion Model with Seasonality and Fear Effect. J Nonlinear Sci 33, 86 (2023). https://doi.org/10.1007/s00332-023-09938-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00332-023-09938-6

Keywords

Mathematics Subject Classification

Search

Navigation