The Influence of Additive Allee Effect and Periodic Harvesting to the Dynamics of Leslie-Gower Predator-Prey Model

Hasan S. Panigoro, Emli Rahmi, Novianita Achmad, Sri Lestari Mahmud


In this paper, the influence of additive Allee effect in prey and periodic harvesting in predator to the dynamics of the Leslie-Gower predator-prey model is proposed. We first simplify the model to the non-dimensional system by scaling the variable and transform the model into an autonomous system. If the effect Allee is weak, we have at most two equilibrium points, else if the Allee effect is strong, at most four equilibrium points may exist. Furthermore, the behavior of the system around equilibrium points is investigated. In the end, we give numerical simulations to support theoretical results.


Leslie-Gower; Allee Effect; Periodic Harvesting; Non-autonomous

Full Text:

PDF [English]


A. Suryanto, I. Darti, H. S. Panigoro, and A. Kilicman, “A Fractional-Order Predator–Prey Model with Ratio-Dependent Functional Response and Linear Harvesting,” Mathematics, vol. 7, no. 11, p. 1100, Nov. 2019, DOI: 10.3390/math7111100.

M. Moustafa, M. H. Mohd, A. I. Ismail, and F. A. Abdullah, “Stage Structure and Refuge Effects in the Dynamical Analysis of a Fractional Order Rosenzweig-MacArthur Prey-Predator Model,” Prog. Fract. Differ. Appl., vol. 5, no. 1, pp. 49–64, 2019, DOI: 10.18576/pfda/050106.

J. Ghosh, B. Sahoo, and S. Poria, “Prey-Predator Dynamics with Prey Refuge Providing Additional Food to Predator,” Chaos, Solitons and Fractals, vol. 96, pp. 110–119, 2017, DOI: 10.1016/j.chaos.2017.01.010.

N. Hasan, R. Resmawan, and E. Rahmi, “Analisis Kestabilan Model Eko-Epidemiologi dengan Pemanenan Konstan pada Predator,” J. Mat. Stat. dan Komputasi, vol. 16, no. 2, p. 121, Dec. 2019, DOI: 10.20956/jmsk.v16i2.7317.

H. S. Panigoro, A. Suryanto, W. M. Kusumawinahyu, and I. Darti, “Dynamics of a Fractional-Order Predator-Prey Model with Infectious Diseases in Prey,” Commun. Biomath. Sci., vol. 2, no. 2, pp. 105–117, 2019, DOI: 10.5614/cbms.2019.2.2.4.

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, no. 3&4, pp. 219–234, 1960, DOI: 10.2307/2333294.

A. Suryanto, I. Darti, and S. Anam, “Stability Analysis of a Fractional Order Modified Leslie-Gower Model with Additive Allee Effect,” Int. J. Math. Math. Sci., vol. 2017, no. 11, pp. 1–9, 2017, DOI: 10.1155/2017/8273430.

P. C. Tobin, L. Berec, and A. M. Liebhold, “Exploiting Allee Effects for Managing Biological Invasions,” Ecol. Lett., vol. 14, no. 6, pp. 615–624, 2011, DOI: 10.1111/j.1461-0248.2011.01614.x.

E. Rahmi and H. S. Panigoro, “Pengaruh Pemanenan terhadap Model Verhulst dengan Efek Allee,” SEMIRATA MIPAnet 2017, no. 1, pp. 105–112, 2017.

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. Equations, vol. 2019, no. 1, pp. 1–15, 2019, DOI: 10.1186/s13662-019-2311-1.

Y. Cai, C. Zhao, W. Wang, and J. Wang, “Dynamics of a Leslie-Gower Predator-Prey model With Additive Allee Effect,” Appl. Math. Model., vol. 39, no. 7, pp. 2092–2106, 2015, DOI: 10.1016/j.apm.2014.09.038.

Y. Liu, Z. Liu, and R. Wang, “Bogdanov-Takens Bifurcation with Codimension Three of a Predator-Prey System Suffering the Additive Allee Effect,” Int. J. Biomath., vol. 10, no. 3, pp. 1–24, 2017, DOI: 10.1142/S1793524517500449.

X. Liu, G. Fan, and T. Zhang, “Evolutionary Dynamics of Single Species Model with Allee Effect,” Phys. A Stat. Mech. its Appl., vol. 526, 2019, DOI: 10.1016/j.physa.2019.04.010.

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, 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 and Fractals, vol. 73, pp. 36–63, 2015, DOI: 10.1016/j.chaos.2014.12.007.

B. Dennis, “Allee Effects: Population Growth, Critical Density, and the Chance of Extinction,” Nat. Resour. Model., vol. 3, no. 4, pp. 481–538, 1989, DOI: 10.1111/j.1939-7445.1989.tb00119.x.

P. A. Stephens and W. J. Sutherland, “Consequences of the Allee Effect for Behavior, Ecology and Conservation,” Trends Ecol. Evol., vol. 14, no. 10, pp. 401–405, 1999, DOI: 10.1016/S0169-5347(99)01684-5.

P. Aguirre, E. González-Olivares, and E. Sáez, “Two Limit Cycles in a Leslie-Gower Predator-Prey Model with Additive Allee Effect,” Nonlinear Anal. Real World Appl., vol. 10, no. 3, pp. 1401–1416, 2009, DOI: 10.1016/j.nonrwa.2008.01.022.

P. Aguirre, E. Gonzalez-Olivares, and E. Saez, “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, 2009, DOI: 10.1137/090750688.

H. S. Panigoro and E. Rahmi, “Modifikasi Sistem Predator-Prey: Dinamika Model Leslie-Gower dengan Daya Dukung yang Tumbuh Logistik,” in SEMIRATA MIPAnet, 2017, pp. 94–103.


Copyright (c) 2020 H.S.Panigoro, E.Rahmi, N.Achmad, S.L.Mahmud

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Jambura Journal of Mathematics has been indexed by

>>>More Indexing<<<

Creative Commons License

Jambura Journal of Mathematics (e-ISSN: 2656-1344) by Department of Mathematics Universitas Negeri Gorontalo is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. Powered by Public Knowledge Project OJS. 

Editorial Office

Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Negeri Gorontalo
Jl. Prof. Dr. Ing. B. J. Habibie, Moutong, Tilongkabila, Kabupaten Bone Bolango, Gorontalo, Indonesia