Complex Dynamics of a Discrete-Time Food Chain Model

IF 1.9 4区 数学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
Sarbari Karmakar, Ruma Kumbhakar, Shilpa Garai, Fatemeh Parastesh, S. Jafari, Nikhil Pal
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引用次数: 0

Abstract

In a food chain, the role of intake patterns of predators is very influential on the survival and extinction of the interacting species as well as the entire dynamics of the ecological system. In this study, we investigate the affluent and intricate dynamics of a simple three-species food chain model in a discrete-time framework by analyzing the parameter plane of the system with simultaneous changes of two crucial parameters, the predation rates of middle and top predators. From the theoretical viewpoint, we study the model by determining the fixed points’ biological feasibility and local asymptotic stability criteria, and performing some analyses of local bifurcations, namely, transcritical, flip, and Neimark–Sacker bifurcations. Here, we initiate the numerical simulation by plotting the changes of the prey population density in terms of a vital parameter of the system, and we observe the switching among different dynamical behaviors of the system. We also draw some phase portraits and plot the time series solutions to show the diverse characteristics of the system dynamics. Further, we move one step ahead to explore the intricate dynamical scenarios appearing in the parameter plane by forming Lyapunov exponent and isoperiodic diagrams. In the parameter plane of the system, we see the emergence of innumerable Arnold tongues. All these Arnold tongues are organized along a particular direction, and the beautiful arrangement of these tongues forms several kinds of period-adding sequences. The study sheds more light on various types of multistability occurring in the model system. We see the coexistence of three periodic attractors in the parameter plane. In this study, the most striking observation is the coexistence of four periodic attractors, which occurs infrequently in ecological systems. We draw the basins of attraction for the tristable and tetrastable attractors, which are complex Wada basins. A system with Wada basin is very sensitive to initial conditions and more erratic in nature than a system with fractal basin. Also, we plot the density of all interacting species in terms of the predation rates of middle and top predators and observe the variation in the population densities of all species with the variability of these two key parameters. In the parameter plane created by the simultaneous changes of two parameters, the system exhibits a variety of intricate and subtle dynamics, which cannot be found by changing only a single parameter.
离散时间食物链模型的复杂动力学
在食物链中,捕食者的摄入模式对相互作用物种的生存和灭绝以及整个生态系统的动态都有很大影响。在本研究中,我们在离散时间框架下,通过分析系统的参数平面,研究了一个简单的三物种食物链模型在中间捕食者和顶端捕食者捕食率这两个关键参数同时变化时的富裕和复杂动态。从理论的角度,我们通过确定固定点的生物可行性和局部渐近稳定性标准来研究该模型,并进行了一些局部分岔分析,即跨临界分岔、翻转分岔和 Neimark-Sacker 分岔。在此,我们通过绘制猎物种群密度随系统重要参数的变化曲线来启动数值模拟,并观察系统不同动态行为之间的切换。我们还绘制了一些相位图和时间序列解,以显示系统动态的不同特征。此外,我们还通过形成 Lyapunov 指数图和等周期图,进一步探索参数平面上出现的错综复杂的动力学情景。在系统的参数平面上,我们看到出现了无数的阿诺德舌头。所有这些阿诺德舌头都沿着特定的方向排列,这些舌头的优美排列形成了多种周期添加序列。这项研究进一步揭示了模型系统中出现的各种多稳态性。我们在参数平面上看到了三种周期吸引子的共存。在这项研究中,最引人注目的观察结果是四个周期吸引子的共存,而这在生态系统中并不常见。我们绘制了三可吸引子和四可吸引子的吸引盆地,它们是复杂的和田盆地。具有和田盆地的系统对初始条件非常敏感,其性质比具有分形盆地的系统更加不稳定。此外,我们还根据中间捕食者和顶级捕食者的捕食率绘制了所有相互作用物种的密度图,并观察到所有物种的种群密度随这两个关键参数的变化而变化。在同时改变两个参数所形成的参数平面上,系统呈现出各种错综复杂的微妙动态,而这是只改变一个参数所无法发现的。
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来源期刊
International Journal of Bifurcation and Chaos
International Journal of Bifurcation and Chaos 数学-数学跨学科应用
CiteScore
4.10
自引率
13.60%
发文量
237
审稿时长
2-4 weeks
期刊介绍: The International Journal of Bifurcation and Chaos is widely regarded as a leading journal in the exciting fields of chaos theory and nonlinear science. Represented by an international editorial board comprising top researchers from a wide variety of disciplines, it is setting high standards in scientific and production quality. The journal has been reputedly acclaimed by the scientific community around the world, and has featured many important papers by leading researchers from various areas of applied sciences and engineering. The discipline of chaos theory has created a universal paradigm, a scientific parlance, and a mathematical tool for grappling with complex dynamical phenomena. In every field of applied sciences (astronomy, atmospheric sciences, biology, chemistry, economics, geophysics, life and medical sciences, physics, social sciences, ecology, etc.) and engineering (aerospace, chemical, electronic, civil, computer, information, mechanical, software, telecommunication, etc.), the local and global manifestations of chaos and bifurcation have burst forth in an unprecedented universality, linking scientists heretofore unfamiliar with one another''s fields, and offering an opportunity to reshape our grasp of reality.
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