Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Dynamics of a modified Leslie-Gower predation model considering a generalist predator and the hyperbolic functional response

1 Pontificia Universidad Católica de Valparaíso, Chile;
2 Facultad de Educación, Universidad de Las Américas, Chile;
3 Departamento de Matemática, Física y Estadítica, Facultad de Ciencias Básicas, Universidad Catolica del Maule, Talca, Chile

Special Issues: Recent Advances in Mathematical Population Dynamics

In the ecological literature,many models for the predator-prey interactions have been well formulated but partially analyzed.Assuming this analysis to be true and complete,some authors use that results to study a more complex relationship among species (food webs).Others employ more sophisticated mathematical tools for the analysis,without further questioning.The aim of this paper is to extend,complement and enhance the results established in an earlier article referred to a modified Leslie-Gower model.In that work,the authors proved only the boundedness of solutions,the existence of an attracting set,and the global stability of a single equilibrium point at the interior of the first quadrant.In this paper,new results for the same model are proven,establishing conditions in the parameter space for which up two positive equilibria exist.Assuming there exists a unique positive equilibrium point,we have proved,the existence of:i) a separatrix curve Σ,dividing the trajectories in the phase plane,which can have different ω-limit,ii) a subset of the parameter space in which two concentric limit cycles exist,the innermost unstable and the outermost stable.Then,there exists the phenomenon of tri-stability,because simultaneously,it has:a local stable positive equilibrium point, a stable limit cycle,and an attractor equilibrium point over the vertical axis.Therefore,we warn the model studied have more rich and interesting properties that those shown that earlier papers.Numerical simulations and a bifurcation diagram are given to endorse the analytical results.
  Article Metrics


1. P. Turchin, Complex population dynamics. A theoretical/empirical synthesis, Monographs in Population Biology 35, Princeton University Press, (2003).

2. A. A. Berryman, A. P. Gutierrez and R. Arditi, Credible, parsimonious and useful predator-prey models-a reply to Abrams, Gleeson, and Sarnelle. Ecology, 76 (1995) 1980-1985.

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

4. P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234.

5. R. M. May, Stability and complexity in model ecosystems (2nd edition), Princeton University Press, (2001).

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

7. C. Arancibia-Ibarra and E. Gonzáxlez-Olivares, A modified Leslie-Gower predator-prey model with hyperbolic functional response and Allee effect on prey, In R. Mondaini (Ed.) BIOMAT 2010 International Symposium on Mathematical and Computational Biology, World Scientific Co. Pte. Ltd., Singapore, (2011), 146-162.

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

9. E. Gonzáxlez-Olivares, J. Mena-Lorca, A. Rojas-Palma, et al., Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey, Appl. Math. Model., 35 (2011), 366-381.

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

11. A. Singh and S. Gakkhar, Stabilization of modified Leslie-Gower prey-predator model, Differ. Equ. Dyn. Syst., (2014), 239-249.

12. S. Yu, Global asymptotic stability of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Discrete Dyn. Nat. Soc., 2012, 208167.

13. M. Liu, C. Du and M. Deng, Persistence and extinction of a modified Leslie-Gower Holling-type Ⅱ stochastic predator-prey model with impulsive toxicant input in polluted environments, Nonl. Anal. Hybrid Sys., 27 (2018) 177-190.

14. R. Yuan, W. Jiang and Y. Wang, Saddle-node-Hopf bifurcation in a modified Leslie-Gower predator-prey model with time-delay and prey harvesting, J. Math. Anal. Appl., 422 (2015), 1072-1090.

15. Z. Zhao, L. Yang and L. Chen, Impulsive perturbations of a predator-prey system with modified Leslie-Gower and Holling type Ⅱ schemes, J. App. Math. Comput., 35 (2011), 119-134.

16. P. Feng and Y. Kang, Dynamics of a modified Leslie-Gower model with double Allee effects, Nonlinear Dyn., 80 (2015), 1051-1062.

17. E. Gonzáxlez-Olivares, B. Gonzáxlez-Yañez, R. Becerra-Klix, et al., Multiple stable states in a model based on predator-induced defenses, Ecol. Compl., 32 (2017), 111-120.

18. Q. Yue, Dynamics of a modified Leslie-Gower predator-prey model with Holling-type Ⅱ schemes and a prey refuge, Springer Plus, 5 (2016), 461.

19. I. El Harraki, R. Yafia, A. Boutoulout, et al., The effect of non-selective harvesting in predator-prey model with modified Leslie-Gower and Holling type Ⅱ schemes, Discontinuity Nonlinearity Complexity, 7 (2018), 413-427.

20. A. D. Bazykin, Nonlinear Dynamics of interacting populations, World Scientific Publishing Co. Pte. Ltd., (1998).

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

22. W. M. Getz, A hypothesis regarding the abruptness of density dependence and the growth rate populations, Ecology, 77 (1996), 2014-2026.

23. D. K. Arrowsmith and C. M. Place, Dynamical Systems. Differential equations, maps and chaotic behaviour, Chapman and Hall, (1992).

24. K. Q. Lan and C. R. Zhu, Phase portraits, Hopf bifurcations and limit cycles of the Holling-Tanner models for predator-prey interactions, Nonlinear Anal. Real World Appl., 12 (2011), 1961-1973.

25. E. Sáez and E. Gonzáxlez-Olivares, Dynamics on a predator-prey model, SIAM J. Appl. Math., 59 (1999), 1867-1878.

26. J. T. Tanner, The stability and the intrinsic growth rate of prey and predator population, Ecology, 56 (1975), 855-867.

27. Y. Zhu and K. Wang, Existence and global attractivity of positive periodic solutions for a predator-prey model with modified Leslie-Gower Holling-type Ⅱ schemes, J. Math. Anal. Appl., 384 (2011), 400-408.

28. H. I. Freedman, Deterministic Mathematical Model in Population Ecology, Marcel Dekker, (1980).

29. E. Gonzáxlez-Olivares, L. M. Gallego-Berrxío, B. Gonzáxlez-Yañez, et al., Consequences of weak Allee effect on prey in the May-Holling-Tanner predator-prey model, Math. Meth. Appl. Sci., 38 (2015), 5183-5196.

30. B. Gonzáxlez-Yañez, E. Gonzáxlez-Olivares and J. Mena-Lorca, Multistability on a Leslie-Gower Type predator-prey model with nonmonotonic functional response, In R. Mondaini and R. Dilao (eds.), BIOMAT 2006-International Symposium on Mathematical and Computational Biology, World Scientific Co. Pte. Ltd., (2007), 359-384.

31. S. Vexliz-Retamales and E. Gonzáxlez-Olivares, Dynamics of a Gause type prey-predator model with a rational nonmonotonic consumption function, In R. Mondaini (ed), Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology (BIOMAT-2003), E-Papers Serviços Editoriais Ltda., Rio de Janeiro, 2 (2004), 181-192.

32. C. Chicone, Ordinary differential equations with applications (2nd edition), Texts in Applied Mathematics 34, Springer, (2006).

33. F. Dumortier, J. Llibre and J. C. Artéxs, Qualitative theory of planar differential systems, Springer, (2006).

34. E. Gonzáxlez-Olivares, C. Arancibia-Ibarra, B. Gonzáxlez-Yañez, et al., Bifurcation analysis of the Holling-Tanner predation model considering alternative food for predator, Mathematical Biosciences and Engineering, 16 (2019), 4274-4298.

35. L. Perko, Differential equations and dynamical systems (Third Edition) Springer, (2001).

36. Y. A. Kuznetsov, emphElements of applied bifurcation theory (3rd ed) Springer-Verlag, (2004).

37. P. Monzón, Almost global attraction in planar systems, Syst. Control Lett., 54 (2005), 753-758.

38. A. Rantzer, A dual to Lyapunov's stability theorem, Syst. Control Lett., 42 (2001), 161-168.

39. V. A. Gaiko, Global Bifurcation Theory and Hilbert's Sixteenth Problem, in:Mathematics and its Applications, Kluwer Academic Publishers, 559 (2003).

40. V. A. Gaiko and C. Vuik, Global dynamics in the Leslie-Gower model with the Allee effect, Int. J. Bif. Chaos 28 (2018), 1850151-1850160.

41. S. Wolfram, Mathematica:A System for Doing Mathematics by Computer (2nd edition), Wolfram Research, Addison Wesley, (1991).

42. A. Dhooge, W. Govaerts and Y. Kuznetsov, Matcont:a matlab package for numerical bifurcation analysis of ODES, ACM Trans. Math. Soft. (TOMS), 29 (2003), 141-164.

43. C. Arancibia-Ibarra, The basins of attraction in a modified May-Holling-Tanner predator-prey model with Allee effect, Nonl. Anal., 185 (2019), 15-28.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved