Analytical bifurcation behaviors of a host–parasitoid model with Holling type III functional response
Tóm tắt
This topic presents a study on a host–parasitoid model with a Holling type III functional response. In population dynamics, when host density rises, the parasitoid response initially accelerates due to the parasitoid’s improved searching efficiency. However, above a certain density threshold, the parasitoid response will reach a saturation level due to the influence of reducing the handling time. Thus, we incorporated a Holling type III functional response into the model to characterize such a phenomenon. The dynamics of the current model are discussed in this paper. We first obtained the existence and local stability conditions of the positive fixed point of the model. Furthermore, we investigated the bifurcation behaviors at the positive fixed point. More specifically, we used bifurcation theory and the center manifold theorem to prove that the model possess both period doubling and Neimark–Sacker bifurcations. Then, the chaotic behavior of the model, in the sense of Marotto, is proven. Furthermore, we apply a state-delayed feedback control strategy to control the complex dynamics of the present model. Finally, numerical examples are provided to support our analytic results.
Tài liệu tham khảo
Allen, L.J.: Introduction to Mathematical Biology. Prentice Hall, New Jersey (2007)
Brauer, F., Castillo-Chavez, C.: Mathematical Models in Population Biology and Epidemiology, vol. 2. Springer (2012)
Elaydi, S: Dynamics of first-order difference equations, An Introduction to Difference Equations, pp. 1–55, (2005)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42. Springer Science and Business Media (2013)
Mohamad, S., Naim, A.: Discrete-time analogues of integrodifferential equations modelling bidirectional neural networks. J. Comput. Appl. Math. 138(1), 1–20 (2002). https://doi.org/10.1016/S0377-0427(01)00366-1
Nicholson, A. J., Bailey, V.A: The balance of animal populations.: part I. in Proceedings of the zoological society of London. vol 105, pp. 551–598. Wiley Online Library (1935). https://doi.org/10.1111/j.1096-3642.1935.tb01680.x.
Yousef, A.: Stability and further analytical bifurcation behaviors of Moran–Ricker model with delayed density dependent birth rate regulation. J. Comput. Appl. Math. 355, 143–161 (2019). https://doi.org/10.1016/j.cam.2019.01.012
Yousef, A., Salman, S., Elsadany, A.: Stability and bifurcation analysis of a delayed discrete predator-prey model. Int. J. Bifurc. Chaos. 28(9), 1850116 (2018). https://doi.org/10.1142/S021812741850116X
Thompson, W.: La theorie mathematique de I’action des parasites entomophages et le facteur du hasard. Annu Fac Sci Mars 2, 69–89 (1924)
Yousef, A., Rida, S., Arafat, S.: Stability, analytic bifurcation structure and chaos control in a mutual interference host–parasitoid model. Int. J. Bifurc. Chaos. 30(15), 2050237 (2020). https://doi.org/10.1142/S0218127420502375
Liu, H., Zhang, K., Ye, Y., Wei, Y., Ma, M.: Dynamic complexity and bifurcation analysis of a host–parasitoid model with Allee effect and Holling type III functional response. Adv. Differ. Equ. 2019(1), 1–20 (2019). https://doi.org/10.1186/s13662-019-2430-8
Wu, D., Zhao, H.: Global qualitative analysis of a discrete host–parasitoid model with refuge and strong Allee effects. Math. Methods Appl. Sci. 41(5), 2039–2062 (2018). https://doi.org/10.1002/mma.4731
Din, Q., Hussain, M.: Controlling chaos and Neimark–Sacker bifurcation in a host–parasitoid model. Asian J. Control 21, 1202–1215 (2019). https://doi.org/10.1002/asjc.1809
Din, Q., Gumus, O.A., Khalil, H.: Neimark–Sacker bifurcation and chaotic behaviour of a modified host–parasitoid model. Zeitschrift fur Naturforschung A 72, 25–37 (2017). https://doi.org/10.1515/zna-2016-0335
Liu, X., Chu, Y., Liu, Y.: Bifurcation and chaos in a host–parasitoid model with a lower bound for the host. Adv. Differ. Equ. 2018, 31 (2018). https://doi.org/10.1186/s13662-018-1476-3
Ringel, M., Rees, M., Godfray, H.: The evolution of diapause in a coupled host–parasitoid system. J. Theor. Biol. 194, 195–204 (1998). https://doi.org/10.1006/jtbi.1998.0754
Zhao, M., Zhang, L., Zhu, J.: Dynamics of a host–parasitoid model with prolonged diapause for parasitoid. Commun. Nonlinear Sci. Numer. Simul. 16, 455–462 (2011). https://doi.org/10.1016/j.cnsns.2010.03.011
Liu, H., Li, Z., Gao, M., Dai, H., Liu, Z.: Dynamics of a host–parasitoid model with Allee effect for the host and parasitoid aggregation. Ecol. Complex. 6, 337–345 (2009)
Zhang, L., Zhao, M.: Dynamic complexities in a hyperparasitic system with prolonged diapause for host. Chaos Solit. Fract. 42, 1136–1142 (2009). https://doi.org/10.1016/j.chaos.2009.03.007
Holling, C.: S: Some characteristics of simple types of predation and parasitism. Can. Entomol. 91, 385–398 (1959). https://doi.org/10.4039/Ent91385-7
Hassell, M.: The Spatial and Temporal Dynamics of Host–Parasitoid Interactions. OUP Oxford (2000)
van Baalen, M., Krivan, V., van Rijn, P.C.J., Sabelis, M.W.: Alternative food, switching predators, and the persistence of predator prey systems. Am Nat 157(5), 512–524 (2001). https://doi.org/10.1086/319933
Yang, X.: Uniform persistence and periodic solutions for a discrete predator-prey system with delays. J. Math. Anal. Appl. 316, 161–177 (2006). https://doi.org/10.1016/j.jmaa.2005.04.036
Dhar, J., Singh, H., Bhatti, H.S.: Discrete-time dynamics of a system with crowding effect and predator partially dependent on prey. Appl. Math. Comput. 252, 324–335 (2015). https://doi.org/10.1016/j.amc.2014.12.021
Choo, Y.: An elementary proof of the Jury test for real polynomials. Automatica 47, 249–252 (2011). https://doi.org/10.1016/j.automatica.2010.10.040
Kuznetsov, Y.: A: Elements of Applied Bifurcation Theory, vol. 112. Springer Science and Business Media (2013)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2. Springer Science and Business Media (2003)
Marotto, F.R.: On redefining a snap-back repeller. Chaos Solitons Fract. 25, 25–28 (2005). https://doi.org/10.1016/j.chaos.2004.10.003
Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196 (1990). https://doi.org/10.1103/PhysRevLett.64.1196
Romeiras, F.J., Grebogi, C., Ott, E., Dayawansa, W.: Controlling chaotic dynamical systems. Phys. D 58, 165–192 (1992). https://doi.org/10.1016/0167-2789(92)90107-X
Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. CRC Press (1998)
He, Z., Jiang, X.: Bifurcation and chaotic behaviour of a discrete-time variable-territory predator-prey model. IMA J. Appl. Math. 76, 899–918 (2011). https://doi.org/10.1093/imamat/hxr006
Tang, S., Chen, L.: Chaos in functional response host–parasitoid ecosystem models. Chaos Solit. Fract. 13, 875–884 (2002)