Skip to main content
SpringerLink
Log in

Dynamic Interplay: Cooperative Hunting Strategies in a Predator-Prey Model with Smith Growth Dynamics

  • Published:
Brazilian Journal of Physics Aims and scope Submit manuscript

Abstract

In this research paper, we delve into the critical examination of interactions between predators and prey within ecological systems. Drawing on principles from theoretical biology and mathematical ecology, our study introduces a predator-prey model that incorporates the Smith growth rate of prey and Holling type II functional response, while uniquely considering the influence of hunting cooperation between predators. We rigorously explore various aspects such as positivity, boundedness, local stability, and different bifurcation types, including transcritical, saddle-node, Hopf, cusp, and Bogdanov-Takens bifurcation. Both theoretical analyses and graphical representations are employed to offer a comprehensive understanding of the system dynamics. To unravel the complexity of the interactions, we present two-parametric bifurcation diagrams, utilizing a and K as bifurcation parameters. The first quadrant of the \(a-K\) plane is systematically partitioned into seven regions, demarcated by distinct bifurcation curves. Bistable dynamics are exhibited in the ecological system. A notable observation stemming from our investigation is the vulnerability of the predator population to extinction at low carrying capacity values. Furthermore, our numerical findings underscore the enhanced stability of the system with an increase in hunting cooperation. This research contributes valuable insights into the intricate dynamics of predator-prey interactions, shedding light on the significance of ecological parameters and cooperative behaviors in shaping the stability and persistence of these systems.

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

Similar content being viewed by others

Data Availability

No datasets were generated or analysed during the current study.

References

  1. R.M. Pringle, T.R. Kartzinel, T.M. Palmer, T.J. Thurman, K. Fox-Dobbs, C.C. Xu, M.C. Hutchinson, T.C. Coverdale, J.H. Daskin, D.A. Evangelista, K.M. Gotanda, Predator-induced collapse of niche structure and species coexistence. Nature 570(7759), 58–64 (2019)

    Article  ADS  Google Scholar 

  2. F.E. Smith, Population dynamics in Daphnia magna and a new model for population growth. Ecology 44(4), 651–663 (1963)

    Article  Google Scholar 

  3. M. Sivakumar, M. Sambath, K. Balachandran, Stability and Hopf bifurcation analysis of a diffusive predator-prey model with Smith growth. Int. J. Biomath. 8(01), 1550013 (2015)

    Article  MathSciNet  Google Scholar 

  4. X. Han, C. Lei, Bifurcation and turing instability analysis for a space-and time-discrete predator-prey system with Smith growth function. Chaos, Solitons & Fractals 166, 112910 (2023)

    Article  MathSciNet  Google Scholar 

  5. C. Packer, D. Scheel, A.E. Pusey, Why lions form groups: food is not enough. Am. Nat. 136(1), 1–19 (1990)

    Article  Google Scholar 

  6. D. Scheel, C. Packer, Group hunting behaviour of lions: a search for cooperation. Anim. Behav. 41(4), 697–709 (1991)

    Article  Google Scholar 

  7. P.A. Schmidt, L.D. Mech, Wolf pack size and food acquisition. Am. Nat. 150(4), 513–517 (1997)

    Article  Google Scholar 

  8. S. Creel, N.M. Creel, Communal hunting and pack size in African wild dogs. Lycaon pictus. Animal Behaviour 50(5), 1325–1339 (1995)

    Article  Google Scholar 

  9. S.A. Foster, Acquisition of a defended resource: a benefit of group foraging for the neotropical wrasse, Thalassoma lucasanum. Environ. Biol. Fishes 19, 215–222 (1987)

    Article  ADS  Google Scholar 

  10. N.C. Pati, G.C. Layek, N. Pal, Bifurcations and organized structures in a predator-prey model with hunting cooperation. Chaos, Solitons & Fractals 140, 110184 (2020)

    Article  MathSciNet  Google Scholar 

  11. S. Pal, N. Pal, S. Samanta, J. Chattopadhyay, Effect of hunting cooperation and fear in a predator-prey model. Ecol. Complex. 39, 100770 (2019)

    Article  Google Scholar 

  12. U. Ghosh, S. Sarkar, B. Mondal, Study of stability and bifurcation of three species food chain model with non-monotone functional response. Int. J. Appl. Comput. Math. 7, 1–24 (2021)

    Article  MathSciNet  Google Scholar 

  13. C.S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation. The Memoirs of the Entomological Society of Canada 97(S45), 5–60 (1965)

    Article  Google Scholar 

  14. L. Perko, Differential equations and dynamical systems, vol. 7 (Springer Science & Business Media, 2013)

    Google Scholar 

  15. Y.J. Gong, J.C. Huang, Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting. Acta Math. Appl. Sin. Engl. Ser. 30(1), 239–244 (2014)

    Article  MathSciNet  Google Scholar 

  16. U. Ghosh, A.A. Thirthar, B. Mondal, P. Majumdar, Effect of fear, treatment, and hunting cooperation on an eco-epidemiological model: memory effect in terms of fractional derivative. Iranian J. Sci. Technol. Trans. A Sci. 46(6), 1541–1554 (2022)

    Article  MathSciNet  Google Scholar 

  17. B. Mondal, S. Sarkar, U. Ghosh, A study of a prey-generalist predator system considering hunting cooperation and fear effects under interval uncertainty. J. Uncertain Syst. 16(02), 2350001 (2023)

    Article  Google Scholar 

  18. B. Mondal, S. Sarkar, U. Ghosh, Complex dynamics of a generalist predator-prey model with hunting cooperation in predator. Eur. Phys. J. Plus 137(1), 43 (2021)

    Article  Google Scholar 

  19. B. Mondal, A. Sarkar, S.S. Santra, D. Majumder, T. Muhammad, Sensitivity of parameters and the impact of white noise on a generalist predator-prey model with hunting cooperation. Eur. Phys. J. Plus 138(1), 1070 (2023)

    Article  Google Scholar 

  20. B. Mondal, M.S. Rahman, S. Sarkar, U. Ghosh, Studies of dynamical behaviours of an imprecise predator-prey model with Holling type II functional response under interval uncertainty. Eur. Phys. J. Plus 137(1), 1–20 (2022)

    Article  Google Scholar 

Download references

Funding

None.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kalyani, Kalyani, 741235, India

    Titli Maiti & Avijit Sarkar

  2. Department of Applied Mathematics, University of Calcutta, Kolkata, 700009, India

    Bapin Mondal

Authors
  1. Titli Maiti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bapin Mondal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Avijit Sarkar
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

T.M.-Formal analysis,Writing– original draft. B.M.-Conceptualization, Methodology, Writing– review & editing, Numerical simulation, A.S.-Validation, Supervision.

Corresponding author

Correspondence to Bapin Mondal.

Ethics declarations

Competing Interests

The authors declare no competing interests.

Additional information

Publisher's Note

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

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

Maiti, T., Mondal, B. & Sarkar, A. Dynamic Interplay: Cooperative Hunting Strategies in a Predator-Prey Model with Smith Growth Dynamics. Braz J Phys 54, 82 (2024). https://doi.org/10.1007/s13538-024-01450-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13538-024-01450-w

Keywords

Search

Navigation