Dynamical Analysis in A Leslie-Gower Model Involving Allee Effect and Non-Linear Harvesting


Adin Lazuardy Firdiansyah(1*)

(1) State Islamic Institute of Madura, Indonesia
(*) Corresponding Author

Abstract


The fishing industry has fished the Atlantic cod population for a millennium. As fishing became greater, the Atlantic cod population declined. During this crisis, fishing declined drastically, but the species is still struggling to bounce back. This is due to the growing number of sharks and other North Atlantic invertebrates. Based on this phenomenon, a Leslie-Gower model involving additive Allee effects and non-linear harvesting is considered to describe population dynamics. Further research is done on the existence and stability of equilibrium points as well as the positivity, permanence, and boundedness of solutions. It was found that the permanence requirement was used as a condition to ensure the two populations lived together. Numerical simulations are performed in the last section to bolster the analysis's findings.

Keywords


Allee effect; Leslie-Gower model; non-linear harvesting

References


G. Morhua, “Atlantic Cod,” Oceana. Accessed: Dec. 06, 2023. [Online]. Available: https://oceana.org/marine-life/atlantic-cod/

A. L. Firdiansyah, “Effect of Fear in Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response Incorporating Prey Refuge,” IJCSAM, vol. 7, no. 2, p. 56, Aug. 2021, doi: 10.12962/j24775401.v7i2.8718.

Y. Cai, C. Zhao, W. Wang, and J. Wang, “Dynamics of a Leslie–Gower predator–prey model with additive Allee effect,” Applied Mathematical Modelling, vol. 39, no. 7, pp. 2092–2106, Apr. 2015, doi: 10.1016/j.apm.2014.09.038.

J. Bascompte, “Extinction Thresholds: Insights from Simple Models,” Annales Zoologici Fennici, vol. 40, pp. 99–114, 2003.

F. Courchamp, T. Clutton-Brock, and B. Grenfell, “Inverse density dependence and the Allee effect,” Trends in Ecology & Evolution, vol. 14, no. 10, pp. 405–410, Oct. 1999, doi: 10.1016/S0169-5347(99)01683-3.

L. Berec, E. Angulo, and F. Courchamp, “Multiple Allee Effects and Population Management,” Trends in Ecology & Evolution, vol. 22, no. 4, pp. 185–191, Apr. 2007, doi: 10.1016/j.tree.2006.12.002.

P. A. Stephens and W. J. Sutherland, “Consequences of the Allee effect for behaviour, ecology and conservation,” Trends in Ecology & Evolution, vol. 14, no. 10, pp. 401–405, Oct. 1999, doi: 10.1016/S0169-5347(99)01684-5.

H. S. Panigoro, E. Rahmi, N. Achmad, and S. L. Mahmud, “The Influence of Additive Allee Effect and Periodic Harvesting to the Dynamics of Leslie-Gower Predator-Prey Model,” Jambura J. Math, vol. 2, no. 2, pp. 87–96, Mar. 2020, doi: 10.34312/jjom.v2i2.4566.

P. Feng and Y. Kang, “Dynamics of a modified Leslie–Gower model with double Allee effects,” Nonlinear Dyn, vol. 80, no. 1–2, pp. 1051–1062, Apr. 2015, doi: 10.1007/s11071-015-1927-2.

P. J. Pal and T. Saha, “Qualitative analysis of a predator–prey system with double Allee effect in prey,” Chaos, Solitons & Fractals, vol. 73, pp. 36–63, Apr. 2015, doi: 10.1016/j.chaos.2014.12.007.

Y. Ye, H. Liu, Y. Wei, K. Zhang, M. Ma, and J. Ye, “Dynamic study of a predator-prey model with Allee effect and Holling type-I functional response,” Adv Differ Equ, vol. 2019, no. 1, p. 369, Dec. 2019, doi: 10.1186/s13662-019-2311-1.

P. Aguirre, E. Gonzalez-Olivares, and E. Saez, “Two limit cycles in a Leslie–Gower predator–prey model with additive Allee effect,” Nonlinear Analysis, vol. 10, pp. 1401–1416, 2009.

P. Aguirre, E. González-Olivares, and E. Sáez, “Three Limit Cycles in a Leslie–Gower Predator-Prey Model with Additive Allee Effect,” SIAM J. Appl. Math., vol. 69, no. 5, pp. 1244–1262, Jan. 2009, doi: 10.1137/070705210.

B. Ghosh and T. K. Kar, “Sustainable use of prey species in a prey–predator system: Jointly determined ecological thresholds and economic trade-offs,” Ecological Modelling, vol. 272, pp. 49–58, Jan. 2014, doi: 10.1016/j.ecolmodel.2013.09.013.

C. Lu, X. Liu, and Z. Li, “The dynamics and harvesting strategies of a predator-prey system with Allee effect on prey,” MATH, vol. 8, no. 12, pp. 28897–28925, 2023, doi: 10.3934/math.20231481.

X. Liu and Q. Huang, “Analysis of optimal harvesting of a predator-prey model with Holling type IV functional response,” Ecological Complexity, vol. 42, p. 100816, Mar. 2020, doi: 10.1016/j.ecocom.2020.100816.

S. G. Mortoja, P. Panja, and S. K. Mondal, “Stability Analysis of Plankton–Fish Dynamics with Cannibalism Effect and Proportionate Harvesting on Fish,” Mathematics, vol. 11, no. 13, p. 3011, Jul. 2023, doi: 10.3390/math11133011.

D. Wu, H. Zhao, and Y. Yuan, “Complex dynamics of a diffusive predator–prey model with strong Allee effect and threshold harvesting,” Journal of Mathematical Analysis and Applications, vol. 469, no. 2, pp. 982–1014, Jan. 2019, doi: 10.1016/j.jmaa.2018.09.047.

Z. Shang and Y. Qiao, “Multiple bifurcations in a predator–prey system of modified Holling and Leslie type with double Allee effect and nonlinear harvesting,” Mathematics and Computers in Simulation, vol. 205, pp. 745–764, Mar. 2023, doi: 10.1016/j.matcom.2022.10.028.

C. Wei and L. Chen, “Periodic solution and heteroclinic bifurcation in a predator–prey system with Allee effect and impulsive harvesting,” Nonlinear Dyn, vol. 76, no. 2, pp. 1109–1117, Apr. 2014, doi: 10.1007/s11071-013-1194-z.

P. H. Leslie and J. C. Gower, “The properties of a stochastic model for the predator-prey type of interaction between two species,” Biometrika, vol. 47, pp. 219–234, 1960.

M. K. Singh, B. S. Bhadauria, and B. K. Singh, “Qualitative Analysis of a Leslie-Gower Predator-Prey System with Nonlinear Harvesting in Predator,” International Journal of Engineering Mathematics, vol. 2016, pp. 1–15, Oct. 2016, doi: 10.1155/2016/2741891.

J. D. Murray, Mathematical Biology I: An Introduction, 3rd ed., vol. 17. Washington: Springer.

A. L. Firdiansyah and D. Rosikhoh, “A Fractional-Order Leslie-Gower Model with Fear and Allee Effect,” CAUCHY, vol. 7, no. 4, pp. 521–534, May 2023, doi: 10.18860/ca.v7i4.17336.

F. Chen, “On a nonlinear nonautonomous predator–prey model with diffusion and distributed delay,” Journal of Computational and Applied Mathematics, vol. 180, no. 1, pp. 33–49, Aug. 2005, doi: 10.1016/j.cam.2004.10.001.

P. J. Pal, S. Sarwardi, T. Saha, and P. K. Mandal, “MEAN SQUARE STABILITY IN A MODIFIED LESLIE-GOWER AND HOLLING-TYPE II PREDATOR-PREY MODEL,” Journal Applied Mathematics & Informatics, vol. 29, no. 3–4, pp. 781–802, 2011.




DOI: https://doi.org/10.15575/kubik.v9i2.38272

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