Abstract
In this paper, we investigate the dynamical behavior of a modified May–Holling–Tanner predator–prey model by considering the Allee effect in the prey and alternative food sources for the predator. The model is analyzed theoretically as well as numerically to determine all local codimension one and two bifurcations. Using the local parametrization method and Hopf bifurcation theorem, we analyze the Hopf bifurcation and compute its corresponding normal form coefficient to reveal its criticality. We specially derive normal form of the system near Bogdanov–Takens and generalized Hopf bifurcations, to determine possible bifurcation scenarios near each bifurcation. For Bogdanov–Takens, we further determine a set of parameters for which curves of saddle-node, Hopf and Homoclinic bifurcations can be observed. By numerical continuation technique, we compute several curves of equilibria and bifurcations, and detect different bifurcation points on these curves. For the computed bifurcation points, we numerically compute their corresponding normal form coefficients. We specially compute a family of limit cycles and limit point cycle bifurcation on this family.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Hsu, S.B.: On global stability of a predator-prey system. Math. Biosci. 1–10, 39 (1978)
Braza, P.A.: Predator-prey dynamics with square root functional responses. Nonlinear Anal. Real World Appl. 1837–1843, 13 (2012)
Yan, D., Cao, H., Xu, X., Wang, X.: Hopf bifurcation for a predator-prey model with age structure. Phys. A 1–15, 526 (2019)
Arancibia-Ibarra, C.: The basins of attraction in a modified May–Holling–Tanner predator-prey model with Allee affect. Nonlinear Anal. 15–28, 185 (2019)
Arrowsmith, D., Chapman, C.: Dynamical systems: differential equations, maps and chaotic behaviour. Comput. Math. Appl. 1–330, 32 (1996)
Banerjee, M.: Turing and non-Turing patterns in two-dimensional prey-predator models. In: Applications of Chaos and Nonlinear Dynamics in Science and Engineering. Springer, Berlin, Heidelberg, pp. 257–280 (2015)
Kozlova, I., Singh, M., Easton, A., Ridland, P.: Spider mite predator-prey model. Math. Comput. Model. 1287–1298 (2005)
Arancibia-Ibarra, C., GonzAalez-Olivares, E.: The Holling–Tanner model considering an alternative food for predator. In: Proceedings of the 2015 International Conference on Computational and Mathematical Methods in Science and Engineering CMMSE, pp. 130–141 (2015)
Andersson, M., Erlinge, S.: Influence of predation on rodent populations. Oikos 29(3), 591–597 (1977)
Aziz-Alaoui, M., Daher, M.: Boundedness and global stability for a predator-prey model with modified Leslie–Gower and Holling-type II schemes. Appl. Math. Lett. 1069–1075, 16 (2003)
Arancibia-Ibarra, C., Flores, J.: Modelling and analysis of a modified May-Holling-Tanner predator-prey model with Allee effect in the prey and an alternative food source for the predator. Math. Biosci. Eng. 17(6), 8052–8073 (2020)
Arancibia-Ibarra, C., Flores, J., Bode, M., Pettet, G., van Heijster, P.: A modiffed May-Holling-Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator. Discrete Contin. Dyn. Syst. Ser. B 22(11), 1–20 (2017)
Hamzi, B., Kang, W., Barbot, J.P.: Analysis and control of Hopf bifurcations. SIAM J. Control Optim. 42(6), 2200–2220 (2004)
Hamzi, B.: Quadratic stabilization of nonlinear control systems with a double- zero bifurcation. In: IFAC Proceedings, pp 161–166 (2001)
Hamzi, B., Lamb, J.S.W., Lewis, D.: A characterization of normal forms for control systems. J. Dyn. Control Syst. 21(2), 273–84 (2015)
Fattahpour, H., Zangeneh, H.R.Z., Nagata, W.: Dynamics of rodent population with two predators. Bull. Iran. Math. Soc. 00, 45 (2019)
Kramer, A., Berec, L., Drake, J.: Allee effects in ecology and evolution. J. Anim. Ecol. 7–10, 87 (2018)
Verdy, A.: Modulation of predator-prey interactions by the Allee effect. Ecol. Model. 1098–1107, 221 (2010)
Arancibia-Ibarra, D.C., Gonzalez-Olivares, E.: A modified Leslie–Gower predator-prey model with hyperbolic functional response and Allee effect on prey. In: BIOMAT 2010 International Symposium on Mathematical and Computational Biology, pp. 146–162 (2011)
Harley, K., van Heijster, P., Marangell, R., Pettet, G., Wechselberger, M.: Existence of traveling wave solutions for a model of tumor invasion. SIAM J. Appl. Dyn. Syst. 13, 1–26 (2014)
Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (2004)
Hu, D., Cao, H.: Stability and bifurcation analysis in a predator and prey system with Michaelis–Menten type predator harvesting. Nonlinear Anal. Real World Appl. 58–82, 33 (2017)
Perko, L.: Differential equations and dynamical systems. Springer, NewYork, Berlin, (2001)
Chen, J., Huang, J., Ruan, S., Wang, J.: Bifurcations of invariant tori in predator-prey models with seasonal prey harvesting. SIAM J. Appl. Math. 73, 1876–1905 (2013)
Huang, J., Gong, Y., Chen, J.: Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting. Int. J. Bifurc. Chaos 23(10), 1–24 (2013)
Allgower, E.L., Georg, K.: Numerical Continuation Methods: An Introduction. Springer, Berlin (1990)
Wittmann, M.J., Stuis, H., Metzler, D.: Genetic Allee effects and their interaction with ecological Allee effects. J. Animal Ecol. 87(1), 11–23 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Majid Gazor.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Khanghahi, M.J., Ghaziani, R.K. Bifurcation Analysis of a Modified May–Holling–Tanner Predator–Prey Model with Allee Effect. Bull. Iran. Math. Soc. 48, 3405–3437 (2022). https://doi.org/10.1007/s41980-022-00698-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41980-022-00698-9