| Citation: |
Zuzhi Cai, Fengde Chen, Lijuan Chen, Zhong Li. DYNAMICAL ANALYSIS OF A DISCRETE COMMENSALISM MODEL WITH MICHAELIS–MENTEN TYPE HARVESTING[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1531-1568. doi: 10.11948/20260123
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DYNAMICAL ANALYSIS OF A DISCRETE COMMENSALISM MODEL WITH MICHAELIS–MENTEN TYPE HARVESTING
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Abstract
In this work, we formulate and analyze a discrete-time commensalism model in which the first species is subject to Michaelis–Menten type harvesting. We first investigate the existence, stability, and global attractivity of all feasible fixed points. Our results demonstrate that when the two species coexist stably, increasing the harvesting intensity leads to the extinction of the harvested species. We then establish the conditions under which various bifurcations occur, including transcritical, pitchfork, and fold bifurcations, with particular emphasis on the period-doubling route to chaos. In addition, a comprehensive analysis of the global dynamics is conducted. Numerical simulations highlight the critical influence of the commensalism strength: Moderate commensalism ensures stable coexistence of species and increases population density, whereas excessive commensalism destabilizes the system, triggering a period-doubling bifurcation cascade to chaos. Furthermore, the intrinsic stability of the host species is shown to profoundly modulate the dynamical complexity of the commensal species. Under certain conditions, instability of the host species can advance the onset of chaos in the commensal species.
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Keywords:
- Michaelis–Menten type harvesting /
- commensalism /
- global attractivity /
- bifurcation /
- chaos
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References
[1] R. Ahmed, M. Rafaqat, I. Siddique and M. A. Arefin, Complex dynamics and chaos control of a discrete-time predator–prey model, Discrete Dynamics in Nature and Society, 2023, 2023(1), 8873611. [2] M. Alsubhi, R. Ahmed, I. Alraddadi, F. Alsharif and M. Imran, Stability and bifurcation analysis of a discrete-time plant-herbivore model with harvesting effect, AIMS Mathematics, 2024, 9(8), 20014–20042. doi: 10.3934/math.2024976 [3] B. Chen, The influence of commensalism on a Lotka–Volterra commensal symbiosis model with Michaelis–Menten type harvesting, Advances in Difference Equations, 2019, 2019(1), 43. doi: 10.1186/s13662-019-1989-4 [4] F. Chen, Y. Chen, Z. Li and L. Chen, Note on the persistence and stability property of a commensalism model with Michaelis–Menten harvesting and Holling type Ⅱ commensalistic benefit, Applied Mathematics Letters, 2022, 134, 108381. doi: 10.1016/j.aml.2022.108381 [5] J. Chen, X. He and F. Chen, The influence of fear effect to a discrete-time predator–prey system with predator has other food resource, Mathematics, 2021, 9(8), 865. doi: 10.3390/math9080865 [6] Y. Chong, A. J. Kashyap, S. Chen and F. Chen, Dynamics analysis of a discrete-time commensalism model with additive Allee for the host species, Axioms, 2023, 12(11), 1031. doi: 10.3390/axioms12111031 [7] C. W. Clark and M. Mangel, Aggregation and fishery dynamics: A theoretical study of schooling and the purse seine tuna fisheries, Fishery Bulletin, 1979, 77(2), 317–337. [8] J. M. Cushing, R. F. Costantino, B. Dennis, R. A. Desharnais and S. M. Henson, Chaos in Ecology: Experimental Nonlinear Dynamics, Academic Press, San Diego, 2003. [9] Q. Din, Complexity and chaos control in a discrete-time prey–predator model, Communications in Nonlinear Science and Numerical Simulation, 2017, 49, 113–134. doi: 10.1016/j.cnsns.2017.01.025 [10] A. Ditta, P. A. Naik, R. Ahmed and Z. Huang, Exploring periodic behavior and dynamical analysis in a harvested discrete-time commensalism system, International Journal of Dynamics and Control, 2025, 13(2), 63. doi: 10.1007/s40435-024-01551-z [11] K. Fang and X. Chen, Modeling a Lotka–Volterra commensalism system with prey subject to both Allee effects and linear harvesting, IAENG International Journal of Applied Mathematics, 2025, 55(10), 3367–3375. [12] X. Feng, Y. Liu, S. Ruan and J. Yu, Periodic dynamics of a single species model with seasonal Michaelis–Menten type harvesting, Journal of Differential Equations, 2023, 354, 237–263. doi: 10.1016/j.jde.2023.01.014 [13] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. [14] T. He and Y. Zhang, Stability and bifurcation analysis of a predator–prey model with Allee effect and predator harvesting, International Journal of Bifurcation and Chaos, 2025, 35(01), 2550007. doi: 10.1142/S0218127425500075 [15] X. He, Z. Zhu, J. Chen and F. Chen, Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect, Open Mathematics, 2022, 20(1), 646–665. doi: 10.1515/math-2022-0055 [16] M. Imran, M. B. Almatrafi and R. Ahmed, Stability and bifurcation analysis of a discrete predator–prey system of Ricker type with harvesting effect, Communications in Mathematical Biology and Neuroscience, 2024, 2024, 11. [17] S. Jawad, Study the dynamics of commensalism interaction with Michaels–Menten type prey harvesting, Al-Nahrain Journal of Science, 2022, 25(1), 45–50. [18] S. Jawad and S. Kazim Hassan, Bifurcation analysis of commensalism interaction and harvesting on food chain model, Brazilian Journal of Biometrics, 2023, 41(3), 218–233. doi: 10.28951/bjb.v41i3.609 [19] H. Jiang and T. D. Rogers, The discrete dynamics of symmetric competition in the plane, Journal of Mathematical Biology, 1987, 25(6), 573–596. doi: 10.1007/BF00275495 [20] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed., Springer-Verlag, New York, 2004. [21] Q. Li, F. Chen, L. Chen and Z. Li, Dynamical analysis of a discrete amensalism system with Michaelis–Menten type harvesting for the second species, Qualitative Theory of Dynamical Systems, 2024, 23(Suppl. 1), 279. [22] S. Linge and H. P. Langtangen, Programming for Computations–Python: A Gentle Introduction to Numerical Simulations with Python 3.6, 2nd ed., Springer, Cham, 2020. [23] Y. Liu, X. Feng, S. Ruan and J. Yu, Periodic dynamics of a single species model with seasonal Michaelis–Menten type harvesting, Ⅱ: Existence of two periodic solutions, Journal of Differential Equations, 2024, 388, 253–285. doi: 10.1016/j.jde.2024.01.004 [24] M. G. Mortuja, M. K. Chaube and S. Kumar, Dynamic analysis of a predator–prey system with nonlinear prey harvesting and square root functional response, Chaos, Solitons & Fractals, 2021, 148, 111071. [25] A. K. Pal, Controlling chaotic dynamics of a delayed Hassell–Varley type predator–prey model with non-linear harvesting efforts in prey by using imprecise biological parameters, Results in Control and Optimization, 2024, 14, 100361. doi: 10.1016/j.rico.2023.100361 [26] R. R. Patra and S. Maitra, Dynamics of stability, bifurcation and control for a commensal symbiosis model, International Journal of Dynamics and Control, 2024, 12(7), 2369–2384. doi: 10.1007/s40435-023-01367-3 [27] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd ed., CRC Press, Boca Raton, 1998. [28] K. Sarkar and B. Mondal, Bifurcation and dynamic analysis of prey–predator model with combined nonlinear harvesting, International Journal of Biomathematics, 2024, 18, 2450010. [29] V. S. Sharma, A. Singh and P. Malik, Bifurcation patterns in a discrete predator–prey model incorporating ratio-dependent functional response and prey harvesting, Qualitative Theory of Dynamical Systems, 2024, 23(2), 74. doi: 10.1007/s12346-023-00929-2 [30] N. Song, J. Li and S. Zhu, Dynamics of a discrete one-predator two-prey system with Michaelis–Menten-type prey harvesting and prey refuge, Mathematical Methods in the Applied Sciences, 2024, 47(14), 11565–11601. doi: 10.1002/mma.10144 [31] G. C. Sun and H. Sun, Analysis on symbiosis model of two populations, Journal of Weinan Normal University, 2013, 28(9), 6–8. [32] A. Tassaddiq, R. Ahmed and A. Ditta, Exploring stability and bifurcation in a discretized commensalism model, Nonlinear Dynamics, 2025, 113(20), 28463–28475. doi: 10.1007/s11071-025-11530-4 [33] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd ed., Springer-Verlag, New York, 2003. [34] L. Xu, Y. Xue, Q. Lin and C. Lei, Global attractivity of symbiotic model of commensalism in four populations with Michaelis–Menten type harvesting in the first commensal populations, Axioms, 2022, 11(7), 337. doi: 10.3390/axioms11070337 [35] W. Yao and X. Li, Bifurcation difference induced by different discrete methods in a discrete predator–prey model, Journal of Nonlinear Modeling and Analysis, 2022, 4(1), 64–79. [36] Q. Zhou, F. Chen and S. Lin, Complex dynamics analysis of a discrete amensalism system with a cover for the first species, Axioms, 2022, 11(8), 365. doi: 10.3390/axioms11080365 [37] Q. Zhou, S. Lin, F. Chen and R. Wu, Positive periodic solution of a discrete Lotka–Volterra commensal symbiosis model with Michaelis–Menten type harvesting, WSEAS Transactions on Mathematics, 2022, 21, 515–523. doi: 10.37394/23206.2022.21.57 -
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Zuzhi Cai, Fengde Chen, Lijuan Chen, Zhong Li. DYNAMICAL ANALYSIS OF A DISCRETE COMMENSALISM MODEL WITH MICHAELIS–MENTEN TYPE HARVESTING[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1531-1568. doi: 10.11948/20260123
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Figure 1.
(a) Global attractivity of
$ E_{22}\left(x_{22}^*, \frac{\delta}{\beta}\right) $ $ E_1\left(0, \frac{\delta}{\beta}\right) $ -
Figure 2.
Bifurcation diagrams in the
$ \alpha $ $ x $ $ \beta = 1 $ $ \gamma = 2 $ $ \delta = 1 $ $ \beta = 1 $ $ \gamma = 1 $ $ \delta = 1 $ -
Figure 3.
Bifurcation diagrams at
$ E_1 $ $ \alpha $ $ x $ $ \beta = 1 $ $ \gamma = 1 $ $ \delta = 1 $ $ \beta = 1 $ $ \gamma = 2 $ $ \delta = 1 $ -
Figure 4.
Period-doubling bifurcation at
$ E_{22} $ $ \beta $ $ x $ $ \alpha = 0.8 $ $ \gamma = 0.5 $ $ \delta = 1.6 $ $ (3.47, 2.68) $ -
Figure 5.
Period-doubling bifurcation at
$ E_{22} $ $ \delta $ $ x $ $ \alpha = 0.8 $ $ \beta = 1.3 $ $ \gamma = 0.5 $ $ (0.5, 0.5) $ -
Figure 6.
Global attractivity of
$ E_1(0, \frac{\delta}{\beta}) $ -
Figure 7.
Global attractivity of
$ E_{22}(x_{22}^*, \frac{\delta}{\beta}) $ -
Figure 8.
Time series of species
$ x $ -
Figure 9.
Bifurcation diagram and maximum Lyapunov exponent as functions of
$ \beta $ $ (\alpha, \delta, \gamma) = (1.8, 0.6, 1.3) $ $ \beta $ $ x $ -
Figure 10.
Stability of
$ E_{22}(x_{22}^*, \frac{\delta}{\beta}) $ -
Figure 11.
Global attractivity of
$ E_{22}(x_{22}^*, \frac{\delta}{\beta}) $ -
Figure 12.
Time series of species
$ x $ -
Figure 13.
Bifurcation diagram and maximum Lyapunov exponent as functions of
$ \beta $ $ (\alpha, \delta, \gamma) = (1, 0.6, 1.3) $ -
Figure 14.
Bifurcation diagram and maximum Lyapunov exponent as functions of
$ \beta $ $ y $ $ (\alpha, \gamma, \delta) = (1.8, 1.3, 2.2) $ -
Figure 15.
Bifurcation diagrams of species
$ x $ $ \beta $ $ x $ $ \delta $ $ (\alpha, \gamma) = (0.8, 1) $