Search Advanced

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 5: 8632-8665. doi: 10.3934/mbe.2023379
Previous Article Next Article
Research article

Bifurcation analysis in a Holling-Tanner predator-prey model with strong Allee effect

  • 1.
    School of Mathematics and Statistics, Fuzhou University, Fuzhou, Fujian 350108, PR China
  • 2.
    College of Mathematics and Data Science, Minjiang University, Fuzhou, Fujian 350108, PR China
  • Received: 30 December 2022 Revised: 10 February 2023 Accepted: 23 February 2023 Published: 06 March 2023

  • In this paper, we analyze the bifurcation of a Holling-Tanner predator-prey model with strong Allee effect. We confirm that the degenerate equilibrium of system can be a cusp of codimension 2 or 3. As the values of parameters vary, we show that some bifurcations will appear in system. By calculating the Lyapunov number, the system undergoes a subcritical Hopf bifurcation, supercritical Hopf bifurcation or degenerate Hopf bifurcation. We show that there exists bistable phenomena and two limit cycles. By verifying the transversality condition, we also prove that the system undergoes a Bogdanov-Takens bifurcation of codimension 2 or 3. The main conclusions of this paper complement and improve the previous paper [30]. Moreover, numerical simulations are given to verify the theoretical results.

    Citation: Yingzi Liu, Zhong Li, Mengxin He. Bifurcation analysis in a Holling-Tanner predator-prey model with strong Allee effect[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 8632-8665. doi: 10.3934/mbe.2023379

    Related Papers:

    [1] Huiying Zhang, Jiayan Lin, Lan Zhou, Jiahui Shen, Wenshun Sheng . Facial age recognition based on deep manifold learning. Mathematical Biosciences and Engineering, 2024, 21(3): 4485-4500. doi: 10.3934/mbe.2024198
    [2] Mubashir Ahmad, Saira, Omar Alfandi, Asad Masood Khattak, Syed Furqan Qadri, Iftikhar Ahmed Saeed, Salabat Khan, Bashir Hayat, Arshad Ahmad . Correction: Facial expression recognition using lightweight deep learning modeling. Mathematical Biosciences and Engineering, 2023, 20(6): 10675-10677. doi: 10.3934/mbe.2023472
    [3] R Nandhini Abiram, P M Durai Raj Vincent . Identity preserving multi-pose facial expression recognition using fine tuned VGG on the latent space vector of generative adversarial network. Mathematical Biosciences and Engineering, 2021, 18(4): 3699-3717. doi: 10.3934/mbe.2021186
    [4] Basem Assiri, Mohammad Alamgir Hossain . Face emotion recognition based on infrared thermal imagery by applying machine learning and parallelism. Mathematical Biosciences and Engineering, 2023, 20(1): 913-929. doi: 10.3934/mbe.2023042
    [5] Jia-Gang Qiu, Yi Li, Hao-Qi Liu, Shuang Lin, Lei Pang, Gang Sun, Ying-Zhe Song . Research on motion recognition based on multi-dimensional sensing data and deep learning algorithms. Mathematical Biosciences and Engineering, 2023, 20(8): 14578-14595. doi: 10.3934/mbe.2023652
    [6] Shuai Cao, Biao Song . Visual attentional-driven deep learning method for flower recognition. Mathematical Biosciences and Engineering, 2021, 18(3): 1981-1991. doi: 10.3934/mbe.2021103
    [7] Yuanyao Lu, Kexin Li . Research on lip recognition algorithm based on MobileNet + attention-GRU. Mathematical Biosciences and Engineering, 2022, 19(12): 13526-13540. doi: 10.3934/mbe.2022631
    [8] C Willson Joseph, G. Jaspher Willsie Kathrine, Shanmuganathan Vimal, S Sumathi., Danilo Pelusi, Xiomara Patricia Blanco Valencia, Elena Verdú . Improved optimizer with deep learning model for emotion detection and classification. Mathematical Biosciences and Engineering, 2024, 21(7): 6631-6657. doi: 10.3934/mbe.2024290
    [9] Yufeng Qian . Exploration of machine algorithms based on deep learning model and feature extraction. Mathematical Biosciences and Engineering, 2021, 18(6): 7602-7618. doi: 10.3934/mbe.2021376
    [10] Chelsea Harris, Uchenna Okorie, Sokratis Makrogiannis . Spatially localized sparse approximations of deep features for breast mass characterization. Mathematical Biosciences and Engineering, 2023, 20(9): 15859-15882. doi: 10.3934/mbe.2023706

  • In this paper, we analyze the bifurcation of a Holling-Tanner predator-prey model with strong Allee effect. We confirm that the degenerate equilibrium of system can be a cusp of codimension 2 or 3. As the values of parameters vary, we show that some bifurcations will appear in system. By calculating the Lyapunov number, the system undergoes a subcritical Hopf bifurcation, supercritical Hopf bifurcation or degenerate Hopf bifurcation. We show that there exists bistable phenomena and two limit cycles. By verifying the transversality condition, we also prove that the system undergoes a Bogdanov-Takens bifurcation of codimension 2 or 3. The main conclusions of this paper complement and improve the previous paper [30]. Moreover, numerical simulations are given to verify the theoretical results.




    [1] S. X. Yan, D. X. Jia, T. H. Zhang, S. L. Yuan, Pattern dynamic in a diffusive predator-prey model with hunting cooperations, Chaos Solitons Fract., 130 (2020), 109428. https://doi.org/10.1016/j.chaos.2019.109428 doi: 10.1016/j.chaos.2019.109428
    [2] D. Tang, Y. M. Chen, Predator-prey systems in open advective heterogeneous environments with Holling-Tanner interaction term, J. Differ. Equations, 334 (2022), 280–308. https://doi.org/10.1016/j.jde.2022.06.022 doi: 10.1016/j.jde.2022.06.022
    [3] W. Ding, W. Z. Huang, Global dynamics of a ratio-dependent Holling-Tanner predator-prey system, J. Math. Anal. Appl., 460 (2018), 458–475. https://doi.org/10.1016/j.jmaa.2017.11.057 doi: 10.1016/j.jmaa.2017.11.057
    [4] S. L. Yuan, D. M. Wu, G. J. Lan, H. Wang, Noise-induced transitions in a nonsmooth predator-prey model with stoichiometric constraints, Bull. Math. Biol., 55 (2020), 2250086. https://doi.org/10.1007/s11538-020-00733-y doi: 10.1007/s11538-020-00733-y
    [5] S. Q. Zhang, T. H. Zhang, S. L. Yuan, Dynamics of a stochastic predator-prey model with habitat complexity and prey aggregation, Ecol. Compl., 45 (2021), 100889. https://doi.org/10.1016/j.ecocom.2020.100889 doi: 10.1016/j.ecocom.2020.100889
    [6] A. A. Berryman, A. P. Gutierrez, R. Arditi, Credible, parsimonious and useful predator-prey models-A reply to Abrams, Gleeson, and Sarnelle, Ecology, 76 (1995), 1980–1985. https://doi.org/10.2307/1940728 doi: 10.2307/1940728
    [7] P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Princeton University Press, 2003. https://doi.org/10.1515/9781400847280
    [8] J. X. Zhao, Y. F. Shao, Bifurcations of a prey-predator system with fear, refuge and additional food, Math. Biosci. Eng., 20 (2023), 3700–3720. https://doi.org/10.3934/mbe.2023173 doi: 10.3934/mbe.2023173
    [9] Z. L. Zhu, M. X. He, Z. Li, F. D. Chen, Stability and bifurcation in a Logistic model with Allee effect and feedback control, Int. J Bifurcat. Chaos, 30 (2020), 2050231. https://doi.org/10.1142/s0218127420502314 doi: 10.1142/s0218127420502314
    [10] M. A. Han, J. Llibre, J. M. Yang, On uniqueness of limit cycles in general Bogdanov-Takens Bifurcation, Int. J. Bifurcat. Chaos, 28 (2018), 1850115. https://doi.org/10.1142/s0218127418501158 doi: 10.1142/s0218127418501158
    [11] T. Qiao, Y. L. Cai, S. M. Fu, W. M. Wang, Stability and Hopf bifurcation in a predator-prey model with the cost of anti-predator behaviors, Int. J. Bifurcat. Chaos, 29 (2019), 1950185. https://doi.org/10.1142/s0218127419501852 doi: 10.1142/s0218127419501852
    [12] P. H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213–245. https://doi.org/10.2307/2332342 doi: 10.2307/2332342
    [13] P. H. Leslie, J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219–234. https://doi.org/10.2307/2333294 doi: 10.2307/2333294
    [14] Z. L. Zhu, Y. M. Chen, Z. Li, F. D. Chen, Stability and bifurcation in a Leslie-Gower predator-prey model with Allee effect, Int. J. Bifurcat. Chaos, 32 (2022), 2250040. https://doi.org/10.1142/s0218127422500407 doi: 10.1142/s0218127422500407
    [15] W. Q. Yin, M. X. He, Z. Li, F. D. Chen, Modeling the Allee effect in the Lesile-Gower predator-prey system incorporating a prey refuge, Int. J. Bifurcat. Chaos, 32 (2022), 2250086. https://doi.org/10.1142/s0218127422500869 doi: 10.1142/s0218127422500869
    [16] C. Arancibia-Ibarra, J. Flores, Dynamics of a Leslie-Gower predator-prey model with Holling type II functional response, Allee effect and a generalist predator, Math. Comput. Simulat., 188 (2021), 1–22. https://doi.org/10.1016/j.matcom.2021.03.035 doi: 10.1016/j.matcom.2021.03.035
    [17] E. González-Olivares, C. Arancibia-Ibarra, A. Rojas-Palma, B. González-Ya˜nez, Bifurcations and multistability on the May-Holling-Tanner predation model considering alternative food for the predators, Math. Biosci. Eng., 16 (2019), 4274–4298. https://doi.org/10.3934/mbe.2019213 doi: 10.3934/mbe.2019213
    [18] R. Sivasamy, K. Sathiyanathan, K. Balachandran, Dynamics of a modified Leslie-Gower model with gestation effect and nonlinear harvesting, J. Appl. Anal. Comput., 9 (2019), 747–764. https://doi.org/10.11948/2156-907x.20180165 doi: 10.11948/2156-907x.20180165
    [19] J. Song, Qualitative analysis of a predator-prey system with ratio-dependent and modified Leslie-Gower functional response, J. Nonlinear Model. Anal., 2 (2020), 317–332. https://doi.org/10.12150/jnma.2020.317 doi: 10.12150/jnma.2020.317
    [20] C. Arancibia-Ibarra, The basins of attraction in a modified May-Holling-Tanner predator-prey model with Allee affect, Nonlinear Anal., 185 (2019), 15–28. https://doi.org/10.1016/j.na.2019.03.004 doi: 10.1016/j.na.2019.03.004
    [21] C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European Pine Sawfly, Canad. Entomol., 91 (1959), 293–320. https://doi.org/10.4039/ent91293-5 doi: 10.4039/ent91293-5
    [22] E. Sáez, E. González-Olivares, Dynamics of a predator-prey model, SIAM J. Appl. Math., 59 (1999), 1867–1878. https://doi.org/10.1137/S0036139997318457 doi: 10.1137/S0036139997318457
    [23] W. C. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, USA, 1931. https://doi.org/10.5962/bhl.title.7313
    [24] Y. L. Cai, M. Banerjee, Y. Kang, W. M. Wang, Spatiotemporal complexity in a predator-prey model with weak Allee effects, Math. Biosci. Eng., 11 (2014), 1247–1274. https://doi.org/10.3934/mbe.2014.11.1247 doi: 10.3934/mbe.2014.11.1247
    [25] K. Manna, M. Banerjee, Stability of Hopf-bifurcating limit cycles in a diffusion-driven prey-predator system with Allee effect and time delay, Math. Biosci. Eng., 16 (2019), 2411–2446. https://doi.org/10.3934/mbe.2019121 doi: 10.3934/mbe.2019121
    [26] E. González-Olivares, J. Mena-Lorca, A. Rojas-Palma, J. Flores, Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey, Appl. Math. Model., 35 (2011), 366–381. https://doi.org/10.1016/j.apm.2010.07.001 doi: 10.1016/j.apm.2010.07.001
    [27] Y. J. Li, M. X. He, Z. Li, Dynamics of a ratio-dependent Leslie-Gower predator-prey model with Allee effect and fear effect, Math. Comput. Simulat., 201 (2022), 417–439. https://doi.org/10.1016/j.matcom.2022.05.017 doi: 10.1016/j.matcom.2022.05.017
    [28] N. Martínez-Jeraldo, P. Aguirre, Allee effect acting on the prey species in a Leslie-Gower predation model, Nonlinear Anal. Real World Appl., 45 (2019), 895–917. https://doi.org/10.1016/j.nonrwa.2018.08.009 doi: 10.1016/j.nonrwa.2018.08.009
    [29] C. Arancibia-Ibarra, J. Flores, 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 (2020), 8052–8073. https://doi.org/10.3934/mbe.2020408 doi: 10.3934/mbe.2020408
    [30] C. Arancibia-Ibarra, J. Flores, G. Pettet, P. Heijster, A Holling-Tanner predator-prey model with strong Allee effect, Int. J. Bifurcat. Chaos, 21 (2019), 1930032. https://doi.org/10.1142/s0218127419300325 doi: 10.1142/s0218127419300325
    [31] J. C. Huang, Y. J. Gong, J. Chen, Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, Int. J. Bifurcat. Chaos, 10 (2013), 1350164. https://doi.org/10.1142/s0218127413501642 doi: 10.1142/s0218127413501642
    [32] L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1996. https://doi.org/10.1007/978-1-4684-0392-3
    [33] M. Lu, J. C. Huang, S. G. Ruan, P. Yu, Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Differ. Equations, 267 (2019), 1859–1898. https://doi.org/10.1016/j.jde.2019.03.005 doi: 10.1016/j.jde.2019.03.005
    [34] Y. F. Dai, Y. L. Zhao, B. Sang, Four limit cycles in a predator-prey system of Leslie type with generalized Holling type III functional response, Nonlinear Anal. Real World Appl., 50 (2019), 218–239. https://doi.org/10.1016/j.nonrwa.2019.04.003 doi: 10.1016/j.nonrwa.2019.04.003
    [35] J. C. Huang, S. H. Liu, S. G. Ruan, X. N. Zhang, Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting, Commun. Pure Appl. Anal., 15 (2016), 1041–1055. https://doi.org/10.3934/cpaa.2016.15.1041 doi: 10.3934/cpaa.2016.15.1041
    [36] C. Z. Li, J. Q. Li, Z. E. Ma, Codimension 3 B-T bifurcations in an epidemic model with a nonlinear incidenc, Discrete Contin. Dynam. Systems, 20 (2015), 1107–1116. https://doi.org/10.3934/dcdsb.2015.20.1107 doi: 10.3934/dcdsb.2015.20.1107
    [37] S. N. Chow, C. Li, D. Wang, Normal Forms and Bifurcation of Planar Vector, Cambridge University Press, Cambridge, 1994. https://doi.org/10.1017/CBO9780511665639
    [38] F. Dumortier, R. Roussarie, J. Sotomayor, K. Zoladek, Bifurcation of Planar Vector Fields, Nilpotent Singularities and Abelian Integrals, Lect. Notes Math., vol. 1480, Springer-Verlag, Berlin, 1991. https://doi.org/10.1007/BFb0098353
    [39] F. Dumortier, R. Roussarie, J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergod. Theory Dyn. Syst. 7 (1987), 375–413. https://doi.org/10.1017/s0143385700004119

  • This article has been cited by:

    1. Jassim AlMulla, Mohammad Tariqul Islam, Hamada R. H. Al-Absi, Tanvir Alam, Rahul Gomes, SoccerNet: A Gated Recurrent Unit-based model to predict soccer match winners, 2023, 18, 1932-6203, e0288933, 10.1371/journal.pone.0288933
    2. Chaolin Tang, Dong Zhang, Qichuan Tian, Convolutional Neural Network–Bidirectional Gated Recurrent Unit Facial Expression Recognition Method Fused with Attention Mechanism, 2023, 13, 2076-3417, 12418, 10.3390/app132212418
    3. Edy Winarno, Anindita Septiarini, Wiwien Hadikurniawati, Hamdani Hamdani, The Hybrid Features and Supervised Learning for Batik Pattern Classification, 2024, 17, 1556-4673, 1, 10.1145/3631131
    4. Samla Salim, R. Sarath, BREAST CANCER DETECTION AND CLASSIFICATION USING HISTOPATHOLOGICAL IMAGES BASED ON OPTIMIZATION-ENABLED DEEP LEARNING, 2024, 36, 1016-2372, 10.4015/S101623722350028X
    5. Shengfu Zhang, Zhongjie Xiao, Facial Expression Recognition Using a Semantic-Based Bottleneck Attention Module, 2024, 20, 1552-6283, 1, 10.4018/IJSWIS.352418
    6. Haoyu Zhou, Lingfeng Sang, Jingjing Luo, Hongbo Wang, Yongfei Feng, Xueze Zhang, Li Chen, Development of a multifunctional nursing bed system for in-bed position recognition and automatic repositioning, 2024, 238, 0954-4062, 5437, 10.1177/09544062231223878
    7. Fen Wang, Kim Ju Kyoung, Leather Defect Detection Method in Clothing Design Based on TDENet, 2023, 11, 2169-3536, 104890, 10.1109/ACCESS.2023.3308493
    8. A Çelik, E Tekin, Classification of Hatchery Eggs Using a Machine Learning Algorithm Based on Image Processing Methods: A Comparative Study, 2024, 26, 1806-9061, 10.1590/1806-9061-2023-1882
    9. Jian‐Ting Shi, Gui‐Xu Qu, Zhi‐Jun Li, UCSwin‐UNet model for medical image segmentation based on cardiac haemangioma, 2024, 18, 1751-9659, 3302, 10.1049/ipr2.13175
    10. P Karthikeyan, Kirutheesvar S, Sivakumar S, 2024, Facial Emotion Recognition for Enhanced Human-Computer Interaction using Deep Learning and Temporal Modeling with BiLSTM, 979-8-3315-0440-3, 1791, 10.1109/ICOSEC61587.2024.10722687
    11. Muhammad Hameed Siddiqi, Irshad Ahmad, Yousef Alhwaiti, Faheem Khan, Facial Expression Recognition for Healthcare Monitoring Systems Using Neural Random Forest, 2025, 29, 2168-2194, 30, 10.1109/JBHI.2024.3482450
    12. Kun Zheng, Li Tian, Jinling Cui, Junhua Liu, Hui Li, Jing Zhou, Junjie Zhang, An Adaptive Thresholding Method for Facial Skin Detection in HSV Color Space, 2025, 25, 1530-437X, 3098, 10.1109/JSEN.2024.3506579
    13. Muhammed Telceken, Devrim Akgun, Sezgin Kacar, Kübra YESİN, Metin Yıldız, Can artificial intelligence understand our emotions? Deep learning applications with face recognition, 2025, 1046-1310, 10.1007/s12144-025-07375-0
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
4.4

Metrics

Article views(2830) PDF downloads(277) Cited by(6)

Preview PDF
Download XML
Export Citation
Article outline
Abstract
References

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog