Complex dynamics in a discrete-time model of two competing prey with a shared predator

Debasis Mukherjee

Abstract


This paper concentrates on the study of a discrete time model of two competing prey with a shared predator. The condition for the existence and local stability of positive fixed point are derived. By using an iteration scheme and the comparison principle of difference equations, it is possible to obtain the sufficient condition for global stability of the positive fixed point. The sufficient criterion for Neimark-Sacker bifurcation and flip bifurcation are established. The system admits chaotic dynamics for a certain choice of the system parameters which is controlled by applying hybrid control method. The intra-specific competition among predators and the intrinsic growth rate of prey species have major impact for different bifurcation. For continuous system, handling time spent for prey population plays an important role for obtaining limit cycle behaviour. The decrease amount of this rate makes the system stable. Global convergence of the solutions to the coexistence equilibrium point is possible for a particular choice of system parameters. The obtained results for discrete system are verified through numerical simulations. Also some diagrams are presented for continuous system.

Keywords


Predator-prey model; stability; bifurcation; chaos control

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References


A. Klebanoff and A. Hastings, "Chaos in one-predator, two-prey models: cGeneral results from bifurcation theory," Mathematical Biosciences, vol. 122, no. 2, pp. 221–233, 1994. DOI:10.1016/0025-5564(94)90059-0

M. F. Elettreby, "Two-prey one-predator model," Chaos, Solitons & Fractals, vol. 39, no. 5, pp. 2018–2027, 2009. DOI:10.1016/j.chaos.2007.06.058

M. Fan, B. Zhang, and M. Y. Li, "Mechanisms for stable coexistence in insect community," Mathematical Biosciences and Engineering, vol. 7, no. 3, pp. 603–622, 2010. DOI:10.3934/mbe.2010.7.603

D. Mukherjee, "The effect of refuge and immigration in a predator–prey system in the presence of a competitor for the prey," Nonlinear Analysis: Real World Applications, vol. 31, pp. 277–287, 2016. DOI:10.1016/j.nonrwa.2016.02.004

R. Colucci, E´ . Diz-Pita, and M. V. Otero-Espinar, "Dynamics of a Two Prey and One Predator System with Indirect Effect,"Mathematics, vol. 9, no. 4, p. 436, 2021. DOI:10.3390/math9040436.

E´ . Diz-Pita and M. V. Otero-Espinar, "Predator–Prey Models: A Review of Some Recent Advances," Mathematics, vol. 9, no. 15, p. 1783, 2021. DOI:10.3390/math9151783

S. Gakkhar and R. K. Naji, "Existence of chaos in two-prey, one-predator system," Chaos, Solitons & Fractals, vol. 17, no. 4, pp. 639–649, 2003. DOI:10.1016/S0960-0779(02)00473-3

D. Mukherjee, "Co-existence of competing prey with a shared predator," Mathematical and Computer Modelling of Dynamical Systems, vol. 11, no. 1, pp. 111–121, 2005. DOI:10.1080/13873950500052538

J. P. Tripathi, S. Abbas, and M. Thakur, "Local and global stability analysis of a two prey one predator model with help," Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 9, pp. 3284-3297, 2014. DOI:10.1016/j.cnsns.2014.02.003

A. Bhattacharya and A. K. Pal, "Complex dynamics of delay induced two-prey one-predator model with Beddington-DeAngelis response function," Nonlinear Studies, vol. 28, no. 2, 2021.

M. R. Khalaf and S. J. Majeed, "Dynamical analysis of a twon prey-one predator model," Asia Pacific Journal of Mathematics, vol. 11, p. 91, 2024. DOI:10.28924/APJM/11-91

H. N. Agiza et al., "Chaotic dynamics of a discrete prey–predator model with Holling type II," Nonlinear Anal. Real World Appl, vol. 10, no. 1, pp. 116–129, 2009. DOI:10.1016/j.nonrwa.2007.08.029

Q. Din, "Complexity and chaos control in a discrete-time prey-predator model," Communication in Nonlinear Science and Numerical Simulations, vol. 49, pp. 113–134, 2017. DOI:10.1016/j.cnsns.2017.01.025

M. E. Elettreby, T. Nabil, and A. Khawagi, "Stability and Bifurcation Analysis of a Discrete Predator-Prey Model with Mixed Holling Interaction," Computer Modeling in Engineering & Sciences, vol. 122, no. 3, pp. 907–921, 2020. DOI:10.32604/cmes.2020.08664

Z. M. He and X. Lai, "Bifurcations and chaotic behavior of a discrete-time predator-prey system," Nonlinear Analysis RWA, vol. 12, no. 1, pp. 403–417, 2011. DOI:10.1016/j.nonrwa.2010.06.026

M. Zhao, Z. Xuan, and C. Li, "Dynamics of a discrete-time predator-prey system," Advances in Difference Equations, vol. 2016, no. 1, p. 191, 2016. DOI:10.1186/s13662-016-0903-6

Z. He and B. Li, "Complex dynamic behavior of a discrete-time predator-prey system of Holling-III type," Advances in Difference Equations, vol. 2014, no. 1, pp. 1–12, 2014. DOI:10.1186/1687-1847-2014-180

P. Santra, G. S. Mahapatra, and G. Phaijoo, "Bifurcation and Chaos of a Discrete Predator-Prey Model with Crowley–Martin Functional Response Incorporating Proportional Prey Refuge," Mathematical Problems in Engineering, vol. 2020, pp. 1–18, 2020. DOI:10.1155/2020/5309814

H. Seno, "A discrete prey–predator model preserving the dynamics of a structurally unstable Lotka–Volterra model," Journal of Difference Equations and Applications, vol. 13, no. 12, pp. 1155–1170, 2007. DOI:10.1080/10236190701464996

J. Chen, X. He, and F. Chen, "The Influence of Fear Effect to a Discrete-Time Predator-Prey System with Predator Has Other Food Resource," Mathematics, vol. 9, no. 8, p. 865, 2021. DOI:10.3390/math9080865.

D. Mukherjee, "Global stability and bifurcation analysis in a discrete-time two prey-one predator model with help," International Journal of Modeling and Simulation, vol. 43, no. 5, pp. 752–763, 2023. DOI:10.1080/02286203.2022.2121676

D. Mukherjee, "Dynamics of A Discrete-Time Ecogenetic Predator-Prey Model," Communication in Biomathematical Sciences, vol. 5, no. 2, pp. 161–169, 2022. DOI:10.5614/cbms.2022.5.2.5

R. Banerjee et al., "In the presence of fear and refuge: Permanence, bifurcation and chaos control of a discrete-time ecological system," International Journal of Modeling, Simulation, and Scientific Computing, vol. 14, no. 03, 2023. DOI:10.1142/S1793962323500095.

D. Mukherjee, "Global Stability and Bifurcation Analysis in a Discrete-Time Two Predator-One Prey Model with Michaelis-Menten Type Prey Harvesting," Communications in Advanced Mathematical Sciences, vol. 6, no. 1, pp. 1–18, 2023. DOI:10.33434/cams.1171482

D. Mukherjee, "Qualitative Study of a Discrete-Time Harvested Fishery Model in the Presence of Toxicity," Journal of Mathematical Sciences and Modelling, vol. 6, no. 2, pp. 65–75, 2023. DOI:10.33187/jmsm.1177403

C. Mondal, D. Kesh, and D. Mukherjee, "Bifurcation and global stability of a discrete prey–predator model with saturated prey refuge," Mathematical Methods in Applied Sciences, vol. 46, no. 17, pp. 18354–18374, 2024. DOI:10.1002/mma.9562

C. Mondal, D. Kesh, and D. Mukherjee, "Global stability and bifurcation analysis of an infochemical induced three species discrete-time phytoplankton–zooplankton model," Chaos, Solitons and Fractals, vol. 176, p. 114136, 2023. DOI:10.1016/j.chaos.2023.114136

E. A. Grove and G. Ladas, Periodicities in nonlinear difference equations (Vol. 4), CRC Press, Boca Raton, 2004.

G. Y. Chen and Z. D. Teng, "On the stability in a discrete two-species competition system," Journal of Applied Mathematics and Computing, vol. 38, no. 1–2, pp. 25–39, 2012. DOI:10.1007/s12190-010-0460-1

L. Wang and M. Wang, Ordinary Difference Equations, XinJiang University Press, Urmuqi, 1989.

G. Wen, "Criterion to identify Hopf bifurcations in maps of arbitrary dimension," Physical Review E, vol. 72, no. 2, p. 026201, 2005. DOI:10.1103/PhysRevE.72.026201

G. Wen, S. Chen, and Q. Jin, "A new criterion of period-dubling bifurcation in maps and its application to an internal impact shaker," Journal of Sound Vibration, vol. 311, pp. 212–223, 2008.

X. S. Luo et al., "Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems," Chaos, Solitons & Fractals, vol. 18, no. 4, pp. 775–783, 2003. DOI:10.1016/S0960-0779(03)00028-6

Q. Din, "Bifurcation analysis and chaos control in discrete-time glycolysis models," Journal of Mathematical Chemistry, vol. 56, no. 3, pp. 904–931, 2018. DOI:10.1007/s10910-017-0839-4

Q. Din, T. Donchev, and D. Kolev, "Stability, bifurcation analysis and chaos control in chlroine dioxide-iodine-malonic acid reaction," MATCH Communications in Mathematical and in Computer Chemistry, vol. 79, pp. 577–606, 2018.

Q. Din, and U. Saeed, "Bifurcation analysis and chaos control in a host-parasitoid model," Mathematical Methods in Applied Sciences, vol. 40, no. 14, pp. 5391–5406, 2017. DOI:10.1002/mma.4395

Y. Zou et al., "Shrimp structure and associated dynamics in parametrically excited oscillators," International Journal of Bifurcation and Chaos, vol. 16, no. 12, pp. 3567–3579, 2006. DOI:10.1142/S0218127406016987




DOI: https://doi.org/10.37905/jjbm.v5i2.27453

Copyright (c) 2024 Debasis Mukherjee

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