Skip to main content
SpringerLink
Log in

Dynamics of a Class of Modified Leslie-Gower Predator-Prey Model with Strong Allee Effect on Prey and Non-monotonic Rational Functional Response

  • Conference paper
  • First Online:
Modeling, Simulation and Optimization

Part of the book series: Smart Innovation, Systems and Technologies ((SIST,volume 206))

Abstract

In this paper, we have modified Leslie-Gower prey-predator model with strong Allee effect on prey and non-monotonic rational functional response. We have demonstrated the existence of axial equilibria and its stability. Extinction of the all species is possible in our considered model as extinction equilibrium point is always stable. Our considered model exhibited atmost two coexistence equilibria. We have provided the stability properties of interior equilibrium points with the help of graphical Jacobian method. Our proposed model reported bi-stability behavior between trivial and coexistence equilibrium points, and trivial equilibrium point is the global attractor under some parameter choice. We have reported a comprehensive study of the global dynamics of our proposed model. In the course of bifurcation analysis, we have provided the co-dimension two bifurcation diagram by considering Allee as a bifurcation parameter. On co-dimension two bifurcation plane, our proposed model reported rich dynamics in small parametric range. We have reported all possible local and global bifurcations, namely saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation and Homoclinic bifurcation, respectively. Numerical examples are performed to validate the analytical findings.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Aguirre, P., González-Olivares, E., Sáez, E.: Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect. Nonlinear Analys. Real World Appl. 10, 1401–1416 (2009)

    Article  MathSciNet  Google Scholar 

  2. Aguirre, P., González-Olivares, E., Sáez, E.: Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect. SIAM J. Appl. Math. 69(5), 1244–1269 (2009)

    Article  MathSciNet  Google Scholar 

  3. Aziz-Alaoui, M.A., Daher Okiye, M.: Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett. 16, 1069–1075 (2003)

    Google Scholar 

  4. Bazykin, A.D., Berezovskaya, F.S., Isaev, A.S., Khlebopros, R.G.: Dynamics of forest insect density: bifurcation approach. J. Theoret. Biol. 186, 267–278 (1997)

    Article  Google Scholar 

  5. Bazykin, A.D.: Nonlinear Dynamics of Interacting Populations. World Scientific (1998)

    Google Scholar 

  6. Chicone, C.: Ordinary Differential Equations with Applications (2nd edn.), Texts in Applied Mathematics, vol. 34, Springer (2006)

    Google Scholar 

  7. Clark, C.W.: Mathematical Bioeconomics: The Mathematics of Conservation, 3nd ed. John Wiley and Sons Inc. (2010)

    Google Scholar 

  8. Courchamp, F., Clutton-Brock, T., Grenfell, B.: Inverse dependence and the Allee effect. Trends Ecol. Evol. 14, 405–410 (1999)

    Article  Google Scholar 

  9. Courchamp, F., Berec, L., Gascoigne, J.: Allee Effects in Ecology and Conservation. Oxford University Press (2008)

    Google Scholar 

  10. González-Olivares, E., Mena-Lorca, J., Rojas-Palma, A., Flores, J.D.: Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey. Appl. Math. Modell. 35, 366–381 (2011)

    Google Scholar 

  11. González-Olivares, E., Gallego-Berrío, L.M., González-Yañez, B., Rojas-Palma, A.: Consequences of weak Allee effect on prey in the may-holling-tanner predator-prey model. Math. Methods Appl. Sci. 38, 5183–5196 (2015)

    Article  MathSciNet  Google Scholar 

  12. González-Olivares, E., Flores, J.D.: Consequences in an open access fishery model considering multiple Allee effects. J. Bio. Syst. 23(01), 101–121 (2015)

    Article  Google Scholar 

  13. González-Yañez, B., González-Olivares, E., Mena-Lorca, J.: Multistability on a leslie-gower type predator-prey model with nonmonotonic functional response. In: Mondaini, R., Dilao, R. (eds.) BIOMAT 2006—International Symposium on Mathematical and Computational Biology. World Scientific Co. Pte. Ltd., pp. 359–384 (2007)

    Google Scholar 

  14. Korobeinikov, A.: A Lyapunov function for Leslie-Gower predator-prey models. Appl. Math. Lett. 14, 697–699 (2001)

    Article  MathSciNet  Google Scholar 

  15. Leslie, P.H.: Some further notes on the use of matrices in Population Mathematics. Biometrika 35, 213–245 (1948)

    Article  MathSciNet  Google Scholar 

  16. Liermann, M., Hilborn, R.: Depensation: evidence, models and implications. Fish Fish. 2, 33–58 (2001)

    Article  Google Scholar 

  17. May, R.M.: Stability and Complexity in Model Ecosystems, 2nd edn. Princeton University Press (2001)

    Google Scholar 

  18. Mena-Lorca, J., González-Olivares, E., González-Ya, B.: The leslie-gower predator-prey model with Allee effct on prey: A simple model with a rich and interesting dynamics. In: Mondaini R. (ed.), Proceedings of the 2006 International Symposium on Mathematical and Computational Biology BIOMAT 2006, E-papers Servicios Editoriales Ltda., Rio de Janeiro, pp. 105–132 (2007)

    Google Scholar 

  19. Sáez, E., González-Olivares, E.: Dynamics on a predator-prey model. SIAM J. Appl. Math. 59, 1867–1878 (1999)

    Article  MathSciNet  Google Scholar 

  20. Stephens, P.A., Sutherland, W.J.: Consequences of the Allee effect for behaviour, ecology and conservation. Trends Ecolog. Evol. 14, 401–405 (1999)

    Article  Google Scholar 

  21. Stephens, P.A., Sutherland, W.J., Freckleton, R.P.: What is the Allee effect? Oikos 87, 185–190 (1999)

    Article  Google Scholar 

  22. Taylor, R.J.: Predation. Chapman and Hall (1984)

    Google Scholar 

  23. Turchin, P.: Complex Population Dynamics. A Theoretical/Empirical Synthesis, Monographs in Population Biology 35, Princeton University Press (2003)

    Google Scholar 

  24. van Voorn, G.A.K., Hemerik, L., Boer, M.P., Kooi, B.W.: Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee. Math. Biosci. 209, 451–469 (2007)

    Article  MathSciNet  Google Scholar 

  25. Wang, J., Shi, J., Wei, J.: Predator-prey system with strong Allee effect in prey. J. Mathe. Biol. 62, 291–331 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Udai Kumar’s research is supported by research fellowship from MHRD, Government of India. Partha Sarathi Mandal’s research is supported by SERB, DST project [grant: YSS/2015/001548].

Author information

Authors and Affiliations

  1. Department of Mathematics, NIT Patna, Patna, Bihar, 800005, India

    Udai Kumar & Partha Sarathi Mandal

Authors
  1. Udai Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Partha Sarathi Mandal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Udai Kumar .

Editor information

Editors and Affiliations

  1. Department of Mechanical Engineering, National Institute of Technology, Silchar, Assam, India

    Biplab Das

  2. Department of Computer Science and Engineering, National Institute of Technology Silchar, Silchar, Assam, India

    Ripon Patgiri

  3. National Institute of Technology Silchar, Silchar, India

    Sivaji Bandyopadhyay

  4. Aurel Vlaicu University of Arad, Arad, Romania

    Valentina Emilia Balas

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kumar, U., Sarathi Mandal, P. (2021). Dynamics of a Class of Modified Leslie-Gower Predator-Prey Model with Strong Allee Effect on Prey and Non-monotonic Rational Functional Response. In: Das, B., Patgiri, R., Bandyopadhyay, S., Balas, V.E. (eds) Modeling, Simulation and Optimization. Smart Innovation, Systems and Technologies, vol 206. Springer, Singapore. https://doi.org/10.1007/978-981-15-9829-6_41

Download citation

Publish with us

Policies and ethics

Search

Navigation