Primary: 92D25, 34C; Secondary: 58F14, 58F21.

Export file:

Format

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

Content

• Citation Only
• Citation and Abstract

Uniqueness of limit cycles and multiple attractors in a Gause-typepredator-prey model with nonmonotonic functional response and Allee effecton prey

1. Grupo de Ecología Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Valparaíso
2. Department of Mathematics, The University of South Dakota, Vermillion, SD 57069-2390

## Abstract    Related pages

The main purpose of this work is to analyze a Gause type predator-prey modelin which two ecological phenomena are considered: the Allee effect affectingthe prey growth function and the formation of group defence by prey in orderto avoid the predation.
We prove the existence of a separatrix curves in the phase plane, determinedby the stable manifold of the equilibrium point associated to the Alleeeffect, implying that the solutions are highly sensitive to the initialconditions.
Trajectories starting at one side of this separatrix curve have theequilibrium point $(0,0)$ as their $\omega$-limit, while trajectoriesstarting at the other side will approach to one of the following threeattractors: a stable limit cycle, a stable coexistence point or the stableequilibrium point $(K,0)$ in which the predators disappear andprey attains their carrying capacity.
We obtain conditions on the parameter values for the existence of one or twopositive hyperbolic equilibrium points and the existence of a limit cyclesurrounding one of them. Both ecological processes under study, namely thenonmonotonic functional response and the Allee effect on prey, exert astrong influence on the system dynamics, resulting in multiple domains ofattraction.
Using Liapunov quantities we demonstrate the uniqueness of limit cycle, whichconstitutes one of the main differences with the model where the Alleeeffect is not considered. Computer simulations are also given in support ofthe conclusions.
Figure/Table
Supplementary
Article Metrics

Citation: Eduardo González-Olivares, Betsabé González-Yañez, Jaime Mena-Lorca, José D. Flores. Uniqueness of limit cycles and multiple attractors in a Gause-typepredator-prey model with nonmonotonic functional response and Allee effecton prey. Mathematical Biosciences and Engineering, 2013, 10(2): 345-367. doi: 10.3934/mbe.2013.10.345

References

• 1. in "Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology" (ed. R. Mondaini ), E-Papers Serviços Editoriais Ltda, Río de Janeiro, 2 (2004), 206-217.
• 2. SIAM Journal on Applied Mathematics, 69 (2009), 1244-1262.
• 3. Chapman and Hall, 1992.
• 4. World Scientific, 1998.
• 5. Trends in Ecology and Evolution, 22 (2007), 185-191.
• 6. Journal of Theoretical Biology, 218 (2002), 375-394.
• 7. (2nd edition), Texts in Applied Mathematics 34, Springer, 2006.
• 8. (2nd edition), John Wiley and Sons, 1990.
• 9. Cambridge University Press, 2007.
• 10. in "Differential Equations Model" (eds. M. Braun, C. S. Coleman and D. Drew ), Springer Verlag, (1983), 279-297.
• 11. Journal of Mathematical Biology, 36 (1997), 149-168.
• 12. SIAM Journal on Applied Mathematics, 46 (1986), 630-642.
• 13. Trends in Ecology and Evolution, 14 (1999), 405-410.
• 14. Oxford University Press, 2008.
• 15. Springer, 2006.
• 16. Marcel Dekker, 1980.
• 17. Bulletin of Mathematical Biology, 48 (1986), 493-508.
• 18. Mathematics an its applications, 559, Kluwer Academic Publishers, 2003.
• 19. Journal of Applied Ecology, 41 (2004), 801-810.
• 20. Discrete and Continuous Dynamical Systems, 6 (2006), 525-534.
• 21. in "Proceedings of the 2006 International Symposium on Mathematical and Computational Biology" (ed. R. Mondaini), E-papers Serviços Editoriais Ltda. Rio de Janeiro, (2007), 53-71.
• 22. Nonlinear Analysis: Real World and Applications, 12 (2011), 2931-2942.
• 23. Bulletin of Mathematical Biology, 73 (2011), 1378-1397.
• 24. Applied Mathematical Modelling, 35 (2011), 366-381.
• 25. in "Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology" (ed. R. Mondaini), E-Papers Serviços Editoriais Ltda, Río de Janeiro, 2 (2004), 358-373.
• 26. Journal of Mathematical Biology, 60 (2010), 59-74.
• 27. Cambridge University Press, 2001.
• 28. Mathematical Biosciences, 88 (1988), 67-84.
• 29. Fish and Fisheries, 2 (2001), 33-58.
• 30. (3rd ed), Texts in Applied Mathematics 7, Springer-Verlag, 2001.
• 31. in "Proceedings of the 2006 International Symposium on Mathematical and Computational Biology" (ed. R. Mondaini), E-papers Serviços Editoriais Ltda., (2007), 295-321.
• 32. SIAM Journal of Applied Mathematics, 61 (2001), 1445-1472.
• 33. Trends in Ecology and Evolution, 14 (1999), 401-405.
• 34. Oikos, 87 (1999), 185-190.
• 35. Chapman and Hall, 1984.
• 36. Monographs in Population Biology 35, Princeton University Press, 2003.
• 37. Mathematical Biosciences, 209 (2007), 451-469.
• 38. in "Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology" (ed. R. Mondaini), E-Papers Serviços Editoriais Ltda, Río de Janeiro, 2 (2004), 181-192.
• 39. Journal of Mathematical Biology, 62 (2011), 291-331.
• 40. (2nd edition), Wolfram Research, Addison Wesley, 1991.
• 41. SIAM Journal on Applied Mathematics, 48 (1988), 592-606.
• 42. International Journal of Bifurcation and Chaos, 11 (2001), 2123-2131.
• 43. Nonlinearity, 16 (2003), 1185-1201.
• 44. SIAM Journal on Applied Mathematics, 63 (2002), 636-682.
• 45. Applied Mathematics and Computation, 217 (2010), 3542-3556.

• 1. Ruiwen WU, Xiuxiang LIU, Dynamics of a predator-prey system with a mate-finding Allee effect on prey, TURKISH JOURNAL OF MATHEMATICS, 2017, 41, 585, 10.3906/mat-1411-8
• 2. Ruiwen Wu, Xiuxiang Liu, Dynamics of a Predator-Prey System with a Mate-Finding Allee Effect, Abstract and Applied Analysis, 2014, 2014, 1, 10.1155/2014/673424
• 3. J. Leonel Rocha, Abdel-Kaddous Taha, D. Fournier-Prunaret, Allee’s dynamics and bifurcation structures in von Bertalanffy’s population size functions, Journal of Physics: Conference Series, 2018, 990, 012011, 10.1088/1742-6596/990/1/012011
• 4. Pablo Aguirre, José D. Flores, Eduardo González-Olivares, Bifurcations and global dynamics in a predator–prey model with a strong Allee effect on the prey, and a ratio-dependent functional response, Nonlinear Analysis: Real World Applications, 2014, 16, 235, 10.1016/j.nonrwa.2013.10.002
• 5. J. Leonel Rocha, Abdel-Kaddous Taha, Danièle Fournier-Prunaret, Big bang bifurcations in von Bertalanffy’s dynamics with strong and weak Allee effects, Nonlinear Dynamics, 2016, 84, 2, 607, 10.1007/s11071-015-2510-6
• 6. Abdel-Kaddous Taha, Danièle Fournier-Prunaret, J. Leonel Rocha, Strong and weak Allee effects and chaotic dynamics in Richards' growths, Discrete and Continuous Dynamical Systems - Series B, 2013, 18, 9, 2397, 10.3934/dcdsb.2013.18.2397
• 7. Pablo Aguirre, A general class of predation models with multiplicative Allee effect, Nonlinear Dynamics, 2014, 78, 1, 629, 10.1007/s11071-014-1465-3
• 8. Nicole Martínez-Jeraldo, Pablo Aguirre, Allee effect acting on the prey species in a Leslie–Gower predation model, Nonlinear Analysis: Real World Applications, 2019, 45, 895, 10.1016/j.nonrwa.2018.08.009
• 9. Viviana Rivera, Pablo Aguirre, Study of a Tritrophic Food Chain Model with Non-differentiable Functional Response, Acta Applicandae Mathematicae, 2019, 10.1007/s10440-019-00239-3