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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2023, Vol. 40 ›› Issue (4): 634-646.doi: 10.3969/j.issn.1005-3085.2023.04.009

• • 上一篇    下一篇

具有斑块结构和多时滞的随机Nicholson-型模型的动力学分析

刘  荣1,  张凤琴2   

  1. 1. 山西财经大学应用数学学院，太原  030006;
    2. 运城学院数学与信息技术学院，运城 044000
  • 收稿日期:2022-01-28 接受日期:2022-06-24 出版日期:2023-08-15 发布日期:2023-10-15
  • 通讯作者: 张凤琴 E-mail: zhafq@263.net
  • 基金资助:
    国家自然科学基金(12071418; 12001341).

Dynamics of Stochastic Nicholson Model with Patch Structure and Multiple Delays

LIU Rong1,  ZHANG Fengqin2   

  1. 1. School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006;
    2. School of Mathematics and Information Technology, Yuncheng University, Yuncheng 044000
  • Received:2022-01-28 Accepted:2022-06-24 Online:2023-08-15 Published:2023-10-15
  • Contact: F. Zhang. E-mail address: zhafq@263.net
  • Supported by:
    The National Natural Science Foundation of China (12071418; 12001341).

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摘要：

针对带斑块结构和多时滞的随机Nicholson-型模型，通过构造合适的Lyapunov函数，证明了该模型全局正解的存在唯一性。通过对其构造不同的Lyapunov函数并利用Chebyshev不等式、Borel-Cantelli引理以及指数鞅不等式理论，讨论模型解的随机最终有界性、样本Lyapunov指数的非正性等有关性质。在所有时滞都相等的条件下，利用Burkholder-Davis-Gundy不等式和强大数定律，给出各个斑块的物种都灭绝的充分条件。最后，给出数值模拟结果：斑块之间的相互作用有利于物种的生存，且时滞越大物种灭绝越慢。所获结果推广和改进了相关文献的部分结果，如去掉了相应文献中全局正解的存在唯一性定理的条件，缩小了相关文献中样本的李亚普诺夫指数的界等。

关键词: 随机Nicholson-型模型, 斑块结构, 多时滞, Lyapunov函数, 灭绝

Abstract:

The stochastic Nicholson model with patch structure and multiple delays is considered. Firstly, the existence of a unique global positive solution is established by constructing a proper Lyapunov function. Next, by constructing suitable stochastic Lyapunov functions, and using Chebyshev inequality, Borel-Cantelli lemma and exponential martingale inequality, the properties of the solution, such as stochastic ultimate boundedness and non-positivity of sample Lyapunov exponents, etc., are studied. Then, under the condition that all time delays are equal, the sufficient conditions for the extinction of species in each patch are given by using the Burkholder-Davis-Gundy inequality and a strong law of large numbers. Finally, some numerical results are presented, which shows that the interaction between patches has the advantage of population survival, and the greater the time delay, the slower the extinction of species. The results generalize and improve the previous related results, such as removing the condition of the existence and uniqueness theorem of global positive solutions and narrowing the boundary of the sample Lyapunov exponents in related literature.

Key words: stochastic Nicholson model, patch structure, multiple delays, Lyapunov function, extinction

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