A Fractional-Order Leslie-Gower Model with Fear and Allee Effect

Adin Lazuardy Firdiansyah, Dewi Rosikhoh

Abstract


In this manuscript, we investigate the dynamics behavior of a fractional-order Leslie-Gower model by considering fear effect in the prey and Allee effect in predators. Firstly, all possible equilibrium points are analyzed by identifying the conditions of their existence and local stability. Here, we find four equilibrium points, where two local stable points and two unstable points. Furthermore, we also investigate the stability changing caused by Hopf bifurcation when the order of derivative changes. Finally, we perform several simulations to support our analysis results.        


Keywords


Hopf Bifurcation; Leslie-Gower Model; Local Stability

Full Text:

PDF

References


H. S. Panigoro, R. Resmawan, A. T. R. Sidik, N. Walangadi, A. Ismail, and C. Husuna, “A Fractional-Order Predator-Prey Model with Age Structure on Predator and Nonlinear Harvesting on Prey,” Jambura J. Math., vol. 4, no. 2, pp. 355–366, 2022, doi: 10.34312/jjom.v4i2.15220.

S. Pal, S. Majhi, S. Mandal, and N. Pal, “Role of fear in a predator-prey model with beddington-deangelis functional response,” Zeitschrift fur Naturforsch. - Sect. A J. Phys. Sci., vol. 74, no. 7, pp. 591–595, 2019, doi: 10.1515/zna-2018-0449.

H. Zhang, Y. Cai, S. Fu, and W. Wang, “Impact of the fear effect in a prey-predator model incorporating a prey refuge,” Appl. Math. Comput., vol. 356, no. 61373005, pp. 328–337, 2019, doi: 10.1016/j.amc.2019.03.034.

A. L. Firdiansyah and Nurhidayati, “Dynamics in two competing predators-one prey system with two types of Holling and fear effect,” Jambura J. Biomath., vol. 2, no. 2, pp. 58–67, 2021, doi: 10.34312/jjbm.v2i2.11264.

X. Wang, Y. Tan, Y. Cai, and W. Wang, “Impact of the fear effect on the stability and bifurcation of a leslie-gower predator-prey model,” Int. J. Bifurc. Chaos, vol. 30, no. 14, pp. 1–13, 2020, doi: 10.1142/S0218127420502107.

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, 2020, doi: 10.34312/jjom.v2i2.4566.

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.

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

E. Rahmi, I. Darti, A. Suryanto, Trisilowati, and H. S. Panigoro, “Stability Analysis of a Fractional-Order Leslie-Gower Model with Allee Effect in Predator,” J. Phys. Conf. Ser., vol. 1821, no. 1, 2021, doi: 10.1088/1742-6596/1821/1/012051.

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

L. K. Beay and M. Saija, “A Stage-Structure Rosenzweig-MacArthur Model with Effect of Prey Refuge,” Jambura J. Biomath., vol. 1, no. 1, pp. 1–7, 2020, doi: 10.34312/jjbm.v1i1.6891.

A. L. Firdiansyah, “Effect of Prey Refuge and Harvesting on Dynamics of Eco-epidemiological Model with Holling Type III,” Jambura J. Math., vol. 3, no. 1, pp. 16–25, 2021, doi: 10.34312/jjom.v3i1.7281.

F. Fitriah, A. Suryanto, and N. Hidayat, “Numerical Study of Predator-Prey Model with Beddington-DeAngelis Functional Response and Prey Harvesting,” J. Trop. Life Sci., vol. 5, no. 2, pp. 105–109, 2015, doi: 10.11594/jtls.05.02.09.

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

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

A. J. Terry, “Predator–prey models with component Allee effect for predator reproduction,” J. Math. Biol., vol. 71, no. 6–7, pp. 1325–1352, 2015, doi: 10.1007/s00285-015-0856-5.

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. 0, p. 9, 2017, doi: 10.1155/2017/8273430.

S. K. Sasmal and J. Chattopadhyay, “An eco-epidemiological system with infected prey and predator subject to the weak Allee effect,” Math. Biosci., vol. 246, no. 2, pp. 260–271, 2013, doi: 10.1016/j.mbs.2013.10.005.

B. R. Noon and K. S. McKelvey, “Management of the spotted owl: A case history in conservation biology,” Annu. Rev. Ecol. Syst., vol. 27, pp. 135–162, 1996, doi: 10.1146/annurev.ecolsys.27.1.135.

R. K. Upadhyay and S. Mishra, “Population dynamic consequences of fearful prey in a spatiotemporal predator-prey system,” Math. Biosci. Eng., vol. 16, no. 1, pp. 338–372, 2019, doi: 10.3934/mbe.2019017.

L. Y. Zanette, A. F. White, M. C. Allen, and M. Clinchy, “Perceived predation risk reduces the number of offspring songbirds produce per year,” Science (80-. )., vol. 334, no. 6061, pp. 1398–1401, 2011, doi: 10.1126/science.1210908.

A. L. Firdiansyah, “A Fractional-Order Food Chain Model with Omnivore and Anti-Predator,” Commun. Biomath. Sci., vol. 5, no. 2, pp. 121–136, 2023, doi: 10.5614/cbms.2022.5.2.2.

H. S. Panigoro, A. Suryanto, W. M. Kusumawinahyu, and I. Darti, “Dynamics of an Eco-Epidemic Predator – Prey Model Involving Fractional Derivatives with Power-Law and Mittag-Leffler Kernel,” Symmetry (Basel)., vol. 13, no. 785, pp. 1–29, 2021, doi: https://doi.org/10.3390/sym13050785.

E. Rahmi, I. Darti, A. Suryanto, and Trisilowati, “A Modified Leslie–Gower Model Incorporating Beddington–Deangelis Functional Response, Double Allee Effect and Memory Effect,” Fractal Fract., vol. 5, no. 3, 2021, doi: 10.3390/fractalfract5030084.

P. Panja, “Dynamics of a fractional order predator-prey model with intraguild predation,” Int. J. Model. Simul., vol. 39, no. 4, pp. 256–268, 2019, doi: 10.1080/02286203.2019.1611311.

H. L. Long, Z. Cheng, Y. Jiang, and Z. Teng, “Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge,” J. Appl. Math. Comput., 2016, doi: 10.1007/s12190-016-1017-8.

M. A. Aziz-Alaoui and M. Daher Okiye, “Boundedness and Global Stability for a Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes,” Appl. Math. Lett., vol. 16, no. 7, pp. 1069–1075, 2003, doi: 10.1016/S0893-9659(03)90096-6.

S. Yu, “Global asymptotic stability of a predator-prey model with modified Leslie-Gower and Holling-type II schemes,” Discret. Dyn. Nat. Soc., vol. 2012, p. 8, 2012, doi: 10.1155/2012/208167.

S. Yu, “Global stability of a modified Leslie-Gower model with Beddington-DeAngelis functional response,” Adv. Differ. Equations, vol. 2014, no. 1, pp. 1–14, 2014, doi: 10.1186/1687-1847-2014-84.

W. Cresswell, “Predation in bird populations,” J. Ornithol., vol. 152, pp. S251–S263, 2011, doi: 10.1007/s10336-010-0638-1.

S. Samaddar, M. Dhar, and P. Bhattacharya, “Effect of fear on prey-predator dynamics: Exploring the role of prey refuge and additional food,” Chaos, vol. 30, no. 6, pp. 1–18, 2020, doi: 10.1063/5.0006968.

J. D. Murray, Mathematical Biology I: An Introduction, 3rd ed., vol. 17. Washington: Springer, 2002.

H. S. Panigoro, A. Suryanto, W. M. Kusumawinahyu, and I. Darti, “A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel,” Axioms, vol. 9, no. 122, pp. 1–22, 2020, doi: 10.3390/axioms9040122.

I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Beijing: Springer, 2011.

D. Matignon, “Stability Results for Fractional Differential Equations with Applications to Control Processing,” Comput. Eng. Syst. Appl., vol. 2, pp. 963–968, 1996.

X. Li and R. Wu, “Hopf Bifurcation Analysis of a New Commensurate Fractional-Order Hyperchaotic System,” Nonlinear Dyn., vol. 78, no. 1, pp. 279–288, 2014, doi: 10.1007/s11071-014-1439-5.

K. A. I. Diethelm, N. J. Ford, and A. D. Freed, “A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equation,” Nonlinear Dyn., vol. 29, no. 1–4, pp. 3–22, 2002, doi: https://doi.org/https://doi.org/10.1023/A:1016592219341.




DOI: https://doi.org/10.18860/ca.v7i4.17336

Refbacks

  • There are currently no refbacks.


Copyright (c) 2023 Adin Lazuardy Firdiansyah

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

Editorial Office
Mathematics Department,
Universitas Islam Negeri Maulana Malik Ibrahim Malang
Gajayana Street 50 Malang, East Java, Indonesia 65144
Faximile (+62) 341 558933
e-mail: cauchy@uin-malang.ac.id

Creative Commons License
CAUCHY: Jurnal Matematika Murni dan Aplikasi is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.