Dynamics of a Fractional Order Harvested Predator-Prey Model Incorporating Fear Effect and Refuge
Keywords:
Caputo fractional derivative, stability analysis, harvesting, fear effect, prey refuge
Abstract
This study presents a fractional-order predator-prey dynamics model that considers the impact of fear, refuge, and harvesting on the population, respectively. The proposed model uses the Caputo fractional derivative to successfully obtain the memory effects of this interaction between predators and preys. We prove the existence and uniqueness of solution to ensure the non-negativity and boundedness of the system, which is indispensable for maintaining biologically feasible populations. The stability analysis is conducted on the equilibrium points at local and global levels, explaining the conditions that guarantee these points are stable or lead to periodic dynamics through Hopf bifurcation. To support the analytical results, numerical simulations are provided, which demonstrate the essential roles played by fear, refuge, and harvesting in the survival of prey and the overall dynamics of the system.References
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169, no. 1, pp. 26–39, 2018.
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vol. 11, pp. 3303–3321, 2023.
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Nonlinear Prey Refuge, Mathematics, vol. 10, pp. 2857, 2023.
6. R. Han, G. Mandal, L.N. Guin, S. Chakravarty, Dynamical response of a reaction–diffusion predator–prey system with cooperative
hunting and prey refuge, Journal of Statistical Mechanics: Theory and Experiment, vol. 10, pp. 103502, 2022.
7. J. Ghosh, B. Sahoo, S. Poria, Prey-predator dynamics with prey refuge providing additional food to predator, Chaos, Solitons &
Fractals, vol. 96, pp. 110–119, 2017.
8. S. Wang, Z. Ma, W. Wang, Dynamical behavior of a generalized eco-epidemiological system with prey refuge, Advances in
Difference Equations, vol. 2018, pp. 1–20, 2018.
9. S. N. Raw, B. P. Sarangi, Qualitative and dynamical analysis of a bionomic fishery model with prey refuge, Acta Biotheoretica, vol.
70, pp. 11–48, 2022.
10. M. Rayungsari, A. Suryanto,W. M. Kusumawinahyu, I. Darti, Dynamical analysis of a predator-prey model incorporating predator
cannibalism and refuge, Axioms, vol. 11, pp. 116–133, 2022.
11. H. Deng, F. Chen, Z. Zhu, Z. Li, Dynamic behaviors of Lotka–Volterra predator–prey model incorporating predator cannibalism,
Advances in Difference Equations, vol 2019, pp. 1–17, 2019.
12. F. Zhang, Y. Chen, J. Li, Dynamical analysis of a stage-structured predator-prey model with cannibalism, Mathematical
Biosciences, vol. 307, pp. 33–41, 2019.
13. M. Rayungsari, A. Suryanto, W. M. Kusumawinahyu, I. Darti, Dynamics analysis of a predator–prey fractional-order model
incorporating predator cannibalism and refuge, Frontiers in Applied Mathematics and Statistics,vol. 9, pp. 1122330–1122341, 2023.
14. A. S. Abdulghafour, R. K. Naji, Modeling and analysis of a prey-predator system incorporating fear, predator-dependent refuge,
and cannibalism, Commun. Math. Biol. Neurosci.,vol. 2022, pp. 2052–2541, 2022.
15. S. Roy, P.K. Tiwari, H. Nayak, M. Martcheva, Effects of fear, refuge and hunting cooperation in a seasonally forced eco-epidemic
model with selective predation, The European Physical Journal Plus, vol 137, pp. 528–558, 2022.
16. F.B. Yousef, A. Yousef, C. Maji, Effects of fear in a fractional-order predator-prey system with predator density-dependent prey
mortality, Chaos, Solitons & Fractals, vol. 145, pp. 110711–110720, 2021.
17. X. Wang, L. Zanette, X. Zou, Modelling the fear effect in predator–prey interactions, Journal of mathematical biology, vol. 73, pp.
1179–1204, 2016.
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108163, 2022.
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hunting cooperation and fear effect, International Journal of Dynamics and Control, vol. 11, pp. 2733–2750, 2023.
23. P.K. Tiwari, K.A.N.A. Amri, S. Samanta, Q.J.A. Khan, J Chattopadhyay, A systematic study of autonomous and nonautonomous
predator–prey models with combined effects of fear, migration and switching, Nonlinear Dynamics, vol. 103, pp. 2125–2162, 2021.
24. P. Panday, N. Pal, S. Samanta, J. Chattopadhyay, Stability and bifurcation analysis of a three-species food chain model with fear,
International Journal of Bifurcation and Chaos, vol. 28, pp. 1850009, 2018.
25. K.H. Elliott, G.S. Betini, D.R. Norris, Fear creates an Allee effect: experimental evidence from seasonal populations, Proceedings
of the Royal Society B: Biological Sciences, vol. 284, pp. 20170878, 2017.
26. S. Pal, N. Pal, S. Samanta, J. Chattopadhyay, Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model,
Mathematical Biosciences and Engineering, vol. 16, pp. 5146, 2019.
27. P. Panday, S. Samanta, N. Pal, J. Chattopadhyay, Delay induced multiple stability switch and chaos in a predator–prey model with
fear effect, Mathematics and Computers in Simulation, vol. 172, pp. 134–158, 2020.
28. H. Zhang, Y. Cai, S. Fu, W. Wang, Impact of the fear effect in a prey-predator model incorporating a prey refuge, Applied
Mathematics and Computation, vol. 356, pp. 328–337, 2019.
29. P. Paul, T.K. Kar, E. Das, Reactivity in prey–predator models at equilibrium under selective harvesting efforts, The European
Physical Journal Plus, vol. 136, pp. 510, 2021.
30. B. Das¸bas¸ı, Stability analysis of mathematical model including pathogen-specific immune system response with fractional-order
differential equations, Computational and mathematical methods in medicine, vol. 2018, pp. 1–10, 2018.
31. H.A.A. El-Saka, A.A.M. Arafa, M.I. Gouda, Dynamical analysis of a fractional SIRS model on homogenous networks, Advances
in Difference Equations, vol. 2019, pp. 1–15, 2019.
32. D. Mukherjee, Fear induced dynamics on Leslie-Gower predator-prey system with Holling-type IV functional response, Jambura
Journal of Biomathematics (JJBM), vol 3, pp. 49–57, 2022.
33. E. Rahmi, I. Darti, A. Suryanto, Trisilowati, A modified Leslie–Gower model incorporating Beddington–DeAngelis functional
response, double Allee effect and memory effect, Fractal and Fractional, vol. 5, pp. 84, 2021.
34. A. Suryanto, I. Darti, S. Anam, Stability analysis of a fractional order modified Leslie-Gower model with additive Allee effect,
International Journal of Mathematics and Mathematical Sciences, vol.2017, pp. 1–10, 2017.
35. S. Karthikeyan, P. Ramesh, M. Sambath, Stability analysis of fractional-order predator-prey model with anti-predator behaviour
and prey refuge, Journal of Mathematical Modeling, vol.11, pp.527–546, 2023.
36. A. A. Themairi, M.A. Alqudah, Existence and Uniqueness of Caputo Fractional Predator-Prey Model of Holling-Type II with
Numerical Simulations, Mathematical Problems in Engineering, vol 2021, pp. 1–6, 2021.
37. M. Das, G.P. Samanta, A delayed fractional order food chain model with fear effect and prey refuge, Mathematics and Computers
in Simulation, vol. 178, pp. 218–245, 2020.
38. M. Higazy, S.A.M. Alsallami, S. Abdel-Khalek, A. El-Mesady, Dynamical and structural study of a generalized Caputo fractional
order Lotka-Volterra model, Results in Physics, vol. 37, pp. 105478, 2022.
39. H. Jafari, R.M. Ganji, N.S. Nkomo, Y.P. Lv, A numerical study of fractional order population dynamics model, Results in Physics,
vol.27, pp. 104456, 2021.
40. H.L. Li, L. Zhang, C. Hu, Y.L. Jiang, Z. Teng, Dynamical analysis of a fractional-order predator-prey model incorporating a prey
refuge, Journal of Applied Mathematics and Computing,vol. 57, pp. 435–449, 2017.
41. J. Cresson, A. Szafra´nska, Discrete and continuous fractional persistence problems–the positivity property and applications,
Communications in Nonlinear Science and Numerical Simulation, vol. 44, pp. 424–448, 2017.
42. M. Al-Refai, Y. Luchko, Comparison principles for solutions to the fractional differential inequalities with the general fractional
derivatives and their applications, Journal of Differential Equations, vol. 319, pp. 312–324, 2022.
43. J. Huo, H. Zhao, L. Zhu, The effect of vaccines on backward bifurcation in a fractional order HIV model, Nonlinear Analysis: Real
World Applications, vol. 26, pp. 289–305, 2015.
44. C. Vargas-De-Le´on, Volterra-type Lyapunov functions for fractional-order epidemic systems, Communications in Nonlinear Science
and Numerical Simulation, vol. 24, pp. 75–85, 2015.
45. A. El-Mesady, O. Bazighifan, S. Araci, Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on
Immigration Effect, Journal of Mathematics, vol. 2023, pp. 8795660, 2023.
46. R.C. Mittal, S. Pandit, A numerical algorithm to capture spin patterns of fractional Bloch nuclear magnetic resonance flow models,
Journal of Computational and Nonlinear Dynamics, vol. 14, pp. 81001, 2019.
47. S. Pandit, R.C. Mittal, A numerical algorithm based on scale-3 Haar wavelets for fractional advection dispersion equation,
Engineering Computations, vol. 38, pp. 1706–1724, 2021.
48. R.C. Mittal, S. Pandit, Quasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynamical
systems, Engineering Computations, vol. 35, pp. 1907–1931, 2019.
49. A.M. Diop, J.L. Polleux, C. Algani, S. Mazer, M. Fattah, M.E.L. Bekkali, Design electrical model noise and perform nonlinearities
of SiGe bipolar phototransistor, International Journal of Innovative Research and Scientific Studies, vol.6, pp. 731–740, 2023.
50. G. Krishna, R. Singh, A. Gehlot, P. Singh, S. Rana, S.V. Akram, K. Joshi, An imperative role of studying existing battery datasets
and algorithms for battery management system, Review of Computer Engineering Research, vol 10, pp. 28–39, 2023.
51. Z. Alsalami, Modeling of Optimal Fully Connected Deep Neural Network based Sentiment Analysis on Social Networking Data,
Journal of Smart Internet of Things, vol 2022, pp. 114–132, 2022.
combined effects of fear, refuge, cooperation and harvesting, The European Physical Journal Plus, vol. 137, no. 1, pp. 724–746, 2022.
2. X. Liu, Q. Huang, The dynamics of a harvested predator–prey system with Holling type IV functional response, Biosystems, vol.
169, no. 1, pp. 26–39, 2018.
3. K. Sarkar, B. Mondal, Dynamic analysis of a fractional-order predator–prey model with harvesting, Int. Journal of Dynamics and
Control, vol. 11, pp. 1518–1531, 2023.
4. C. Zhang, X. Li, Dynamics of a Discrete Leslie–Gower Model with Harvesting and Holling-II Functional Response, Mathematics,
vol. 11, pp. 3303–3321, 2023.
5. A.R.M. Jamil, R.K. Naji, Modeling and Analysis of the Influence of Fear on the Harvested Modified Leslie–Gower Model Involving
Nonlinear Prey Refuge, Mathematics, vol. 10, pp. 2857, 2023.
6. R. Han, G. Mandal, L.N. Guin, S. Chakravarty, Dynamical response of a reaction–diffusion predator–prey system with cooperative
hunting and prey refuge, Journal of Statistical Mechanics: Theory and Experiment, vol. 10, pp. 103502, 2022.
7. J. Ghosh, B. Sahoo, S. Poria, Prey-predator dynamics with prey refuge providing additional food to predator, Chaos, Solitons &
Fractals, vol. 96, pp. 110–119, 2017.
8. S. Wang, Z. Ma, W. Wang, Dynamical behavior of a generalized eco-epidemiological system with prey refuge, Advances in
Difference Equations, vol. 2018, pp. 1–20, 2018.
9. S. N. Raw, B. P. Sarangi, Qualitative and dynamical analysis of a bionomic fishery model with prey refuge, Acta Biotheoretica, vol.
70, pp. 11–48, 2022.
10. M. Rayungsari, A. Suryanto,W. M. Kusumawinahyu, I. Darti, Dynamical analysis of a predator-prey model incorporating predator
cannibalism and refuge, Axioms, vol. 11, pp. 116–133, 2022.
11. H. Deng, F. Chen, Z. Zhu, Z. Li, Dynamic behaviors of Lotka–Volterra predator–prey model incorporating predator cannibalism,
Advances in Difference Equations, vol 2019, pp. 1–17, 2019.
12. F. Zhang, Y. Chen, J. Li, Dynamical analysis of a stage-structured predator-prey model with cannibalism, Mathematical
Biosciences, vol. 307, pp. 33–41, 2019.
13. M. Rayungsari, A. Suryanto, W. M. Kusumawinahyu, I. Darti, Dynamics analysis of a predator–prey fractional-order model
incorporating predator cannibalism and refuge, Frontiers in Applied Mathematics and Statistics,vol. 9, pp. 1122330–1122341, 2023.
14. A. S. Abdulghafour, R. K. Naji, Modeling and analysis of a prey-predator system incorporating fear, predator-dependent refuge,
and cannibalism, Commun. Math. Biol. Neurosci.,vol. 2022, pp. 2052–2541, 2022.
15. S. Roy, P.K. Tiwari, H. Nayak, M. Martcheva, Effects of fear, refuge and hunting cooperation in a seasonally forced eco-epidemic
model with selective predation, The European Physical Journal Plus, vol 137, pp. 528–558, 2022.
16. F.B. Yousef, A. Yousef, C. Maji, Effects of fear in a fractional-order predator-prey system with predator density-dependent prey
mortality, Chaos, Solitons & Fractals, vol. 145, pp. 110711–110720, 2021.
17. X. Wang, L. Zanette, X. Zou, Modelling the fear effect in predator–prey interactions, Journal of mathematical biology, vol. 73, pp.
1179–1204, 2016.
18. X. Dong, B. Niu, On a diffusive predator–prey model with nonlocal fear effect, Applied Mathematics Letters,vol. 132 ,pp. 108156–
108163, 2022.
19. Y. Li, M. He, Z. Li, Dynamics of a ratio-dependent Leslie–Gower predator–prey model with Allee effect and fear effect, Mathematics
and Computers in Simulation, vol. 201, pp. 417–439, 2022.
20. R. Yang, D. Jin, Dynamics in a predator-prey model with memory effect in predator and fear effect in prey, Electronic Research
Archive,vol. 30, pp. 1322–1339, 2022.
21. M. Chen, Y. Takeuchi, J.F. Zhang, Dynamic complexity of a modified Leslie–Gower predator–prey system with fear effect,
Communications in Nonlinear Science and Numerical Simulation, vol.119, pp. 107109–107126, 2023.
22. I. Benamara, A. E. Abdllaoui, Bifurcation in a delayed predator–prey model with Holling type IV functional response incorporating
hunting cooperation and fear effect, International Journal of Dynamics and Control, vol. 11, pp. 2733–2750, 2023.
23. P.K. Tiwari, K.A.N.A. Amri, S. Samanta, Q.J.A. Khan, J Chattopadhyay, A systematic study of autonomous and nonautonomous
predator–prey models with combined effects of fear, migration and switching, Nonlinear Dynamics, vol. 103, pp. 2125–2162, 2021.
24. P. Panday, N. Pal, S. Samanta, J. Chattopadhyay, Stability and bifurcation analysis of a three-species food chain model with fear,
International Journal of Bifurcation and Chaos, vol. 28, pp. 1850009, 2018.
25. K.H. Elliott, G.S. Betini, D.R. Norris, Fear creates an Allee effect: experimental evidence from seasonal populations, Proceedings
of the Royal Society B: Biological Sciences, vol. 284, pp. 20170878, 2017.
26. S. Pal, N. Pal, S. Samanta, J. Chattopadhyay, Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model,
Mathematical Biosciences and Engineering, vol. 16, pp. 5146, 2019.
27. P. Panday, S. Samanta, N. Pal, J. Chattopadhyay, Delay induced multiple stability switch and chaos in a predator–prey model with
fear effect, Mathematics and Computers in Simulation, vol. 172, pp. 134–158, 2020.
28. H. Zhang, Y. Cai, S. Fu, W. Wang, Impact of the fear effect in a prey-predator model incorporating a prey refuge, Applied
Mathematics and Computation, vol. 356, pp. 328–337, 2019.
29. P. Paul, T.K. Kar, E. Das, Reactivity in prey–predator models at equilibrium under selective harvesting efforts, The European
Physical Journal Plus, vol. 136, pp. 510, 2021.
30. B. Das¸bas¸ı, Stability analysis of mathematical model including pathogen-specific immune system response with fractional-order
differential equations, Computational and mathematical methods in medicine, vol. 2018, pp. 1–10, 2018.
31. H.A.A. El-Saka, A.A.M. Arafa, M.I. Gouda, Dynamical analysis of a fractional SIRS model on homogenous networks, Advances
in Difference Equations, vol. 2019, pp. 1–15, 2019.
32. D. Mukherjee, Fear induced dynamics on Leslie-Gower predator-prey system with Holling-type IV functional response, Jambura
Journal of Biomathematics (JJBM), vol 3, pp. 49–57, 2022.
33. E. Rahmi, I. Darti, A. Suryanto, Trisilowati, A modified Leslie–Gower model incorporating Beddington–DeAngelis functional
response, double Allee effect and memory effect, Fractal and Fractional, vol. 5, pp. 84, 2021.
34. A. Suryanto, I. Darti, S. Anam, Stability analysis of a fractional order modified Leslie-Gower model with additive Allee effect,
International Journal of Mathematics and Mathematical Sciences, vol.2017, pp. 1–10, 2017.
35. S. Karthikeyan, P. Ramesh, M. Sambath, Stability analysis of fractional-order predator-prey model with anti-predator behaviour
and prey refuge, Journal of Mathematical Modeling, vol.11, pp.527–546, 2023.
36. A. A. Themairi, M.A. Alqudah, Existence and Uniqueness of Caputo Fractional Predator-Prey Model of Holling-Type II with
Numerical Simulations, Mathematical Problems in Engineering, vol 2021, pp. 1–6, 2021.
37. M. Das, G.P. Samanta, A delayed fractional order food chain model with fear effect and prey refuge, Mathematics and Computers
in Simulation, vol. 178, pp. 218–245, 2020.
38. M. Higazy, S.A.M. Alsallami, S. Abdel-Khalek, A. El-Mesady, Dynamical and structural study of a generalized Caputo fractional
order Lotka-Volterra model, Results in Physics, vol. 37, pp. 105478, 2022.
39. H. Jafari, R.M. Ganji, N.S. Nkomo, Y.P. Lv, A numerical study of fractional order population dynamics model, Results in Physics,
vol.27, pp. 104456, 2021.
40. H.L. Li, L. Zhang, C. Hu, Y.L. Jiang, Z. Teng, Dynamical analysis of a fractional-order predator-prey model incorporating a prey
refuge, Journal of Applied Mathematics and Computing,vol. 57, pp. 435–449, 2017.
41. J. Cresson, A. Szafra´nska, Discrete and continuous fractional persistence problems–the positivity property and applications,
Communications in Nonlinear Science and Numerical Simulation, vol. 44, pp. 424–448, 2017.
42. M. Al-Refai, Y. Luchko, Comparison principles for solutions to the fractional differential inequalities with the general fractional
derivatives and their applications, Journal of Differential Equations, vol. 319, pp. 312–324, 2022.
43. J. Huo, H. Zhao, L. Zhu, The effect of vaccines on backward bifurcation in a fractional order HIV model, Nonlinear Analysis: Real
World Applications, vol. 26, pp. 289–305, 2015.
44. C. Vargas-De-Le´on, Volterra-type Lyapunov functions for fractional-order epidemic systems, Communications in Nonlinear Science
and Numerical Simulation, vol. 24, pp. 75–85, 2015.
45. A. El-Mesady, O. Bazighifan, S. Araci, Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on
Immigration Effect, Journal of Mathematics, vol. 2023, pp. 8795660, 2023.
46. R.C. Mittal, S. Pandit, A numerical algorithm to capture spin patterns of fractional Bloch nuclear magnetic resonance flow models,
Journal of Computational and Nonlinear Dynamics, vol. 14, pp. 81001, 2019.
47. S. Pandit, R.C. Mittal, A numerical algorithm based on scale-3 Haar wavelets for fractional advection dispersion equation,
Engineering Computations, vol. 38, pp. 1706–1724, 2021.
48. R.C. Mittal, S. Pandit, Quasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynamical
systems, Engineering Computations, vol. 35, pp. 1907–1931, 2019.
49. A.M. Diop, J.L. Polleux, C. Algani, S. Mazer, M. Fattah, M.E.L. Bekkali, Design electrical model noise and perform nonlinearities
of SiGe bipolar phototransistor, International Journal of Innovative Research and Scientific Studies, vol.6, pp. 731–740, 2023.
50. G. Krishna, R. Singh, A. Gehlot, P. Singh, S. Rana, S.V. Akram, K. Joshi, An imperative role of studying existing battery datasets
and algorithms for battery management system, Review of Computer Engineering Research, vol 10, pp. 28–39, 2023.
51. Z. Alsalami, Modeling of Optimal Fully Connected Deep Neural Network based Sentiment Analysis on Social Networking Data,
Journal of Smart Internet of Things, vol 2022, pp. 114–132, 2022.
Published
2025-01-19
How to Cite
Afiyah, S. N., Fatmawati, Windarto, & Abidemi, A. (2025). Dynamics of a Fractional Order Harvested Predator-Prey Model Incorporating Fear Effect and Refuge. Statistics, Optimization & Information Computing. https://doi.org/10.19139/soic-2310-5070-2341
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