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On the extinction-extinguishing dichotomy for a stochastic Lotka–Volterra type population dynamical system

Author

Listed:
  • Ren, Yan-Xia
  • Xiong, Jie
  • Yang, Xu
  • Zhou, Xiaowen

Abstract

Applying the Foster–Lyapunov type criteria and a martingale method, we study a two-dimensional process (X,Y) arising as the unique nonnegative solution to a pair of stochastic differential equations driven by independent Brownian motions and compensated spectrally positive Lévy random measures. Both processes X and Y can be identified as continuous-state nonlinear branching processes where the evolution of Y is negatively affected by X. Assuming that process X extinguishes, i.e. it converges to 0 but never reaches 0 in finite time, and process Y converges to 0, we identify rather sharp conditions under which the process Y exhibits, respectively, one of the following behaviors: extinction with probability one, extinguishing with probability one or both extinction and extinguishing occurring with strictly positive probabilities.

Suggested Citation

  • Ren, Yan-Xia & Xiong, Jie & Yang, Xu & Zhou, Xiaowen, 2022. "On the extinction-extinguishing dichotomy for a stochastic Lotka–Volterra type population dynamical system," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 50-90.
    • Handle: RePEc:eee:spapps:v:150:y:2022:i:c:p:50-90
    DOI: 10.1016/j.spa.2022.04.005
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    References listed on IDEAS

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    1. Li, Pei-Sen, 2019. "A continuous-state polynomial branching process," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2941-2967.
    2. Hui He & Zenghu Li & Wei Xu, 2018. "Continuous-State Branching Processes in Lévy Random Environments," Journal of Theoretical Probability, Springer, vol. 31(4), pages 1952-1974, December.
    3. Fang, Rongjuan & Li, Zenghu, 2019. "A conditioned continuous-state branching process with applications," Statistics & Probability Letters, Elsevier, vol. 152(C), pages 43-49.
    4. Xiong, Jie & Yang, Xu, 2019. "Existence and pathwise uniqueness to an SPDE driven by α-stable colored noise," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2681-2722.
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