Dynamical Analysis of Discrete-Time Modified Leslie-Gower Predator-Prey with Fear Effect

Anna Silvia Purnomo; Isnani Darti; Agus Suryanto; Wuryansari Muharini Kusumawinahyu

doi:10.31764/jtam.v9i1.26515

Dynamical Analysis of Discrete-Time Modified Leslie-Gower Predator-Prey with Fear Effect

Anna Silvia Purnomo, Isnani Darti, Agus Suryanto, Wuryansari Muharini Kusumawinahyu

Abstract

It has been studied that fear plays a significant role in establishing ecological communities, influencing biodiversity, and preserving ecological balance in predator-prey interactions. In this study, it is proposed a discrete-time predator-prey model that takes the fear effect into account that is derived by using Euler method. Objective of this study is analyzing the model by linearization. Similar to the continuous model properties, the trivial fixed point and the predator-free fixed point are both unstable. The discrete model differs from the continuous model in that the stability of the interior fixed point and the free prey fixed point is affected by the time step size. Using numerical methods, we examine period-doubling bifurcations related to interior fixed point and prey-free point that are impacted by time step size.

Keywords

Fear effect; Leslie-Gower Predator-prey; Discrete-time.

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References

Aziz-Alaoui, M. A., & Daher Okiye, M. (2003). Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes. Applied Mathematics Letters, 16(7), 1069–1075. https://doi.org/10.1016/S0893-9659(03)90096-6

Beretta, E., & Kuang, Y. (2002). Geometric Stability Switch Criteria in Delay Differential Systems with Delay Dependent Parameters. SIAM Journal on Mathematical Analysis, 33(5), 1144–1165. https://doi.org/10.1137/S0036141000376086

Bortuli Junior, A., & Maidana, N. A. (2021). A modified Leslie–Gower predator–prey model with alternative food and selective predation of noninfected prey. Mathematical Methods in the Applied Sciences, 44(5), 3441–3467. https://doi.org/10.1002/mma.6952

Cai, Y., Zhao, C., Wang, W., & Wang, J. (2015). Dynamics of a Leslie–Gower predator–prey model with additive Allee effect. Applied Mathematical Modelling, 39(7), 2092–2106. https://doi.org/10.1016/j.apm.2014.09.038

Chen, J., He, X., & Chen, F. (2021). The Influence of Fear Effect to a Discrete-Time Predator-Prey System with Predator Has Other Food Resource. Mathematics, 9(8), 865. https://doi.org/10.3390/math9080865

Din, Q., Jameel, K., & Shabbir, M. S. (2024). Discrete-Time Predator–Prey System Incorporating Fear Effect: Stability, Bifurcation, and Chaos Control. Qualitative Theory of Dynamical Systems, 23(S1), 285. https://doi.org/10.1007/s12346-024-01145-2

Huang, J., Ruan, S., & Song, J. (2014). Bifurcations in a predator–prey system of Leslie type with generalized Holling type III functional response. Journal of Differential Equations, 257(6), 1721–1752. https://doi.org/10.1016/j.jde.2014.04.024

Khaliq, A., Ibrahim, T. F., Alotaibi, A. M., Shoaib, M., & El-Moneam, M. A. (2022). Dynamical Analysis of Discrete-Time Two-Predators One-Prey Lotka–Volterra Model. Mathematics, 10(21), 4015. https://doi.org/10.3390/math10214015

Li, W., Zhang, C., & Wang, M. (2024). Analysis of the Dynamical Properties of Discrete Predator-Prey Systems with Fear Effects and Refuges. Discrete Dynamics in Nature and Society, 2024, 1–18. https://doi.org/10.1155/2024/9185585

Lotka, A. J. (1920). Analytical Note on Certain Rhythmic Relations in Organic Systems. Proceedings of the National Academy of Sciences, 6(7), 410–415. https://doi.org/10.1073/pnas.6.7.410

Mondal, C., Kesh, D., & Mukherjee, D. (2024). Dynamical behavior of a discrete-time predator–prey system incorporating prey refuge and fear effect. Chinese Journal of Physics, 91, 1004–1026. https://doi.org/10.1016/j.cjph.2024.07.002

Mondal, S., Maiti, A., & Samanta, G. P. (2018). Effects of Fear and Additional Food in a Delayed Predator–Prey Model. Biophysical Reviews and Letters, 13(04), 157–177. https://doi.org/10.1142/S1793048018500091

Pal, S., Gupta, A., Misra, A. K., & Dubey, B. (2023). Complex dynamics of a predator–prey system with fear and memory in the presence of two discrete delays. The European Physical Journal Plus, 138(11), 984. https://doi.org/10.1140/epjp/s13360-023-04614-w

Pal, S., Pal, N., Samanta, S., & Chattopadhyay, J. (2019). Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model. Mathematical Biosciences and Engineering, 16(5), 5146–5179. https://doi.org/10.3934/mbe.2019258

Panigoro, H. S., Resmawan, R., Rahmi, E., Beta, M. A., & Sidik, A. T. R. (2023). The Existence of a Limit-Cycle of a Discrete-Time Lotka-Volterra Model with Fear Effect and Linear Harvesting. E3S Web of Conferences, 400, 03003. https://doi.org/10.1051/e3sconf/202340003003

Rahmi, E., Darti, I., Suryanto, A., & Trisilowati. (2021). A Modified Leslie–Gower Model Incorporating Beddington–DeAngelis Functional Response, Double Allee Effect and Memory Effect. Fractal and Fractional, 5(3), 84. https://doi.org/10.3390/fractalfract5030084

Santra, P. K. (2021). Fear effect in discrete prey-predator model incorporating square root functional response. Jambura Journal of Biomathematics (JJBM), 2(2), 51–57. https://doi.org/10.34312/jjbm.v2i2.10444

Singh, A., & Malik, P. (2021). Bifurcations in a modified Leslie–Gower predator–prey discrete model with Michaelis–Menten prey harvesting. Journal of Applied Mathematics and Computing, 67(1–2), 143–174. https://doi.org/10.1007/s12190-020-01491-9

Singh, M. K., Singh, B. K., Poonam, & Cattani, C. (2023). Under nonlinear prey-harvesting, effect of strong Allee effect on the dynamics of a modified Leslie-Gower predator-prey model. Mathematical Biosciences and Engineering, 20(6), 9625–9644. https://doi.org/10.3934/mbe.2023422

Sun, Y., Zhao, M., & Du, Y. (2023). Multiple bifurcations of a discrete modified Leslie-Gower predator-prey model. Mathematical Biosciences and Engineering, 20(12), 20437–20467. https://doi.org/10.3934/mbe.2023904

Wang, X., Zanette, L., & Zou, X. (2016). Modelling the fear effect in predator–prey interactions. Journal of Mathematical Biology, 73(5), 1179–1204. https://doi.org/10.1007/s00285-016-0989-1

Wang, X., & Zou, X. (2017). Modeling the Fear Effect in Predator–Prey Interactions with Adaptive Avoidance of Predators. Bulletin of Mathematical Biology, 79(6), 1325–1359. https://doi.org/10.1007/s11538-017-0287-0

DOI: https://doi.org/10.31764/jtam.v9i1.26515