Dynamic behaviors of a non-autonomous allelopathic phytoplankton model with threshold inter-inhibition and feedback controls

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This paper investigated a non-autonomous allelopathic phytoplankton model with threshold inhibition and feedback controls. We applied the comparison theorem of differential equations and constructed some suitable Lyapunov functions to obtain sufficient conditions for the permanence and global attractivity of the system. By constructing some suitable Lyapunov-type extinction functions, we further derived sufficient conditions for the extinction of one of the species. The analysis reveals that inter-specific inhibition and toxic substances critically influence the permanence, extinction, and global stability of the two-species system. Some known results are extended.

Keywords

permanence / extinction / global attractivity / phytoplankton / feedback controls

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] Q. L. Wang, Z. J. Liu, Z. X. Li, R. A. Cheke, Existence and global asymptotic stability of positive almost periodic solutions of a two-species competitive system, Int. J. Biomath., 7 (2014), 1450040. https://doi.org/10.1142/S1793524514500405

[2] Q. Wang, Z. Liu, Uniformly asymptotic stability of positive almost periodic solutions for a discrete competitive system, J. Appl. Math., 2013 (2013), 182158. https://doi.org/10.1155/2013/182158

[3] S. Yu, Permanence for a discrete competitive system with feedback controls, Commun. Math. Biol. Neurosci., 2015 (2015), 16. https://doi.org/10.28919/CMBN/2186

[4] Q. Yue, Extinction for a discrete competition system with the effect of toxic substances, Adv. Differ. Equations, 2016 (2016), 1. https://doi.org/10.1186/s13662-015-0739-5

[5] X. D. Xie, Y. L. Xue, R. X. Wu, L. Zhao, Extinction of a two species competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton, Adv. Differ. Equations, 2016 (2016), 258. https://doi.org/10.1186/s13662-016-0974-4

[6] S. M. Chen, F. D. Chen, V. Srivastava, R. D. Parshad, Dynamical analysis of an allelopathic phytoplankton model with fear effect, Qual. Theor. Dyn. Syst., 23 (2024), 189. https://doi.org/10.1007/s12346-024-01047-3

[7] J. Solé, E. García-Ladona, P. Ruardij, M. Estrada, Modelling allelopathy among marine algae, Ecol. Model., 183 (2005), 373-384. https://doi.org/10.1016/j.ecolmodel.2004.08.021

[8] M. Bandyopadhyay, Dynamical analysis of a allelopathic phytoplankton model, J. Biol. Syst., 14 (2006), 205-217. https://doi.org/10.1142/S021833900600174X

[9] Z. Li, F. D. Chen, M. X. He, Global stability of a delay differential equations model of plankton allelopathy, Appl. Math. Comput., 218 (2012), 7155-7163. https://doi.org/10.1016/j.amc.2011.12.083

[10] L. Zhao, B. Qin, F. D. Chen, Dynamics of a discrete allelopathic phytoplankton model with infinite delays and feedback controls, Discrete. Dyn. Nat. Soc., 2020 (2020), 7023075. https://doi.org/10.1155/2020/7023075

[11] B. Bonnard, J. Rouot, Feedback classification and optimal control with applications to the controlled Lotka-Volterra model, Optimization, 74 (2025), 1173-1196. https://doi.org/10.1080/02331934.2024.2392209

[12] A. Muhammadhaji, A. Halik, H. L. Li, Dynamics in a ratio-dependent Lotka-Volterra competitive-competitive-cooperative system with feedback controls and delays, Adv. Differ. Equation, 2021 (2021), 230. https://doi.org/10.1186/s13662-021-03364-2

[13] F. D. Chen, J. H. Yang, L. J. Chen, Note on the persistent property of a feedback control system with delays, Nonlinear Anal. Real World Appl., 11 (2010), 1061-1066. https://doi.org/10.1016/j.nonrwa.2009.01.045

[14] K. Yang, Z. S. Miao, F. D. Chen, X. D. Xie, Influence of single feedback control variable on an autonomous Holling-II type cooperative system, J. Math. Anal. Appl., 435 (2016), 874-888. https://doi.org/10.1016/j.jmaa.2015.10.061

[15] G. W. He, F. D. Chen, Z. Li, L. J. Chen, Dynamical regulation of lotka-volterra competition systems: a novel fear-effect-based feedback control mechanism, Nonlinear Dyn., 113 (2025), 28555-28577. https://doi.org/10.1007/s11071-025-11567-5

[16] J. X. Zhang, X. H. Yang, S. Wang, A three-layer FDM for the Neumann initial-boundary value problem of 2D Kuramoto-Tsuzuki complex equation with strong nonlinear effects, Commun. Nonlinear Sci. Numer. Simul., 152 (2026), 109255. https://doi.org/10.1016/j.cnsns.2025.109255

[17] J. X. Zhang, X. H. Yang, S. Wang, A compact difference method for the 2-D Kuramoto-Tsuzuki complex equation with Neumann boundary characterized by strong nonlinear effects, Comput. Math. Appl., 203 (2026), 1-19. https://doi.org/10.1016/j.camwa.2025.11.013