Analysis of the Stability and Chaotic Dynamics of an Ecological Model

IF 1.7 4区 工程技术 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
Complexity Pub Date : 2024-07-24 DOI:10.1155/2024/1681557
Muhammad Aqib Abbasi, Qamar Din, Olayan Albalawi, Rizwan Niaz, Mohammed Ahmed Alomair, Abdullah Mohammed Alomair
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引用次数: 0

Abstract

Modelling has become an eminent tool in the study of ecological systems. Ecological modelling can help implement sustainable development, mathematical models, and system analysis that explain how ecological processes can promote the sustainable management of resources. In this paper, we also chose a four-dimensional discrete-time Lotka–Volterra ecological model and analyzed its dynamic behavior. In particular, we derived the parametric conditions for the existence of biologically feasible solutions and the stability of the fixed points. We also provided graphs to study the spectrum behavior of all fixed points. In addition, we have seen that when the intrinsic dynamics of the population exceed a certain threshold, the system bifurcates. This particular range of inherent population dynamics depends on the values of other biological parameters and the initial population. We proved that the instability of the model resulted in Neimark–Sacker and period-doubling bifurcations. To confirm these two types of bifurcation, we used bifurcation theory, and to find the direction of bifurcation, we used graphical results. Mainly, through novel periodic plots, we confirm the coexistence of the population and the possible equilibrium states. We apply Marotto’s theorem to verify the existence of chaos in the system. To control the chaos, we use a hybrid control feedback methodology. Finally, we provide numerical examples to illustrate our theoretical results. The outcomes of the numerical simulations show chaotic long-term behavior across an extensive range of parameters.

Abstract Image

生态模型的稳定性和混沌动力学分析
建模已成为研究生态系统的重要工具。生态建模有助于实施可持续发展、数学模型和系统分析,解释生态过程如何促进资源的可持续管理。在本文中,我们也选择了一个四维离散时间 Lotka-Volterra 生态模型,并对其动态行为进行了分析。特别是,我们推导出了生物可行解存在的参数条件以及固定点的稳定性。我们还提供了图形来研究所有固定点的频谱行为。此外,我们还发现,当种群的内在动态超过某个阈值时,系统就会发生分叉。种群固有动态的这一特定范围取决于其他生物参数和初始种群的值。我们证明,模型的不稳定性导致了 Neimark-Sacker 和周期加倍分岔。为了证实这两种分岔,我们使用了分岔理论,为了找到分岔的方向,我们使用了图形结果。主要是通过新颖的周期图,我们确认了种群共存和可能的平衡状态。我们运用马罗托定理验证了系统中混沌的存在。为了控制混沌,我们采用了混合控制反馈方法。最后,我们提供了数值示例来说明我们的理论结果。数值模拟的结果显示,在广泛的参数范围内都存在长期混沌行为。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Complexity
Complexity 综合性期刊-数学跨学科应用
CiteScore
5.80
自引率
4.30%
发文量
595
审稿时长
>12 weeks
期刊介绍: Complexity is a cross-disciplinary journal focusing on the rapidly expanding science of complex adaptive systems. The purpose of the journal is to advance the science of complexity. Articles may deal with such methodological themes as chaos, genetic algorithms, cellular automata, neural networks, and evolutionary game theory. Papers treating applications in any area of natural science or human endeavor are welcome, and especially encouraged are papers integrating conceptual themes and applications that cross traditional disciplinary boundaries. Complexity is not meant to serve as a forum for speculation and vague analogies between words like “chaos,” “self-organization,” and “emergence” that are often used in completely different ways in science and in daily life.
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