Canard Cycles and Homoclinic Orbit of a Leslie–Gower Predator–Prey Model with Allee Effect and Holling Type II Functional Response

Tianyu Shi; Zhenshu Wen

Ayuda

Canard Cycles and Homoclinic Orbit of a Leslie–Gower Predator–Prey Model with Allee Effect and Holling Type II Functional Response

Tianyu Shi ^[1] ; Zhenshu Wen ^[1]
1. [1] Huaqiao University
  
  Huaqiao University
  
  China
Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 5, 2024
Idioma: inglés
DOI: 10.1007/s12346-024-01059-z
Enlaces
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Resumen
- We study dynamics of a fast–slow Leslie–Gower predator–prey system with Allee effect and Holling Type II functional response. More specifically, we show some sufficient conditions to guarantee the existence of two positive equilibria of the system and their location, and then we further fully determine their dynamics. Based on geometric singular perturbation theory and the slow–fast normal form, we determine the associated bifurcation curve and observe canard explosion. Besides, we also find a homoclinic orbit to a saddle with slow and fast segments, in which, the stable and unstable manifolds of the saddle are connected under explicit parameters conditions.
Referencias bibliográficas
- 1. Leslie, P.H.: Some further notes on the use of matrices in population mathematics. Biometrika 35(3/4), 213–245 (1948)
- 2. Leslie, P.: A stochastic model for studying the properties of certain biological systems by numerical methods. Biometrika 45(1–2), 16–31...
- 3. Hsu, S.B., Huang, T.W.: Global stability for a class of predator-prey systems. SIAM J. Appl. Math. 55(3), 763–783 (1995)
- 4. Hsu, S.B., Hwang, T.W.: Hopf bifurcation analysis for a predator–prey system of Holling and Leslie type. Taiwan. J. Math. 3(1), 35–53 (1999)
- 5. Korobeinikov, A.: A Lyapunov function for Leslie–Gower predator–prey models. Appl. Math. Lett. 14(6), 697–699 (2001)
- 6. Yuan, S., Song, Y.: Bifurcation and stability analysis for a delayed Leslie–Gower predator–prey system. IMA J. Appl. Math. 74(4), 574–603...
- 7. Holling, C.S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc....
- 8. Huang, J., Ruan, S., Song, J.: Bifurcations in a predator–prey system of Leslie type with generalized Holling type III functional response....
- 9. Dai, Y., Zhao, Y., Sang, B.: Four limit cycles in a predator–prey system of Leslie type with generalized Holling type III functional response....
- 10. Tiwari, V., Tripathi, J.P., Mishra, S., Upadhyay, R.K.: Modeling the fear effect and stability of nonequilibrium patterns in mutually...
- 11. Tripathi, J.P., Bugalia, S., Jana, D., Gupta, N., Tiwari, V., Li, J., Sun, G.Q.: Modeling the cost of anti-predator strategy in a predator-prey...
- 12. Tripathi, J.P., Abbas, S., Thakur, M.: Dynamical analysis of a prey–predator model with Beddington– DeAngelis type function response incorporating...
- 13. Tripathi, J.P., Abbas, S., Thakur, M.: A density dependent delayed predator-prey model with Beddington–DeAngelis type function response...
- 14. Allee, W., Bowen, E.S.: Studies in animal aggregations: mass protection against colloidal silver among goldfishes. J. Exp. Zool. 61(2),...
- 15. Tripathi, J.P., Mandal, P.S., Poonia, A., Bajiya, V.P.: A widespread interaction between generalist and specialist enemies: the role of...
- 16. Aguirre, P., Gonzalez-Olivares, E., Sáez, E.: Two limit cycles in a Leslie–Gower predator-prey model with additive Allee effect. Nonlinear...
- 17. Aguirre, P., González-Olivares, E., Sáez, E.: Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect. SIAM...
- 18. Aguirre, P.: A general class of predation models with multiplicative Allee effect. Nonlinear Dyn. 78(1), 629–648 (2014)
- 19. Zhu, Z., Chen, Y., Li, Z., Chen, F.: Stability and bifurcation in a Leslie–Gower predator–prey model with Allee effect. Int. J. Bifurc....
- 20. Zu, J., Mimura, M.: The impact of Allee effect on a predator-prey system with Holling type II functional response. Appl. Math. Comput....
- 21. Zu, J.: Global qualitative analysis of a predator-prey system with Allee effect on the prey species. Math. Comput. Simul. 94, 33–54 (2013)
- 22. Wang, C., Zhang, X.: Canards, heteroclinic and homoclinic orbits for a slow–fast predator–prey model of generalized Holling type III....
- 23. Su, W., Zhang, X.: Global stability and canard explosions of the predator–prey model with the Sigmoid functional response. SIAM J. Appl....
- 24. Chen, X., Zhang, X.: Dynamics of the predator-prey model with the Sigmoid functional response. Stud. Appl. Math. 147(1), 300–318 (2021)
- 25. Atabaigi, A.: Canard explosion, homoclinic and heteroclinic orbits in singularly perturbed generalist predator–prey systems. Int. J. Biomath....
- 26. Saha, T., Pal, P.J., Banerjee, M.: Slow-fast analysis of a modified Leslie–Gower model with Holling type I functional response. Nonlinear...
- 27. Zhu, Z., Liu, X.: Canard cycles and relaxation oscillations in a singularly perturbed Leslie–Gower predator-prey model with Allee effect....
- 28. Zhao, L., Shen, J.: Canards and homoclinic orbits in a slow–fast modified May–Holling–Tanner predator-prey model with weak multiple Allee...
- 29. Shi, T., Wen, Z.: Canard cycles and their cyclicity of a fast–slow Leslie–Gower predator–prey model with Allee effect. In: International...
- 30. Chowdhury, P.R., Banerjee, M., Petrovskii, S.: Canards, relaxation oscillations, and pattern formation in a slow–fast ratio-dependent...
- 31. Li, K., Li, S., Wu, K.: Relaxation oscillations and canard explosion of a Leslie–Gower predator-prey system with addictive Allee effect....
- 32. Wen, Z., Shi, T.: Existence and uniqueness of a canard cycle with cyclicity at most two in a singularly perturbed Leslie-Gower predator-prey...
- 33. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)
- 34. Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174(2), 312–368 (2001)
- 35. Kuznetsov, Y.A., Muratori, S., Rinaldi, S.: Homoclinic bifurcations in slow–fast second order systems. Nonlinear Anal. Theory Methods...
- 36. Shen, J.: Canard limit cycles and global dynamics in a singularly perturbed predator–prey system with non-monotonic functional response....