IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v482y2024ics0096300324004338.html
   My bibliography  Save this article

Complex dynamic analysis of a big fish-small fish system by using the Poincaré map

Author

Listed:
  • Cheng, Huidong
  • Li, Wei
  • Zhang, Tonghua

Abstract

In terms of fishery management, if it is not managed carefully, it will lead to the extinction of the species and other unfavorable conditions. In order to strengthen the protection, development and rational utilization of fishery resources, this paper proposes a population capture model with state feedback control based on previous studies, in which the small fish is influenced by the Allee effect, and the catch of the big fish is linearly related with the release of the small fish. Firstly, we give the definition of the Poincaré map of the catch system based on impulsive point series, and study the complex dynamic properties of the map, including single peak, multi-peak functions and multiple jumping points. Moreover, the conditions of existence and the number of jumping points are discussed. At the same time, we analyze its properties including monotonicity, differentiability and fixed point. Secondly, the conditions under which the existence and uniqueness of the order-1 periodic solution are provided, and we address the necessary and sufficient conditions for stability of the order-1 periodic solution. Furthermore, existence of the order-k(k≥2) periodic solution is obtained. Finally, correctness of our theoretical results is illustrated by numerical simulations. The results show that under human control, the density of big fish and small fish will maintain periodic benign changes, and fishery resources are sufficient for sustainable development.

Suggested Citation

  • Cheng, Huidong & Li, Wei & Zhang, Tonghua, 2024. "Complex dynamic analysis of a big fish-small fish system by using the Poincaré map," Applied Mathematics and Computation, Elsevier, vol. 482(C).
  • Handle: RePEc:eee:apmaco:v:482:y:2024:i:c:s0096300324004338
    DOI: 10.1016/j.amc.2024.128972
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300324004338
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.amc.2024.128972?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Guo, Hongjian & Chen, Lansun & Song, Xinyu, 2015. "Qualitative analysis of impulsive state feedback control to an algae-fish system with bistable property," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 905-922.
    2. Tang, Sanyi & Xiao, Yanni & Cheke, Robert A., 2008. "Multiple attractors of host–parasitoid models with integrated pest management strategies: Eradication, persistence and outbreak," Theoretical Population Biology, Elsevier, vol. 73(2), pages 181-197.
    3. Jin Yang & Guangyao Tang & Sanyi Tang, 2018. "Holling-Tanner Predator-Prey Model with State-Dependent Feedback Control," Discrete Dynamics in Nature and Society, Hindawi, vol. 2018, pages 1-18, October.
    4. Yu, Xingwang & Yuan, Sanling & Zhang, Tonghua, 2019. "Survival and ergodicity of a stochastic phytoplankton–zooplankton model with toxin-producing phytoplankton in an impulsive polluted environment," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 249-264.
    5. Sun, Kaibiao & Zhang, Tonghua & Tian, Yuan, 2017. "Dynamics analysis and control optimization of a pest management predator–prey model with an integrated control strategy," Applied Mathematics and Computation, Elsevier, vol. 292(C), pages 253-271.
    6. Liu, Xia & Fan, Guihong & Zhang, Tonghua, 2019. "Evolutionary dynamics of single species model with Allee effect," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 526(C).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lirong Liu & Changcheng Xiang & Guangyao Tang & Yuan Fu, 2019. "Sliding Dynamics of a Filippov Forest-Pest Model with Threshold Policy Control," Complexity, Hindawi, vol. 2019, pages 1-17, November.
    2. Tian, Yuan & Gao, Yan & Sun, Kaibiao, 2022. "Global dynamics analysis of instantaneous harvest fishery model guided by weighted escapement strategy," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    3. Tian, Yuan & Li, Chunxue & Liu, Jing, 2023. "Complex dynamics and optimal harvesting strategy of competitive harvesting models with interval-valued imprecise parameters," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    4. Lifan Chen & Xingwang Yu & Sanling Yuan, 2022. "Effects of Random Environmental Perturbation on the Dynamics of a Nutrient–Phytoplankton–Zooplankton Model with Nutrient Recycling," Mathematics, MDPI, vol. 10(20), pages 1-23, October.
    5. Liao, Tiancai, 2024. "The impact of temperature variation on the algae–zooplankton dynamics with size-selective disturbance," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    6. Yang, Jin & Tang, Sanyi & Tan, Yuanshun, 2016. "Complex dynamics and bifurcation analysis of host–parasitoid models with impulsive control strategy," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 522-532.
    7. Liang, Juhua & Tang, Sanyi & Cheke, Robert A., 2016. "Pure Bt-crop and mixed seed sowing strategies for optimal economic profit in the face of pest resistance to pesticides and Bt-corn," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 6-21.
    8. Liu, Zhi-bin & Liu, Shutang & Wang, Wen & Wang, Da, 2021. "Effect of herd-taxis on the self-organization of a plankton community," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    9. Qi Cai & Yushi Cai & Yali Wen, 2018. "Spatially Differentiated Trends between Forest Pest-Induced Losses and Measures for Their Control in China," Sustainability, MDPI, vol. 11(1), pages 1-16, December.
    10. Zhang, Shengqiang & Yuan, Sanling & Zhang, Tonghua, 2022. "A predator-prey model with different response functions to juvenile and adult prey in deterministic and stochastic environments," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    11. Wang, Huazheng & Jiang, Daqing & Hayat, Tasawar & Alsaedi, Ahmed & Ahmad, Bashir, 2020. "Stationary distribution of stochastic NP ecological model under regime switching," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 549(C).
    12. Zhao, Shengnan & Yuan, Sanling & Zhang, Tonghua, 2022. "The impact of environmental fluctuations on a plankton model with toxin-producing phytoplankton and patchy agglomeration," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    13. Guo, Wenjuan & Zhang, Qimin, 2021. "Explicit numerical approximation for an impulsive stochastic age-structured HIV infection model with Markovian switching," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 86-115.
    14. Zhang, Meng & Zhao, Yi & Chen, Lansun & Li, Zeyu, 2020. "State feedback impulsive modeling and dynamic analysis of ecological balance in aquaculture water with nutritional utilization rate," Applied Mathematics and Computation, Elsevier, vol. 373(C).
    15. Zhenzhen Shi & Yaning Li & Huidong Cheng, 2019. "Dynamic Analysis of a Pest Management Smith Model with Impulsive State Feedback Control and Continuous Delay," Mathematics, MDPI, vol. 7(7), pages 1-15, July.
    16. Kirkow, Velizar & Wang, Hao & Garcia, Pablo Venegas & Ahmed, Shohel & Heggerud, Christopher M., 2022. "Impacts of a changing environment on a stoichiometric producer-grazer system: a stochastic modelling approach," Ecological Modelling, Elsevier, vol. 469(C).
    17. Lu, Minmin & Wang, Yan & Jiang, Daqing, 2021. "Stationary distribution and probability density function analysis of a stochastic HIV model with cell-to-cell infection," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    18. Liu, Qiong & Zhang, Meng & Chen, Lansun, 2019. "State feedback impulsive therapy to SIS model of animal infectious diseases," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 516(C), pages 222-232.
    19. Peipei Wang & Wenjie Qin & Guangyao Tang, 2019. "Modelling and Analysis of a Host-Parasitoid Impulsive Ecosystem under Resource Limitation," Complexity, Hindawi, vol. 2019, pages 1-12, January.
    20. Xu, Chaoqun, 2020. "Probabilistic mechanisms of the noise-induced oscillatory transitions in a Leslie type predator-prey model," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:482:y:2024:i:c:s0096300324004338. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.