A Fractional-Order Predator-Prey Model with Age Structure on Predator and Nonlinear Harvesting on Prey

Hasan S. Panigoro, Resmawan Resmawan, Amelia Tri Rahma Sidik, Nurdia Walangadi, Apon Ismail, Cabelita Husuna


In this manuscript, the dynamics of a fractional-order predator-prey model with age structure on predator and nonlinear harvesting on prey are studied. The Caputo fractional-order derivative is used as the operator of the model by considering its capability to explain the present state as the impact of all of the previous conditions. Three biological equilibrium points are successfully identified including their existing properties. The local dynamical behaviors around each equilibrium point are investigated by utilizing the Matignon condition along with the linearization process. The numerical simulations are demonstrated not only to show the local stability which confirms all of the previous analytical results but also to show the existence of periodic signal as the impact of the occurrence of Hopf bifurcation.


Predator-Prey; Age Structure; Harvesting; Caputo Operator; Dynamics

Full Text:



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,” International Journal of Bifurcation and Chaos, vol. 30, no. 14, pp. 1-13, 2020, doi: https://doi.org/10.1142/S0218127420502107.

J. Liu, B. Liu, P. Lv, and T. Zhang, “An eco-epidemiological model with fear effect and hunting cooperation,” Chaos, Solitons and Fractals, vol. 142, p. 110494, 2021, doi: https: //doi.org/10.1016/j.chaos.2020.110494.

P. K. Santra, “Fear effect in discrete prey-predator model incorporating square root functional response,” Jambura Journal of Biomathematics, vol. 2, no. 2, pp. 51–57, 2021, doi: https://doi.org/https://doi.org/10.34312/jjbm.v2i2.10444.

D. Mukherjee, “Impact of predator fear on two competing prey species,” Jambura Journal of Biomathematics, vol. 2, no. 1, pp. 1–12, 2021, doi: https://doi.org/10.34312/jjbm.v2i1.9249.

H. S. Panigoro and D. Savitri, “Bifurkasi Hopf pada model Lotka-Volterra orde-fraksional dengan efek Allee aditif pada predator,” Jambura Journal of Biomathematics, vol. 1, no. 1, pp. 16–24, 2020, doi: https://doi.org/10.34312/jjbm.v1i1.6908.

H. S. Panigoro and E. Rahmi, “Computational dynamics of a Lotka-Volterra Model with additive Allee effect based on Atangana-Baleanu-fractional derivative,” Jambura Journal of Biomathematics, vol. 2, no. 2, pp. 96–103, 2021, doi: https://doi.org/10.34312/jjbm.v2i2.11886.

M. R. Ali, S. Raut, S. Sarkar, and U. Ghosh, “Unraveling the combined actions of a Holling type III predator–prey model incorporating Allee response and memory effects,” Computational and Mathematical Methods, vol. 3, no. 2, 2021, doi: https://doi.org/10.1002/cmm4.1130.

C. Arancibia-Ibarra, P. Aguirre, J. Flores, and P. van Heijster, “Bifurcation analysis of a predator-prey model with predator intraspecific interactions and ratio-dependent functional response,” Applied Mathematics and Computation, vol. 402, p. 126152, 2021, doi: https://doi.org/10.1016/j.amc.2021.126152.

H. S. Panigoro, A. Suryanto, W. M. Kusumawinahyu, and I. Darti, Global stability of a fractional-order Gause-type predator-prey model with threshold harvesting policy in predator,” Communications in Mathematical Biology and Neuroscience, vol. 2021, no. 2021, p. 63, 2021, doi: https://doi.org/10.28919/cmbn/6118.

A. A. Thirthar, S. J. Majeed, M. A. Alqudah, P. Panja, and T. Abdeljawad, “Fear effect in a predator-prey model with additional food, prey refuge and harvesting on super predator,” Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, vol. 159, p. 112091, 2022, doi: https://doi.org/10.1016/j.chaos.2022.112091.

L. K. Beay and M. Saija, “Dynamics of a stage–structure Rosenzweig–MacArthur model with linear harvesting in prey and cannibalism in predator,” Jambura Journal of Biomathematics, vol. 2, no. 1, pp. 42–50, 2021, doi: https://doi.org/10.34312/jjbm.v2i1.10470.

D. Yan, H. Cao, X. Xu, and X. Wang, “Hopf bifurcation for a predator–prey model with age structure,” Physica A: Statistical Mechanics and its Applications, vol. 526, p. 120953, 2019, doi: https://doi.org/10.1016/j.physa.2019.04.189.

X. Zhang and Z. Liu, “Periodic oscillations in age-structured ratio-dependent predator–prey model with Michaelis–Menten type functional response,” Physica D: Nonlinear Phenomena, vol. 389, pp. 51–63, 2019, doi: https://doi.org/10.1016/j.physd.2018.10.002.

Z. Liu and P. Magal, “Bogdanov–Takens bifurcation in a predator–prey model with age structure,” Zeitschrift fur Angewandte Mathematik und Physik, vol. 72, no. 1, pp. 1–24, 2021, doi: https://doi.org/10.1007/s00033-020-01434-1.

R. P. Gupta and P. Chandra, “Bifurcation Analysis of Modified Leslie-Gower Predator-Prey Model with Michaelis-Menten Type Prey Harvesting,” Journal of Mathematical Analysis and Applications, vol. 398, no. 1, pp. 278–295, 2013, doi: https://doi.org/10.1016/j.jmaa.2012.08.057.

Z. Zhang, R. K. Upadhyay, and J. Datta, “Bifurcation Analysis of a Modified Leslie–Gower Model with Holling Type-IV Functional Response and Nonlinear Prey Harvesting,” Advances in Difference Equations, vol. 2018, no. 1, 2018, doi: https://doi.org/10.1186/s13662-018-1581-3.

X. Yu, Z. Zhu, L. Lai, and F. Chen, “Stability and bifurcation analysis in a single-species stage structure system with Michaelis–Menten-type harvesting,” Advances in Difference Equations, vol. 2020, no. 1, p. 238, 2020, doi: https://doi.org/10.1186/s13662-020-02652-7.

M. Li, B. Chen, and H. Ye, “A bioeconomic differential algebraic predator–prey model with nonlinear prey harvesting,” Applied Mathematical Modelling, vol. 42, pp. 17–28, 2017, doi: https://doi.org/10.1016/j.apm.2016.09.029.

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, vol. 13, no. 5, p. 785, 2021, doi: https://doi.org/10.3390/sym13050785.

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. 4, p. 122, 2020, doi: https://doi.org/10.3390/axioms9040122.

A. Mahata, S. Paul, S. Mukherjee, M. Das, and B. Roy, “Dynamics of Caputo Fractional Order SEIRV Epidemic Model with Optimal Control and Stability Analysis,” International Journal of Applied and Computational Mathematics, vol. 8, no. 1, pp. 1–25, 2022, doi: https://doi.org/10.1007/s40819-021-01224-x.

C. Maji, “Dynamical analysis of a fractional-order predator–prey model incorporating a constant prey refuge and nonlinear incident rate,” Modeling Earth Systems and Environment, vol. 8, no. 1, pp. 47–57, 2022, doi: https://doi.org/10.1007/s40808-020-01061-9.

I. Petras, Fractional-order nonlinear systems: modeling, analysis and simulation. Beijing: Springer London, 2011. ISBN 9788578110796

I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. San Diego CA: Academic Press, 1999. ISBN 0-12 558840-2

M. Caputo, “Linear models of dissipation whose q is almost frequency independent–ii,” Geophysical Journal International, vol. 13, pp. 529–539, 1967, doi: https://doi.org/10.1111/j.1365-246X.1967.tb02303.x.

M. Caputo and M. Fabrizio, “A new definition of fractional derivative without singular kernel,” Progress in Fractional Differentiation and Applications, vol. 1, no. 2, pp. 73–85, 2015, doi: https://doi.org/10.12785/pfda/010201.

A. Atangana and D. Baleanu, “New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model,” Thermal Science, vol. 20, pp. 763–769, 2016, doi: https://doi.org/10.2298/TSCI160111018A.

A. J. Lotka, Elements of Physical Biology. Williams & Wilkins, 1925. ISBN 9780598812681. [Online]. Available: https://books.google.co.id/books?id=lsPQAAAAMAAJ

V. Volterra, “Variations and Fluctuations in the Numbers of Coexisting Animal Species,” in Lecture Notes in Biomathematics. Heidelberg: Springer, 1978, pp. 65–236, doi: https://doi.org/10.1007/978-3-642-50151-7 9.

P.-F. Verhulst, “Notice Sur La Loi Que La Population Poursuit Dans Son Accroissement,” Correspondance math´ematique et physique, vol. 10, pp. 113–121, 1838.

G. Seo and D. L. Deangelis, “A predator-prey model with a holling type I functional response including a predator mutual interference,” Journal of Nonlinear Science, vol. 21, no. 6, pp. 811–833, 2011, doi: https://doi.org/10.1007/s00332-011-9101-6.

A. D. Bazykin, A. I. Khibnik, and B. Krauskopf, Nonlinear Dynamics of Interacting Populations, ser. World Scientific Series on Nonlinear Science Series A. World Scientific, 1998, vol. 11. ISBN 978-981-02-1685-6 Doi: https://doi.org/10.1142/2284.

J. Zhang, L. Zhang, and C. M. Khalique, “Stability and Hopf bifurcation analysis on a Bazykin model with delay,” Abstract and Applied Analysis, vol. 2014, no. 2, 2014, doi: https: //doi.org/10.1155/2014/539684.

E. N. Bodine and A. E. Yust, “Predator–prey dynamics with intraspecific competition and an Allee effect in the predator population,” Letters in Biomathematics, vol. 4, no. 1, pp. 23–38, 2017, doi: https://doi.org/10.1080/23737867.2017.1282843.

L. Pribylov'a, “Regime shifts caused by adaptive dynamics in prey–predator models and 'their relationship with intraspecific competition,” Ecological Complexity, vol. 36, no. May, pp. 48–56, 2018, doi: https://doi.org/10.1016/j.ecocom.2018.06.003.

L. Lai, X. Yu, M. He, and Z. Li, “Impact of Michaelis–Menten type harvesting in a Lotka–Volterra predator–prey system incorporating fear effect,” Advances in Difference Equations, vol. 2020, no. 1, 2020, doi: https://doi.org/10.1186/s13662-020-02724-8.

Y. Li and M. Wang, “Dynamics of a Diffusive Predator-Prey Model with Modified LeslieGower Term and Michaelis-Menten Type Prey Harvesting,” Acta Applicandae Mathematicae, vol. 140, no. 1, pp. 147–172, 2015, doi: https://doi.org/10.1007/s10440-014-9983-z.

X.-P. Yan and C.-H. Zhang, “Global stability of a delayed diffusive predator–prey model with prey harvesting of Michaelis–Menten type,” Applied Mathematics Letters, vol. 114, p. 106904, 2021, doi: https://doi.org/10.1016/j.aml.2020.106904.

D. Matignon, “Stability results for fractional differential equations with applications to control processing,” Computational engineering in systems applications, pp. 963–968, 1996.

E. Ahmed, A. El-Sayed, and H. A. El-Saka, “On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rossler, Chua and Chen systems,” Physics Letters A, vol. 358, no. 1, pp. 1–4, 2006, doi: https://doi.org/10.1016/j.physleta.2006.04.087.

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

DOI: https://doi.org/10.34312/jjom.v4i2.15220

Copyright (c) 2022 Hasan S. Panigoro, Resmawan Resmawan, Amelia Tri Rahma Sidik, Nurdia Walangadi, Apon Ismail, Cabelita Husuna

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
Email: info.jjom@ung.ac.id.