Stationary Distribution and Probability Density for a Stochastic Crime Model

IF 2.1 3区数学 Q1 MATHEMATICS, APPLIED

Mathematical Methods in the Applied Sciences Pub Date : 2025-03-03 DOI:10.1002/mma.10809

Yanling Wu, Guangying Lv

引用次数: 0

Abstract

In this paper, we investigate the dynamic behavior of a criminal model (predator–prey model), in which the predators are the police, and the prey is gang members, and both predators and prey have infectious diseases, infected prey being more susceptible to predation, and infected predators hunting at a reduced rate. We get sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological viewpoint, the existence of a stationary distribution indicates that both the police and gang members will be persistent and coexistent in the long term. What is more, we give the specific expression of the probability density function of the stochastic model around the unique endemic quasi-equilibrium by solving the Fokker-Planck equation. Finally, some numerical simulations are carried out to illustrate the theoretical results.

查看原文本刊更多论文

随机犯罪模型的平稳分布和概率密度

本文研究了一种犯罪模型（捕食者-猎物模型）的动态行为，其中捕食者是警察，被捕食者是团伙成员，捕食者和被捕食者都有传染病，被感染的猎物更容易被捕食，被感染的捕食者狩猎率降低。通过建立一系列合适的Lyapunov函数，得到了该系统正解的遍历平稳分布存在唯一性的充分判据。从生物学的角度来看，固定分布的存在表明警察和帮派成员将长期存在并共存。此外，通过求解Fokker-Planck方程，给出了随机模型在独特的地方性准平衡周围的概率密度函数的具体表达式。最后，通过数值模拟对理论结果进行了验证。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Mathematical Methods in the Applied Sciences 数学-应用数学

CiteScore

4.90

自引率

6.90%

发文量

798

审稿时长

6 months

期刊介绍： Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.