Abstract
In ecology, commensalism and amensalism models are two kinds of important and interesting models. They have attracted much attention of ecologists and mathematicians in recent years. In this paper, we consider a two-species commensalism system with a discrete delay and density dependent birth rates.
First, we investigate the characteristic equation of the proposed system and study the distribution of its roots. We obtain that, when the delay is sufficiently small, the positive equilibrium is locally asymptotically stable, when increases to a critical value, the positive equilibrium loses its stability and a Hopf bifurcation occurs, as continues to increase, a family of periodic solutions bifurcate from the positive equilibrium. Then, by using the normal form theory and the center manifold theorem, we derive the precise formulae to determine the Hopf bifurcation direction and the stability of the bifurcating periodic solutions.
Numerical simulation results are included to support our theoretical analysis. We plot the trajectory graphs on , plane respectively. We also plot phase graphs to illustrate the change of stability of the positive equilibrium and arise of periodic solution. In order to fully validate the occurrence of Hopf bifurcation, we use the numerical continuation package DDE-Biftool to generate bifurcation diagrams and accurately track stability changes of the positive equilibrium and periodic solution with respect to the delay parameter .
The commensalism model we propose considers both density dependent birth rate and time delay, it is of great practical and theoretical significance. The theoretical and numerical analysis that we do on the proposed system can make a supplement to the literature on the dynamics of delay commensalism systems.
2020 MSC: 34K18, 34K20, 92D25
References
Guan X, Chen F. Dynamical analysis of a two species amensalism model with Beddington-DeAngelis functional response and Allee effect on the second species. Nonlinear Anal Real World Appl. 2019; 48: 71-93. https://doi.org/10.1016/j.nonrwa.2019.01.002
Sun G. Qualitative analysis on two populations amensalism model. J Jiamusi Univ. (Natl Sci Ed) 2003; 21(3): 283-6.
Sun G, Wei W. The qualitative analysis of commensal symbiosis model of two populations. Math Theory Appl. 2003; 23(3): 65-68.
Chen B. Dynamic behaviors of a non-selective harvesting Lotka-Volterra amensalism model incorporating partial closure for the populations. Adv Difference Equ. 2018; 2018: 1-14. https://doi.org/10.1186/s13662-018-1555-5
Chen F, Xue Y, Lin Q, Xie X. Dynamic behaviors of a Lotka-Volterra commensal symbiosis model with density dependent birth rate. Adv Difference Equ. 2018; Paper No. 296: 1-14. https://doi.org/10.1186/s13662-018-1758-9
Wu R. A two species amensalism model with non-monotonic functional response. Commun Math Biol Neurosci. 2016; Paper No. 19: 1-10.
Wu R, Li L, Zhou X. A commensal symbiosis model with Holling type functional response. Math Comput Sci. 2016; 16: 364-71. https://doi.org/10.22436/jmcs.016.03.06
Wu R, Zhao L, Lin Q. Stability analysis of a two species amensalism model with Holling II functional response and a cover for the first species. J Nonlinear Funct Anal. 2016; Paper No. 46: 1-15.
Xie X, Chen F, He M. Dynamic behaviors of two species amensalism model with a cover for the first species. Math Comput Sci. 2016; 16(3): 395-401. https://doi.org/10.22436/jmcs.016.03.09
Luo D, Wang Q. Global dynamics of a Beddington-DeAngelis amensalism system with weak Allee effect on the first species. Appl Math Comput. 2021; 408(3): Paper No. 126368 (19 pages). https://doi.org/10.1016/j.amc.2021.126368
Luo D, Wang Q. Global dynamics of a Holling-II amensalism system with nonlinear growth rate and Allee effect on the first species. Int J Bifur Chaos Appl Sci Eng. 2021; 31(3): Paper No. 2150050 (26 pages). https://doi.org/10.1142/S0218127421500504
Wei Z, Xia Y, Zhang T. Stability and bifurcation analysis of a commensal model with additive Allee effect and nonlinear growth rate. Int J Bifur Chaos Appl Sci Eng. 2021; 31(13): Paper No. 2150204 (17 pages). https://doi.org/10.1142/S0218127421502047
Chen F, Zhang M, Han R. Existence of positive periodic solution of a discrete Lotka-Volterra amensalism model. J Shenyang Univ (Natl Sci) 2015; 27(3): 251-4.
Lin Q, Zhou X. On the existence of positive periodic solution of a amensalism model with Holling II functional response. Commun Math Biol Neurosci. 2017; 2017: 1-12. https://doi.org/10.28919/cmbn/2809
Li T, Wang Q. Stability and Hopf bifurcation analysis for a two-species commensalism system with delay. Qual Theory Dyn Syst. 2021; 20(3): 1-20. https://doi.org/10.1007/s12346-021-00524-3
Li T, Wang Q. Bifurcation analysis for two-species commensalism (amensalism) systems with distributed delays. Int J Bifur Chaos Appl Sci Eng. 2022; 32(9): 1-16. https://doi.org/10.1142/S0218127422501334
Qu M. Dynamical analysis of a Beddington-DeAngelis commensalism system with two time delays. J Appl Math Comput. 2023; 69(6): 4111-34. https://doi.org/10.1007/s12190-023-01913-4
Zhang J. Bifurcated periodic solutions in an amensalism system with strong generic delay kernel. Math Methods Appl Sci. 2013; 36(1): 113-24. https://doi.org/10.1002/mma.2575
Zhang Z. Stability and bifurcation analysis for an amensalism system with delays. Math Numer Sinica. 2008; 30(2): 213-24.
Hu X, Li H, Chen F. Bifurcation analysis of a discrete amensalism model. Int J Bifur Chaos Appl Sci Eng. 2024; 34(2): 1-21. https://doi.org/10.1142/S0218127424500202
Li Q, Chen F, Chen L, Li Z. Dynamical analysis of a discrete amensalism system with the Beddington-DeAngelis functional response and fear effect. J Appl Anal Comput. 2025; 15(4): 2089-123. https://doi.org/10.11948/20240399
Xue Y, Chen F, Xie X, Han R. Dynamic behaviors of a discrete commensalism system. Ann Appl Math. 2015; 31(4): 452-61. https://doi.org/10.1155/2015/295483
Li Q, Kashyap A, Zhu Q, Chen F. Dynamical behaviours of discrete amensalism system with fear effects on first species. Math Biosci Eng. 2024; 21(1): 832-60. https://doi.org/10.3934/mbe.2024035
Zhu Q, Chen F, Li Z, Chen L. Global dynamics of two-species amensalism model with Beddington-DeAngelis functional response and fear effect. Int J Bifur Chaos Appl Sci Eng. 2024; 34(6): 1-26. https://doi.org/10.1142/S0218127424500755
He X, Zhu Z, Chen J, Chen F. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect. Open Math. 2022; 20(1): 646-65. https://doi.org/10.1515/math-2022-0055
Zhou Q, Chen Y, Chen S, Chen F. Dynamic analysis of a discrete amensalism model with Allee effect. J Appl Anal Comput. 2023; 13(5): 2416-32. https://doi.org/10.11948/20220332
Zhao M, Du Y. Stability and bifurcation analysis of an amensalism system with Allee effect. Adv. Difference Equ. 2020; 2020: 1-13. https://doi.org/10.1186/s13662-020-02804-9
Chen F, Chen Y, Li Z, Chen L. Note on the persistence and stability property of a commensalism model with Michaelis-Menten harvesting and Holling type II commensalistic benefit. Appl Math Lett. 2022; 134: 1-8. https://doi.org/10.1016/j.aml.2022.108381
Liu H, Yu H, Dai C, Ma Z, Wang Q, Zhao M. Dynamical analysis of an aquatic amensalism model with non-selective harvesting and Allee effect. Math Biosci Eng. 2021; 18(6): 8857-82. https://doi.org/10.3934/mbe.2021437
Liu X, Yue Q. Stability property of the boundary equilibria of a symbiotic model of commensalism and parasitism with harvesting in commensal populations. AIMS Math. 2022; 7(10): 18793-808. https://doi.org/10.3934/math.20221034
Singh M, Poonam. Dynamical study and optimal harvesting of a two-species amensalism model incorporating nonlinear harvesting. Appl Math. 2023; 18(1): 1-16.
Zhao M, Ma Y, Du Y. Global dynamics of an amensalism system with Michaelis-Menten type harvesting. Electron Res Arch. 2023; 31(2): 549-74. https://doi.org/10.3934/era.2023027
Osuna O, Villavicencio-Pulido G. A seasonal commensalism model with a weak Allee effect to describe climate-mediated shifts. Sel Mat. 2024; 11(2): 212-21. https://doi.org/10.17268/sel.mat.2024.02.01
Patra R, Maitra S. Dynamics of stability, bifurcation and control for a commensal symbiosis model. Int J Dyn Control. 2024; 12(7): 2369-84. https://doi.org/10.1007/s40435-023-01367-3
Zhao K. Global asymptotic stability for a classical controlled nonlinear periodic commensalism AG-ecosystem with distributed lags on time scales. Filomat. 2023; 37(29): 9899-911. https://doi.org/10.2298/FIL2329899Z
Wei J, Wang H, Jiang W. Theory and Application of Bifurcation Theory for Delay Differential Equations. Beijing: Science Press; 2012 (in Chinese).
Cunningham W. A nonlinear differential-difference equation of growth. Proc Nat Acad Sci U S A. 1954; 40: 708-13. https://doi.org/10.1073/pnas.40.8.708
Hutchinson G. Circular causal systems in ecology. Ann New York Acad Sci. 1948; 50(4): 221-46. https://doi.org/10.1111/j.1749-6632.1948.tb39854.x
Hale J. Theory of Functional Differential Equations. New York: Springer; 1977. https://doi.org/10.1007/978-1-4612-9892-2
Kuang Y. Delay Differential Equations with Applications in Population Dynamics. Boston, MA: Academic Press; 1993.
Ruan S, Wei J. On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn Contin Discrete Impuls Syst Ser A Math Anal. 2003; 10: 863-74.
Ruan S. Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays. Quart Appl Math. 2001; 59(1): 159-73. https://doi.org/10.1090/qam/1811101
Hassard B, Kazarinoff N, Wan Y. Theory and Applications of Hopf Bifurcation. Cambridge: Cambridge University Press; 1981.
Song Y, Han M, Wei J. Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays. Physica D. 2005; 200(3): 185-204. https://doi.org/10.1016/j.physd.2004.10.010
Wu J. Theory and Applications of Partial Functional Differential Equations. New York: Springer; 1991.
Yi F. Turing instability of the periodic solutions for reaction-diffusion systems with cross-diffusion and the patch model with cross-diffusion-like coupling. J Differential Equations. 2021; 281: 379-410. https://doi.org/10.1016/j.jde.2021.02.006
Kuznetsov Y. Elements of Applied Bifurcation Theory (3rd Ed). New York: Springer-Verlag; 2004. https://doi.org/10.1007/978-1-4757-3978-7
Engelborghs K, Luzyanina T, Roose D. Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Trans Math Software. 2002; 28(1): 1-21. https://doi.org/10.1145/513001.513002
Mohd M, Abdul Rahman N, Abd Hamid N, Yatim Y. Dynamical Systems, Bifurcation Analysis and Applications. Singapore: Springer; 2019. https://doi.org/10.1007/978-981-32-9832-3
Sieber J, Engelborghs K, Luzyanina T, Samaey G, Roose D. DDE-BIFTOOL v.3.1.1 Manual-Bifurcation analysis of delay differential equations. arXiv:1406.7144. http://arxiv.org/abs/1406.7144

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Copyright (c) 2025 Tianyang Li, Qiru Wang