IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i9p1408-d402896.html
   My bibliography  Save this article

A Predator–Prey Two-Sex Branching Process

Author

Listed:
  • Cristina Gutiérrez

    (Department of Mathematics, University of Extremadura, 10071 Cáceres, Spain
    These authors contributed equally to this work.)

  • Carmen Minuesa

    (Department of Mathematics, Autonomous University of Madrid, 28049 Madrid, Spain
    These authors contributed equally to this work.)

Abstract

In this paper, we present the first stochastic process to describe the interaction of predator and prey populations with sexual reproduction. Specifically, we introduce a two-type two-sex controlled branching model. This process is a two-type branching process, where the first type corresponds to the predator population and the second one to the prey population. While each population is described via a two-sex branching model, the interaction and survival of both groups is modelled through control functions depending on the current number of individuals of each type in the ecosystem. In view of their potential for the conservation of species, we provide necessary and sufficient conditions for the ultimate extinction of both species, the fixation of one of them and the coexistence of both of them. Moreover, the description of the present predator–prey two-sex branching process on the fixation events can be performed in terms of the behaviour of a one-type two-sex branching process with a random control on the number of individuals, which is also introduced and analysed.

Suggested Citation

  • Cristina Gutiérrez & Carmen Minuesa, 2020. "A Predator–Prey Two-Sex Branching Process," Mathematics, MDPI, vol. 8(9), pages 1-26, August.
  • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1408-:d:402896
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/9/1408/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/8/9/1408/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Manuel Molina & Manuel Mota & Alfonso Ramos, 2012. "Two-sex Branching Models with Random Control on the Number of Progenitor Couples," Methodology and Computing in Applied Probability, Springer, vol. 14(1), pages 35-48, March.
    2. Durrett, Rick & Mayberry, John, 2010. "Evolution in predator-prey systems," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1364-1392, July.
    3. Kocmoud, Amanda R. & Wang, Hsiao-Hsuan & Grant, William E. & Gallaway, Benny J., 2019. "Population dynamics of the endangered Kemp’s ridley sea turtle following the 2010 oil spill in the Gulf of Mexico: Simulation of potential cause-effect relationships," Ecological Modelling, Elsevier, vol. 392(C), pages 159-178.
    4. Yan, Shuixian & Jia, Dongxue & Zhang, Tonghua & Yuan, Sanling, 2020. "Pattern dynamics in a diffusive predator-prey model with hunting cooperations," Chaos, Solitons & Fractals, Elsevier, vol. 130(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. Mondal, Argha & Hens, Chittaranjan & Mondal, Arnab & Antonopoulos, Chris G., 2021. "Spatiotemporal instabilities and pattern formation in systems of diffusively coupled Izhikevich neurons," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    2. Tian, Yuan & Li, Huanmeng & Sun, Kaibiao, 2024. "Complex dynamics of a fishery model: Impact of the triple effects of fear, cooperative hunting and intermittent harvesting," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 218(C), pages 31-48.
    3. Meng Zhu & Jing Li & Xinze Lian, 2022. "Pattern Dynamics of Cross Diffusion Predator–Prey System with Strong Allee Effect and Hunting Cooperation," Mathematics, MDPI, vol. 10(17), pages 1-20, September.
    4. Shi, Yu & Luo, Xiao-Feng & Zhang, Yong-Xin & Sun, Gui-Quan, 2023. "An indicator of Crohn’s disease severity based on Turing patterns," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    5. Souna, Fethi & Belabbas, Mustapha & Menacer, Youssaf, 2023. "Complex pattern formations induced by the presence of cross-diffusion in a generalized predator–prey model incorporating the Holling type functional response and generalization of habitat complexity e," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 597-618.
    6. Wang, Fatao & Yang, Ruizhi, 2023. "Spatial pattern formation driven by the cross-diffusion in a predator–prey model with Holling type functional response," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    7. Yanfei Du & Ben Niu & Junjie Wei, 2021. "Dynamics in a Predator–Prey Model with Cooperative Hunting and Allee Effect," Mathematics, MDPI, vol. 9(24), pages 1-40, December.
    8. Djilali, Salih & Cattani, Carlo, 2021. "Patterns of a superdiffusive consumer-resource model with hunting cooperation functional response," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    9. Scott, Rebecca L. & Putman, Nathan F. & Beyea, R.Taylor & Repeta, Hallie C. & Ainsworth, Cameron H., 2024. "Modeling transport and feeding of juvenile Kemp's ridley sea turtles on the West Florida shelf," Ecological Modelling, Elsevier, vol. 490(C).
    10. Shivam, & Singh, Kuldeep & Kumar, Mukesh & Dubey, Ramu & Singh, Teekam, 2022. "Untangling role of cooperative hunting among predators and herd behavior in prey with a dynamical systems approach," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    11. Song, Binbin & Zuo, Wenjie, 2023. "Spatiotemporal dynamics of a three-component chemotaxis model for Alopecia Areata," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    12. Pal, Debjit & Kesh, Dipak & Mukherjee, Debasis, 2023. "Qualitative study of cross-diffusion and pattern formation in Leslie–Gower predator–prey model with fear and Allee effects," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    13. Abernethy, Gavin M. & Mullan, Rory & Glass, David H. & McCartney, Mark, 2017. "A multiple phenotype predator–prey model with mutation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 465(C), pages 762-774.
    14. Wang, Henan & Liu, Ping, 2023. "Pattern dynamics of a predator–prey system with cross-diffusion, Allee effect and generalized Holling IV functional response," Chaos, Solitons & Fractals, Elsevier, vol. 171(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:gam:jmathe:v:8:y:2020:i:9:p:1408-:d:402896. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.