| Citation: |
Qingkai Xu, Chunrui Zhang. BIFURCATION ANALYSIS AND CHAOS OF A MODIFIED HOLLING−TANNER MODEL WITH DISCRETE TIME[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3425-3449. doi: 10.11948/20240028
|
BIFURCATION ANALYSIS AND CHAOS OF A MODIFIED HOLLING−TANNER MODEL WITH DISCRETE TIME
-
Abstract
In this paper, we consider a classical Holling-Tanner model with discrete time. The dynamical behavior of the model is given by using both theoretical analysis and numerical simulation. We use the central manifold theorem and bifurcation theory to demonstrate that the system will undergo Hopf bifurcation and Flip bifurcation at the positive equilibrium point. By using Lyapunov exponent, we show that the system can undergo the path from stability to Flip (Hopf) bifurcation to chaos, and then we verify the correctness of the theoretical results via numerical simulations.
-
Keywords:
- Holling-Tanner model /
- Flip bifurcation /
- Hopf bifurcation /
- chaos
-
-
References
[1] C. Arancibia-Ibarra, J. D. Flores, G. Pettet and P. Van Heijster, A Holling–Tanner predator–prey model with strong allee effect, International Journal of Bifurcation and Chaos, 2019, 29(11), 1930032. doi: 10.1142/S0218127419300325 [2] A. Arsie, C. Kottegoda and C. Shan, A predator-prey system with generalized holling type iv functional response and allee effects in prey, Journal of Differential Equations, 2022, 309, 704–740. doi: 10.1016/j.jde.2021.11.041 [3] M. Aziz-Alaoui, Study of a Leslie–Gower-type tritrophic population model, Chaos, Solitons & Fractals, 2002, 14(8), 1275–1293. [4] M. Aziz-Alaoui and M. D. Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and holling-type ii schemes, Applied Mathematics Letters, 2003, 16(7), 1069–1075. doi: 10.1016/S0893-9659(03)90096-6 [5] H. Chen and C. Zhang, Dynamic analysis of a Leslie–Gower-type predator–prey system with the fear effect and ratio-dependent holling iii functional response, Nonlinear Analysis: Modelling and Control, 2022, 27(5), 904–926. [6] J. -M. Chiou, H. -G. Müller and J. -L. Wang, Functional response models, Statistica Sinica, 2004, 675–693. [7] J. M. Cushing, Nonlinear semelparous Leslie models, Mathematical Biosciences & Engineering, 2005, 3(1), 17–36. [8] J. Dawes and M. Souza, A derivation of holling's type i, ii and iii functional responses in predator–prey systems, Journal of Theoretical Biology, 2013, 327, 11–22. doi: 10.1016/j.jtbi.2013.02.017 [9] H. Freedman and R. Mathsen, Persistence in predator-prey systems with ratio-dependent predator influence, Bulletin of Mathematical Biology, 1993, 55(4), 817–827. doi: 10.1016/S0092-8240(05)80190-9 [10] P. Georgescu and Y. -H. Hsieh, Global dynamics of a predator-prey model with stage structure for the predator, SIAM Journal on Applied Mathematics, 2007, 67(5), 1379–1395. doi: 10.1137/060670377 [11] B. Ghanbari and S. Djilali, Mathematical analysis of a fractional-order predator-prey model with prey social behavior and infection developed in predator population, Chaos, Solitons & Fractals, 2020, 138, 109960. [12] C. S. Holling, The functional response of invertebrate predators to prey density, The Memoirs of the Entomological Society of Canada, 1966, 98(S48), 5–86. doi: 10.4039/entm9848fv [13] B. Hong and C. Zhang, Bifurcations and chaotic behavior of a predator-prey model with discrete time, AIMS Mathematics, 2023, 8(6), 13390–13410. doi: 10.3934/math.2023678 [14] S. -B. Hsu and T. -W. Huang, Global stability for a class of predator-prey systems, SIAM Journal on Applied Mathematics, 1995, 55(3), 763–783. doi: 10.1137/S0036139993253201 [15] X. Jia, K. Huang and C. Li, Bifurcation analysis of a modified Leslie–Gower predator–prey system, International Journal of Bifurcation and Chaos, 2023, 33(02), 2350024. doi: 10.1142/S0218127423500244 [16] Y. A. Kuznetsov, I. A. Kuznetsov and Y. Kuznetsov, Elements of Applied Bifurcation Theory, 112, Springer, 1998. [17] T. Li, X. Zhang and C. Zhang, Pattern dynamics analysis of a space–time discrete spruce budworm model, Chaos, Solitons & Fractals, 2024, 179, 114423. [18] X. Li and Y. Liu, Transcritical bifurcation and flip bifurcation of a new discrete ratio-dependent predator-prey system, Qualitative Theory of Dynamical Systems, 2022, 21(4), 122. doi: 10.1007/s12346-022-00646-2 [19] X. Liu and D. Xiao, Complex dynamic behaviors of a discrete-time predator–prey system, Chaos, Solitons & Fractals, 2007, 32(1), 80–94. [20] R. Ma, Y. Bai and F. Wang, Dynamical behavior analysis of a two-dimensional discrete predator-prey model with prey refuge and fear factor, J. Appl. Anal. Comput, 2020, 10(4), 1683–1697. [21] P. S. Mandal and M. Banerjee, Stochastic persistence and stability analysis of a modified Holling–Tanner model, Mathematical Methods in the Applied Sciences, 2013, 36(10), 1263–1280. doi: 10.1002/mma.2680 [22] E. C. Mid and V. Dua, Parameter estimation using multiparametric programming for implicit euler's method based discretization, Chemical Engineering Research and Design, 2019, 142, 62–77. doi: 10.1016/j.cherd.2018.11.032 [23] S. G. Mortoja, P. Panja and S. K. Mondal, Dynamics of a predator-prey model with stage-structure on both species and anti-predator behavior, Informatics in Medicine Unlocked, 2018, 10, 50–57. doi: 10.1016/j.imu.2017.12.004 [24] A. Oaten and W. W. Murdoch, Functional response and stability in predator-prey systems, The American Naturalist, 1975, 109(967), 289–298. doi: 10.1086/282998 [25] M. D. Okiye and M. Aziz-Alaoui, On the dynamics of a predator-prey model with the Holling-Tanner functional response, Mathematical Modelling & Computing in Biology and Medicine, 2002, 1, 270–278. [26] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Physical Review Letters, 1990, 64(8), 821. doi: 10.1103/PhysRevLett.64.821 [27] B. Roy, S. K. Roy, et al., Holling–Tanner model with Beddington–Deangelis functional response and time delay introducing harvesting, Mathematics and Computers in Simulation, 2017, 142, 1–14. doi: 10.1016/j.matcom.2017.03.010 [28] E. Sáez and E. González-Olivares, Dynamics of a predator-prey model, SIAM Journal on Applied Mathematics, 1999, 59(5), 1867–1878. doi: 10.1137/S0036139997318457 [29] G. T. Skalski and J. F. Gilliam, Functional responses with predator interference: Viable alternatives to the holling type ii model, Ecology, 2001, 82(11), 3083–3092. doi: 10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2 [30] J. P. Tripathi, S. Bugalia, V. Tiwari and Y. Kang, A predator–prey model with Crowley–Martin functional response: A nonautonomous study, Natural Resource Modeling, 2020, 33(4), e12287. doi: 10.1111/nrm.12287 [31] C. Wang and X. Li, Stability and Neimark-Sacker bifurcation of a semi-discrete population model, J. Appl. Anal. Comput, 2014, 4(4), 419–435. [32] S. Wiggins, Introduction to Applied Nonlinear Dynamic Systems and Chaos 2D, 2000. [33] C. Xiang, J. Huang and H. Wang, Linking bifurcation analysis of holling–tanner model with generalist predator to a changing environment, Studies in Applied Mathematics, 2022, 149(1), 124–163. doi: 10.1111/sapm.12492 [34] Y. Yang, T. Q. Abdullah, G. Huang and Y. Dong, Mathematical analysis of sir epidemic model with piecewise infection rate and control strategies, Journal of Nonlinear Modeling and Analysis, 2023, 5(3), 524–539. http://jnma.ca ;http://jnma.ijournal.cn .[35] S. Yu, Global stability of a modified Leslie-Gower model with Beddington-Deangelis functional response, Advances in Difference Equations, 2014, 2014, 1–14. [36] S. Yu, et al., Global asymptotic stability of a predator-prey model with modified Leslie-Gower and holling-type ii schemes, Discrete Dynamics in Nature and Society, 2012, 2012. [37] X. Zhang, C. Zhang and Y. Zhang, Pattern dynamics analysis of a time-space discrete Fitzhugh-Nagumo (FHN) model based on coupled map lattices, Computers & Mathematics with Applications, 2024, 157, 92–123. [38] Z. Zhu, Y. Chen, Z. Li and F. Chen, Stability and bifurcation in a Leslie–Gower predator–prey model with allee effect, International Journal of Bifurcation and Chaos, 2022, 32(03), 2250040. doi: 10.1142/S0218127422500407 -
-
Figures(7)
Export File
Citation
Qingkai Xu, Chunrui Zhang. BIFURCATION ANALYSIS AND CHAOS OF A MODIFIED HOLLING−TANNER MODEL WITH DISCRETE TIME[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3425-3449. doi: 10.11948/20240028
Format
Content
DownLoad:
-
Figure 1.
Flip bifurcation appears around the positive point
$ ({{\tilde x}_0}, {{\tilde y}_0}) $ $ (0.309, 0.466) $ $ x $ $ y $ $ x $ $ y $ -
Figure 2.
Phase diagrams at the positive point
$ ({{\tilde x}_0}, {{\tilde y}_0}) $ $ (0.309, 0.466) $ $ (x_0, y_0) \approx (0.3, 0.46666667) $ -
Figure 3.
Hopf bifurcation appears around the positive point
$ ({{\tilde x}_1}, {{\tilde y}_1}) $ $ (0.2285, 0.4785) $ $ x $ $ y $ $ t $ -
Figure 4.
Phase diagrams at the positive point
$ ({{\tilde x}_1}, {{\tilde y}_1}) $ $ (0.2285, 0.4785) $ $ (x_1, y_1) \approx (0.2280, 0.4780) $ -
Figure 5.
$ \delta \in (0, 0.06)- $ $ (0.309, 0.466) $ $ \delta \in (0, 0.06) $ $ \delta $ -
Figure 6.
The phase diagram of the model
$ (1.5) $ $ (0.309, 0.466) $ $ \delta = 0.0569 $ -
Figure 7.
The bifurcation diagram and maximum Lyapunov exponent corresponding to perturbation
$ \delta \in (-0.2, 2.1) $ $ x $