Global dynamics of an amensalism system with Michaelis-Menten type harvesting

Ming Zhao; Yudan Ma; Yunfei Du; Ming Zhao; Yudan Ma; Yunfei Du

doi:10.3934/era.2023027

Electronic Research Archive

2023, Volume 31, Issue 2: 549-574. doi: 10.3934/era.2023027

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Research article

Global dynamics of an amensalism system with Michaelis-Menten type harvesting

1.
School of Science, China University of Geosciences (Beijing), Beijing 100083, China
2.
School of Basic Education, Beijing Institute of Graphic Communication, Beijing 102600, China

Received: 17 September 2022 Revised: 23 October 2022 Accepted: 31 October 2022 Published: 14 November 2022

In this work, a new amensalism system with the nonlinear Michaelis-Menten type harvesting for the second species is studied. Firstly, we clarify topological types for all possible equilibria of the system. Then, the behaviors near infinity and the existence of closed orbits as well as saddle connections of the system are discussed via bifurcation analysis, and the global phase portraits of the model are also illustrated. Finally, for the sake of comparison, we further offer a new complete global dynamics of the model without harvesting. Numerical simulations show that the system with harvesting has far richer dynamics, like preserving the extinction of the first species or approaching the steady-state more slowly. Our research will provide useful information which may help us have a better understanding to the dynamic complexity of amensalism systems with harvesting effects.
- amensalism system,
- Michaelis-Menten type harvesting,
- bifurcation,
- global dynamics
Citation: Ming Zhao, Yudan Ma, Yunfei Du. Global dynamics of an amensalism system with Michaelis-Menten type harvesting[J]. Electronic Research Archive, 2023, 31(2): 549-574. doi: 10.3934/era.2023027

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Abstract

In this work, a new amensalism system with the nonlinear Michaelis-Menten type harvesting for the second species is studied. Firstly, we clarify topological types for all possible equilibria of the system. Then, the behaviors near infinity and the existence of closed orbits as well as saddle connections of the system are discussed via bifurcation analysis, and the global phase portraits of the model are also illustrated. Finally, for the sake of comparison, we further offer a new complete global dynamics of the model without harvesting. Numerical simulations show that the system with harvesting has far richer dynamics, like preserving the extinction of the first species or approaching the steady-state more slowly. Our research will provide useful information which may help us have a better understanding to the dynamic complexity of amensalism systems with harvesting effects.

References

[1]	V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 118 (1926), 558–560. https://doi.org/10.1038/118558a0 doi: 10.1038/118558a0
[2]	A. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925.
[3]	S. Gakkhar, R. K. Naji, Order and chaos in a food web consisting of a predator and two independent preys, Commun. Nonlinear Sci. Numer. Simul., 10 (2005), 105–120. https://doi.org/10.1016/S1007-5704(03)00120-5 doi: 10.1016/S1007-5704(03)00120-5
[4]	M. De la Sen, S. Alonso-Quesada, A control theory point of view on Beverton-Holt equation in population dynamics and some of its generalizations, Appl. Math. Comput., 199 (2008), 464–481. https://doi.org/10.1016/j.amc.2007.10.021 doi: 10.1016/j.amc.2007.10.021
[5]	G. Sun, Qualitative analysis on two populations amensalism model, J. Jiamusi Univ. (Natl. Sci. Ed.), 21 (2003), 283–286. https://doi.org/10.1080/17513750802560346 doi: 10.1080/17513750802560346
[6]	X. Guan, F. Chen, 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., 48 (2019), 71–93. https://doi.org/10.1016/j.nonrwa.2019.01.002 doi: 10.1016/j.nonrwa.2019.01.002
[7]	B. Chen, Dynamic behaviors of a non-selective harvesting Lotka-Volterra amensalism model incorporating partial closure for the populations, Adv. Differ. Equations, 2018 (2018), 1–14. https://doi.org/10.1186/s13662-018-1555-5 doi: 10.1186/s13662-018-1555-5
[8]	F. Chen, M. Zhang, R. Han, Existence of positive periodic solution of a discrete Lotka-Volterra amensalism model, J. Shengyang Univ. (Natl. Sci.), 27 (2015), 251–254. https://doi.org/10.1080/17513750802560346 doi: 10.1080/17513750802560346
[9]	Q. Lin, X. Zhou, On the existence of positive periodic solution of a amensalism model with Holling Ⅱ functional response, Commun. Math. Biol. Neurosci., 2017 (2017), 1–12.
[10]	Y. Liu, L. Zhao, X. Huang, H. Deng, Bioeconomic harvesting of a prey-predator fishery, Adv. Differ. Equations, 2018 (2018), 1–19. https://doi.org/10.1186/s13662-018-1752-2 doi: 10.1186/s13662-018-1752-2
[11]	D. Luo, Q. Wang, Global dynamics of a Beddington-DeAngelis amensalism system with weak Allee effect on the first species, J. Appl. Math. Comput., 68 (2022), 655–680. https://doi.org/10.1007/s12190-021-01533-w doi: 10.1007/s12190-021-01533-w
[12]	D. Luo, Q. Wang, Global dynamics of a Holling-Ⅱ amensalism system with nonlinear growth rate and Allee effect on the first species, Int. J. Bifurcation Chaos, 31 (2021), 2150050. https://doi.org/10.1142/S0218127421500504 doi: 10.1142/S0218127421500504
[13]	J. Zhang, Bifurcated periodic solutions in an amensalism system with strong generic delay kernel, Math. Methods Appl. Sci., 36 (2013), 113–124. https://doi.org/10.1002/mma.2575 doi: 10.1002/mma.2575
[14]	Z. Zhang, Stability and bifurcation analysis for a amensalism system with delays, Math. Numer. Sin., 30 (2008), 213–224. https://doi.org/10.3724/SP.J.1001.2008.01274 doi: 10.3724/SP.J.1001.2008.01274
[15]	Z. Wei, Y. Xia, T. Zhang, Stability and bifurcation analysis of an amensalism model with weak Allee effect, Qual. Theory Dyn. Syst., 19 (2020), 1–15. https://doi.org/10.1007/s12346-020-00341-0 doi: 10.1007/s12346-020-00341-0
[16]	J. Chen, J. Huang, S. Ruan, J. Wang, Bifurcations of invariant tori in predator-prey models with seasonal prey harvesting, SIAM J. Appl. Math., 73 (2013), 1876–1905. https://doi.org/10.1137/120895858 doi: 10.1137/120895858
[17]	D. Xiao, L. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737–753. https://doi.org/10.1137/s0036139903428719 doi: 10.1137/s0036139903428719
[18]	D. Xiao, W. Li, M. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. Math. Anal. Appl., 324 (2006), 14–29. https://doi.org/10.1016/j.jmaa.2005.11.048 doi: 10.1016/j.jmaa.2005.11.048
[19]	T. Kar, K. Chaudhurii, On non-selective harvesting of two competing fish species in the presence of toxicity, Ecol. Modell., 161 (2003), 125–137. https://doi.org/10.1016/S0304-3800(02)00323-X doi: 10.1016/S0304-3800(02)00323-X
[20]	T. Das, R. N. Mukherjee, K. S. Chaudhuri, Bioeconomic harvesting of a prey-predator fishery, J. Biol. Dyn., 3 (2009), 447–462. https://doi.org/10.1080/17513750802560346 doi: 10.1080/17513750802560346
[21]	K. Chakraborty, S. Jana, T. Kar, Bioeconomic harvesting of a prey-predator fishery, Appl. Math. Comput., 218 (2012), 9271–9290. https://doi.org/10.1016/j.amc.2012.03.005 doi: 10.1016/j.amc.2012.03.005
[22]	B. Leard, J. Rebaza, Analysis of predator-prey models with continuous threshold harvesting, Appl. Math. Comput., 217 (2011), 5265–5278. https://doi.org/10.1016/j.amc.2010.11.050 doi: 10.1016/j.amc.2010.11.050
[23]	K. Chakraborty, S. Das, T. Kar, On non-selective harvesting of a multispecies fishery incorporating partial closure for the populations, Appl. Comput. Math., 221 (2013), 581–597. https://doi.org/10.1016/j.amc.2013.06.065 doi: 10.1016/j.amc.2013.06.065
[24]	L. Chen, F. Chen, Global analysis of a harvested predator-prey model incorporating a constant prey refuge, Int. J. Biomath., 3 (2010), 205–223. https://doi.org/10.1142/S1793524510000957 doi: 10.1142/S1793524510000957
[25]	R. May, J. Beddington, C. Clark, S. Holt, R. Laws, Management of multispecies fisheries, Science, 205 (1979), 267–277. https://doi.org/10.1038/277267a0 doi: 10.1038/277267a0
[26]	C. Clark, Aggregation and fishery dynamics: a theoretical study of schooling and the purse seine tuna fisheries, Fish. Bull., 77 (1979), 317–337.
[27]	T. Das, R. Mukherjee, K. Chaudhuri, Bioeconomic harvesting of a prey-predator fishery, J. Biol. Dyn., 3 (2009), 447–462. https://doi.org/10.1080/17513750802560346 doi: 10.1080/17513750802560346
[28]	R. Gupta, P. Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278–295. https://doi.org/10.1016/j.jmaa.2012.08.057 doi: 10.1016/j.jmaa.2012.08.057
[29]	S. Hsu, T. Hwang, Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol., 42 (2001), 489–506. https://doi.org/10.1007/S002850100079 doi: 10.1007/S002850100079
[30]	B. Li, Y. Kuang, Heteroclinic bifurcation in the Michaelis-Menten-type ratio-dependent predator-prey system, SIAM J. Appl. Math., 67 (2007), 1453–1464. https://doi.org/10.1137/060662460 doi: 10.1137/060662460
[31]	R. Yuan, W. Jiang, Y. Wang, Saddle-node-Hopf bifurcation in a modified Leslie-Gower predator-prey model with time-delay and prey harvesting, J. Math. Anal. Appl., 422 (2015), 1072–1090. https://doi.org/10.1016/j.jmaa.2014.09.037 doi: 10.1016/j.jmaa.2014.09.037
[32]	X. Zhang, H. Zhao, Stability and bifurcation of a reaction-diffusion predator-prey model with non-local delay and Michaelis-Menten type prey-harvesting, Int. J. Comput. Math., 93 (2016), 1447–1469. https://doi.org/10.1080/00207160.2015.1056169 doi: 10.1080/00207160.2015.1056169
[33]	D. Hu, H. Cao, Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting, Nonlinear Anal. Real World Appl., 33 (2017), 58–82. https://doi.org/10.1016/j.nonrwa.2016.05.010 doi: 10.1016/j.nonrwa.2016.05.010
[34]	W. Liu, Y. Jiang, Bifurcation of a delayed Gause predator-prey model with Michaelis-Menten type harvesting, J. Theory Biol., 438 (2018), 116–132. https://doi.org/10.1016/j.jtbi.2017.11.007 doi: 10.1016/j.jtbi.2017.11.007
[35]	Z. Zhang, T. Ding, W. Huang, Z. Dong, Qualitative Theory of Differential Equation, Science Press, Beijing, 1997.
[36]	L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 2001. https://doi.org/10.1007/978-1-4613-0003-8

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