Discrete Lotka–Volterra systems with time delay and its stability analysis

IF 2.7 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena Pub Date : 2025-02-10 DOI:10.1016/j.physd.2025.134562

Yusaku Yamamoto , Taisei Yamamoto , Takumi Kuroiwa , Kurumi Oka , Emiko Ishiwata , Masashi Iwasaki

{"title":"Discrete Lotka–Volterra systems with time delay and its stability analysis","authors":"Yusaku Yamamoto , Taisei Yamamoto , Takumi Kuroiwa , Kurumi Oka , Emiko Ishiwata , Masashi Iwasaki","doi":"10.1016/j.physd.2025.134562","DOIUrl":null,"url":null,"abstract":"<div><div>We propose an extension of the discrete-time Lotka–Volterra (dLV) equations describing predator–prey dynamics with time delay <span><math><mi>τ</mi></math></span>. Introducing time delay corresponds to considering multiple generations of each species and gives more expressive power to the model. For example, it becomes possible to model the situation where each individual is eaten only after it has grown up. In this paper, we focus on the system with minimal time delay (<span><math><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow></math></span>) and analyze the stability of the system. In particular, we prove that when the number of species is three, the system exhibits the same asymptotic behavior as the original dLV system. For more general cases with an arbitrary odd number of species, we investigate the local stability of fixed points of the system with the help of the center manifold theory. It is shown that the fixed points that correspond to the asymptotic states of the original dLV system are locally stable.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"474 ","pages":"Article 134562"},"PeriodicalIF":2.7000,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physica D: Nonlinear Phenomena","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167278925000417","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

We propose an extension of the discrete-time Lotka–Volterra (dLV) equations describing predator–prey dynamics with time delay

τ

. Introducing time delay corresponds to considering multiple generations of each species and gives more expressive power to the model. For example, it becomes possible to model the situation where each individual is eaten only after it has grown up. In this paper, we focus on the system with minimal time delay (

τ = 1

) and analyze the stability of the system. In particular, we prove that when the number of species is three, the system exhibits the same asymptotic behavior as the original dLV system. For more general cases with an arbitrary odd number of species, we investigate the local stability of fixed points of the system with the help of the center manifold theory. It is shown that the fixed points that correspond to the asymptotic states of the original dLV system are locally stable.

Abstract Image

查看原文本刊更多论文

离散时滞Lotka-Volterra系统及其稳定性分析

我们提出了描述具有时滞τ的捕食者-猎物动力学的离散时间Lotka-Volterra （dLV）方程的扩展。引入时滞对应于考虑每个物种的多代，使模型更具表现力。例如，有可能模拟每个个体长大后才被吃掉的情况。本文主要研究具有最小时滞（τ=1）的系统，并分析该系统的稳定性。特别地，我们证明了当物种数为3时，系统表现出与原始dLV系统相同的渐近行为。对于具有任意奇数种的更一般情况，我们利用中心流形理论研究了系统不动点的局部稳定性。证明了原dLV系统渐近状态对应的不动点是局部稳定的。

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

Physica D: Nonlinear Phenomena 物理-物理：数学物理

CiteScore

7.30

自引率

7.50%

发文量

213

审稿时长

65 days

期刊介绍： Physica D (Nonlinear Phenomena) publishes research and review articles reporting on experimental and theoretical works, techniques and ideas that advance the understanding of nonlinear phenomena. Topics encompass wave motion in physical, chemical and biological systems; physical or biological phenomena governed by nonlinear field equations, including hydrodynamics and turbulence; pattern formation and cooperative phenomena; instability, bifurcations, chaos, and space-time disorder; integrable/Hamiltonian systems; asymptotic analysis and, more generally, mathematical methods for nonlinear systems.