GLOBAL ASYMPTOTIC STABILITY OF BARNACLE-ALGAE INTERACTION MODEL WITH DELAY

Hui Zhou; Ning Wang

doi:10.11948/20260020

2027 Volume 17 Issue 2

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Hui Zhou, Ning Wang. GLOBAL ASYMPTOTIC STABILITY OF BARNACLE-ALGAE INTERACTION MODEL WITH DELAY[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 851-867. doi: 10.11948/20260020

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Hui Zhou, Ning Wang. GLOBAL ASYMPTOTIC STABILITY OF BARNACLE-ALGAE INTERACTION MODEL WITH DELAY[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 851-867. doi: 10.11948/20260020

GLOBAL ASYMPTOTIC STABILITY OF BARNACLE-ALGAE INTERACTION MODEL WITH DELAY

Hui Zhou^1, ,,
Ning Wang^1,

1.
School of Mathematics and Statistics, Hefei Normal University, Hefei 230026, China

Author Bio: Email: hfnuwn@126.com(N. Wang)
Corresponding author: Email: zhouh16@mail.ustc.edu.cn(H. Zhou)
Fund Project: The authors were supported by National Natural Science Foundation of China (12571170) and Project of Hefei Normal University (2025KY06)

Abstract

Abstract

This work investigates the dynamical behavior of a three-dimensional patch occupancy model that incorporates a discrete time delay into the barnacle-algae interaction. The model framework extends the non-delayed, periodically forced barnacle-algae-mussel system introduced by Benincà in [2]. In the absence of mussels (M = 0) and seasonal forcing, the system studied in [10] reduces to a three-dimensional system, for which global asymptotic stability was previously established. Under biologically motivated assumptions, this paper provides a complete characterization of the global dynamics for the corresponding three-dimensional delayed system. By leveraging the Poincaré-Bendixson Property applicable to type-K competitive systems with delay, we prove the global asymptotic stability of the positive equilibrium whenever it exists. Our analytical results demonstrate that the introduction of a time delay does not disrupt the global convergence behavior observed in the non-delayed system.
- Global asymptotic stability /
- barnacle-algae interaction model /
- orbital stability /
- type-K competitive delayed system
MSC: 37C75, 92D25, 37N25, 34D05

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References

[1]	R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963. Google Scholar
[2]	E. Benincà, B. Ballantine, S. P. Ellner and J. Huisman, Species fluctuations sustained by a cyclic succession at the edge of chaos, Proc. Natl Acad. Sci., USA, 2015, 112, 6389–6394. doi: 10.1073/pnas.1421968112 CrossRef Google Scholar
[3]	E. Benincà, J. Huisman, R. Heerkloss, K. D. Johnk, P. Branco, E. H. Van Nes, M. Scheffer and S. P. Ellner, Chaos in a long-term experiment with a plankton community, Nature, 2008, 451, 822–825. doi: 10.1038/nature06512 CrossRef Google Scholar
[4]	O. N. Bjϕrnstad, Nonlinearity and chaos in ecological dynamics revisited, Proc. Nati. Acad. Sci., USA, 2015, 112, 6252–6253. Google Scholar
[5]	G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc., 1986, 96, 425–430. doi: 10.1090/S0002-9939-1986-0822433-4 CrossRef Google Scholar
[6]	M. Faheem and B. Ghosh, Dynamics of a delayed discrete-time predator prey model proposed from a nonstandard finite difference scheme, J. Compu. Appl. Math., 2025, 458, 116346. doi: 10.1016/j.cam.2024.116346 CrossRef Google Scholar
[7]	M. W. Hirsch, Systems of differential equations which are competitive or cooperative Ⅳ: Structural stability in three dimensional systems, SIAM J. Math. Anal., 1990, 21, 1225–1234. doi: 10.1137/0521067 CrossRef Google Scholar
[8]	M. W. Hirsch and H. L. Smith, Monotone Dynamical Systems, Handbook of Differential Equations: Ordinary Differential Equations, vol. 2, Elsevier, Amsterdam, 2005. Google Scholar
[9]	S.-B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Math. Biosci., 2004, 187, 53–91. doi: 10.1016/j.mbs.2003.07.004 CrossRef Google Scholar
[10]	S.-B. Hsu, Y. Wang and H. Zhou, Analysis of a mathematical model arising from Barnacle-Algae-Mussel interactions, SIAM J. Appl. Math., 2019, 79, 2032–2053. doi: 10.1137/18M1231365 CrossRef Google Scholar
[11]	J. Huisman and F. J. Weissing, Biodiversity of plankton by species oscillations and chaos, Nature, 1999, 402, 407–410. doi: 10.1038/46540 CrossRef Google Scholar
[12]	T. W. Hwang and Y. Kuang, Global analysis of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 2003, 281, 395–401. doi: 10.1016/S0022-247X(02)00395-5 CrossRef Google Scholar
[13]	S. Jain, B. Dubey, Q. Zheng, P. Mathur and V. Pandey, Impact of herd shape and delay in a two strain prey–predator model, Ecological Modell., 2026, 514, 111463. doi: 10.1016/j.ecolmodel.2025.111463 CrossRef Google Scholar
[14]	A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 2009, 71, 75–83. doi: 10.1007/s11538-008-9352-z CrossRef Google Scholar
[15]	A. Korobeinikov, Global stability of a population dynamics model with inhibition and negative feedback, Math. Med. Biol., 2013, 30, 65–72. doi: 10.1093/imammb/dqr027 CrossRef Google Scholar
[16]	A. Kowalewska, J. Krawczyk, M. Bodnar and M. Vela-Perez, Exploring the impact of healthcare capacity and time delays in SIR epidemic model, Math. Compu. Simulation, 2026, 241, 771–782. doi: 10.1016/j.matcom.2025.11.001 CrossRef Google Scholar
[17]	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, New York, Academic Press, 1993. Google Scholar
[18]	P. De Leenheer and H. L. Smith, Virus dynamics, a global analysis, SIAM J. Appl. Math., 2003, 63, 1313–1327. doi: 10.1137/S0036139902406905 CrossRef Google Scholar
[19]	M. Li and J. Muldowney, On Bendixson's criterion, J. Differential Equations, 1993, 106, 27–39. doi: 10.1006/jdeq.1993.1097 CrossRef Google Scholar
[20]	M. Li and J. Muldowney, Global stability for SEIR model in epidemiology, Math. Biosci., 1995, 125, 155–164. doi: 10.1016/0025-5564(95)92756-5 CrossRef Google Scholar
[21]	M. Li and J. Muldowney, A geometric approach to global stability problems, SIAM J. Math. Anal., 1996, 27, 1070–1083. doi: 10.1137/S0036141094266449 CrossRef Google Scholar
[22]	X. Liang and J. Jiang, The dynamical behavior of type-K competitive Kolmogorov systems and its applications to 3-dimensional type-K competitive Lotka-Volterra systems, Nonlinearity, 2003, 16, 785–801. doi: 10.1088/0951-7715/16/3/301 CrossRef Google Scholar
[23]	R. Martin, Logarithmic norms and projections applied to linear differential systems, J. Math. Anal. Appl., 1974, 45, 432–454. doi: 10.1016/0022-247X(74)90084-5 CrossRef Google Scholar
[24]	R. May, Biological populations with nonoverlapping generations, stable points, stable cycles and chaos, Science, 1974, 186, 645–647. doi: 10.1126/science.186.4164.645 CrossRef Google Scholar
[25]	R. May, Simple mathematical models with very complicated dynamics, Nature, 1976, 261, 459–467. doi: 10.1038/261459a0 CrossRef Google Scholar
[26]	J. Muldowney, Dichotomies and asymptotic behavior for linear differential systems, Trans. Amer. Math. Soc., 1984, 283, 465–484. doi: 10.1090/S0002-9947-1984-0737880-1 CrossRef Google Scholar
[27]	J. Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain J. Math., 1990, 20, 857–872. Google Scholar
[28]	K. Salehi and J. Alidousti, The dynamic behavior of time-delayed Leslie-Gower predator-prey model, Appl. Math. Compu., 2026, 508, 129622. Google Scholar
[29]	W. M. Schaffer and M. Kot, Nearly one-dimensional dynamics in an epidemic, J. Theor. Bio., 1985, 112, 403–427. doi: 10.1016/S0022-5193(85)80294-0 CrossRef Google Scholar
[30]	H. L. Smith, Periodic orbits of competitive and cooperative systems, J. Diff. Eqns., 1986, 65, 361–373. doi: 10.1016/0022-0396(86)90024-0 CrossRef Google Scholar
[31]	H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys and Monographs, 41, Amer. Math. Soc., Providence, Rhode Island, 1995. Google Scholar
[32]	H. L. Smith, Monotone dynamical systems: Reflections on new advances and applications, Discrete Contin. Dyn. Syst., 2017, 37, 485–504. doi: 10.3934/dcds.2017020 CrossRef Google Scholar
[33]	H. L. Smith and H. Thieme, Dynamical Systems and Population Persistence, GSM 118, Amer. Math. Soc., Providence, 2011. Google Scholar
[34]	D. Thommandru and S. Kundu, Uncovering ecological thresholds: Effects of external stress driven tipping point and delay in predator–prey system, Chaos, Solitons, Fractals, 2025, 200, 117173. doi: 10.1016/j.chaos.2025.117173 CrossRef Google Scholar