MDPI - Publisher of Open Access Journals

22 pages, 1001 KiB

Open AccessArticle

Complex Dynamics and PID Control Strategies for a Fractional Three-Population Model

by Yan Zhou, Zhuang Cui and Ruimei Li

Mathematics 2024, 12(23), 3793; https://doi.org/10.3390/math12233793 - 30 Nov 2024

Viewed by 756

In recent decades, there have been many studies on Hopf bifurcation and population stability with time delay. However, the stability and Hopf bifurcation of fractional-order population systems with time delay are lower. In this paper, we discuss the dynamic behavior of a fractional-order three-population model with pregnancy delay using Laplace transform of fractional differential equations, stability and bifurcation theory, and MATLAB software. The specific conditions of local asymptotic stability and Hopf bifurcation for fractional-order time-delay systems are determined. A fractional-order proportional–integral–derivative (PID) controller is applied to the three-population food chain system for the first time. The convergent speed and vibration amplitude of the system can be changed by PID control. For example, after fixing the values of the integral control gain

k_{i}

and the differential control gain

k_{d}

, the amplitude of the system decreases and the convergence speed changes as the proportional control gain

k_{p}

decreases. The effectiveness of the PID control strategy in complex ecosystem is proved. The numerical simulation results are in good agreement with the theoretical analysis. The research in this paper has potential application values concerning the management of complex population systems. The bifurcation theory of fractional-order time-delay systems is also enriched. Full article

(This article belongs to the Special Issue Recent Advances in Complex Dynamics in Non-Smooth Systems)

► Show Figures

Figure 1

15 pages, 1153 KiB

Open AccessArticle

Complex Dynamics Analysis and Chaos Control of a Fractional-Order Three-Population Food Chain Model

by Zhuang Cui, Yan Zhou and Ruimei Li

Fractal Fract. 2023, 7(7), 548; https://doi.org/10.3390/fractalfract7070548 - 16 Jul 2023

Cited by 11 | Viewed by 1897

Abstract

The present study investigates the stability analysis and chaos control of a fractional-order three-population food chain model. Previous research has indicated that the predation relationship within a long-established predator–prey system can be influenced by factors such as the prey’s fear of the predator and its carry-over effects. This study examines the state evolution of fractional-order systems and compares their dynamic behavior with integer-order systems. By utilizing the Routh–Hurwitz condition and the stability theory of fractional differential equations, this paper establishes the local stability conditions of the model through the application of the Jacobi matrix and eigenvalue method. Furthermore, the conditions for the Hopf bifurcation generation are determined. Subsequently, chaos control techniques based on the Lyapunov stability theory are employed to stabilize the unstable trajectory at the equilibrium point. The theoretical findings are validated through numerical simulations. These results enhance our understanding of the stability properties and chaos control mechanisms in fractional-order three-population food chain models. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application)

► Show Figures

Figure 1

14 pages, 1012 KiB

Open AccessReview

Resistant Starches and Non-Communicable Disease: A Focus on Mediterranean Diet

by Erika Cione, Alessia Fazio, Rosita Curcio, Paola Tucci, Graziantonio Lauria, Anna Rita Rita Cappello and Vincenza Dolce

Foods 2021, 10(9), 2062; https://doi.org/10.3390/foods10092062 - 1 Sep 2021

Cited by 30 | Viewed by 10245

Abstract

Resistant starch (RS) is the starch fraction that eludes digestion in the small intestine. RS is classified into five subtypes (RS1–RS5), some of which occur naturally in plant-derived foods, whereas the others may be produced by several processing conditions. The different RS subtypes are widely found in processed foods, but their physiological effects depend on their structural characteristics. In the present study, foods, nutrition and biochemistry are summarized in order to assess the type and content of RS in foods belonging to the Mediterranean Diet (MeD). Then, the benefits of RS consumption on health are discussed, focusing on their capability to enhance glycemic control. RS enters the large bowel intestine, where it is fermented by the microbiome leading to the synthesis of short-chain fatty acids as major end products, which in turn have systemic health effects besides the in situ one. It is hoped that this review will help to understand the pros of RS consumption as an ingredient of MeD food. Consequently, new future research directions could be explored for developing advanced dietary strategies to prevent non-communicable diseases, including colon cancer. Full article

(This article belongs to the Special Issue Starch: Properties, Processing, and Functionality in Food Systems)

► Show Figures

Graphical abstract

23 pages, 9456 KiB

Open AccessArticle

Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application

by Sameh Askar, Abdulrahman Al-khedhairi, Amr Elsonbaty and Abdelalim Elsadany

Symmetry 2021, 13(2), 161; https://doi.org/10.3390/sym13020161 - 21 Jan 2021

Cited by 15 | Viewed by 2847

Abstract

Using the discrete fractional calculus, a novel discrete fractional-order food chain model for the case of strong pressure on preys map is proposed. Dynamical behaviors of the model involving stability analysis of its equilibrium points, bifurcation diagrams and phase portraits are investigated. It is demonstrated that the model can exhibit a variety of dynamical behaviors including stable steady states, periodic and quasiperiodic dynamics. Then, a hybrid encryption scheme based on chaotic behavior of the model along with elliptic curve key exchange scheme is proposed for colored plain images. The hybrid scheme combines the characteristics of noise-like chaotic dynamics of the map, including high sensitivity to values of parameters, with the advantages of reliable elliptic curves-based encryption systems. Security analysis assures the efficiency of the proposed algorithm and validates its robustness and efficiency against possible types of attacks. Full article

(This article belongs to the Special Issue Bifurcation and Chaos in Fractional-Order Systems)

► Show Figures

Figure 1

15 pages, 5365 KiB

Open AccessArticle

Dynamics Analysis and Chaotic Control of a Fractional-Order Three-Species Food-Chain System

by Lina Wang, Hui Chang and Yuxia Li

Mathematics 2020, 8(3), 409; https://doi.org/10.3390/math8030409 - 12 Mar 2020

Cited by 9 | Viewed by 3408

Abstract

Based on Hastings and Powell’s research, this paper extends a three-species food-chain system to fractional-order form, whose dynamics are analyzed and explored. The necessary conditions for generating chaos are confirmed by the stability theory of fractional-order systems, chaos is characterized by its phase diagrams, and bifurcation diagrams prove that the dynamic behaviors of the fractional-order food-chain system are affected by the order. Next, the chaotic control of the fractional-order system is realized by the feedback control method with a good effect in a relative short period. The stability margin of the controlled system is revealed by the theory and numerical analysis. Finally, the results of theory analysis are verified by numerical simulations. Full article

► Show Figures

Figure 1

Search Results (5)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (5)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI