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Volume 15
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Qualitative Analysis of a Three-Species Ratio-Dependent Lotka–Volterra Model with Cautious Effect and Feedback Control

by
Changyou Wang
1,2,*,
Qi Shang
2,
Yuanyuan Li
1 and
Peisen Feng
3
1
School of Computer Science, Dianchi College, Kunming 650228, China
2
College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, China
3
SWUFE-UD Institute of Data Science, Southwestern University of Finance and Economics, Chengdu 611130, China
*
Author to whom correspondence should be addressed.
Axioms 2026, 15(9), 705; https://doi.org/10.3390/axioms15090705 (registering DOI)
Submission received: 21 August 2026 / Revised: 11 September 2026 / Accepted: 18 September 2026 / Published: 20 September 2026
(This article belongs to the Section Mathematical Analysis)
Versions Notes

Abstract

This paper presents a qualitative analysis of a three-species ratio-dependent Lotka–Volterra model that incorporates cautious effect and feedback control. These two mechanisms serve as core regulatory factors in population dynamics, and their integration into the classical framework significantly enhances the biological realism of the model. For the persistence property, we apply the comparison principle of differential equations together with inequality techniques to derive sufficient conditions. For the global attractivity, we construct appropriate Lyapunov functionals and employ Lyapunov stability theory to establish sufficient conditions. A numerical example is finally provided to validate the correctness of the proposed theoretical results. Our findings indicate that the combined action of cautious effect and feedback control can effectively reduce extinction risks and suppress excessive population fluctuations in three-species ratio-dependent systems, thereby offering a quantitative basis for precise ecosystem management.
Keywords: cautious effect; feedback control; predator–prey model; ratio-dependence cautious effect; feedback control; predator–prey model; ratio-dependence

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MDPI and ACS Style

Wang, C.; Shang, Q.; Li, Y.; Feng, P. Qualitative Analysis of a Three-Species Ratio-Dependent Lotka–Volterra Model with Cautious Effect and Feedback Control. Axioms 2026, 15, 705. https://doi.org/10.3390/axioms15090705

AMA Style

Wang C, Shang Q, Li Y, Feng P. Qualitative Analysis of a Three-Species Ratio-Dependent Lotka–Volterra Model with Cautious Effect and Feedback Control. Axioms. 2026; 15(9):705. https://doi.org/10.3390/axioms15090705

Chicago/Turabian Style

Wang, Changyou, Qi Shang, Yuanyuan Li, and Peisen Feng. 2026. "Qualitative Analysis of a Three-Species Ratio-Dependent Lotka–Volterra Model with Cautious Effect and Feedback Control" Axioms 15, no. 9: 705. https://doi.org/10.3390/axioms15090705

APA Style

Wang, C., Shang, Q., Li, Y., & Feng, P. (2026). Qualitative Analysis of a Three-Species Ratio-Dependent Lotka–Volterra Model with Cautious Effect and Feedback Control. Axioms, 15(9), 705. https://doi.org/10.3390/axioms15090705

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