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A Predator–Prey Two-Sex Branching Process

Author

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  • 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
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    References listed on IDEAS

    as
    1. Durrett, Rick & Mayberry, John, 2010. "Evolution in predator-prey systems," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1364-1392, July.
    2. 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.
    3. 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.
    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).
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