Dynamic complexity of a discrete predator-prey model with prey refuge and herd behavior
DOI:
https://doi.org/10.21015/vtm.v11i1.1512Abstract
This paper examines a discrete predator-prey model's complex dynamics. Using bifurcation and center manifold theory, we study period-doubling and Neimark-Sacker bifurcations at positive fixed points and their direction. Numerical simulations confirm the theoretical conclusions that the model's dynamics rely on Euler method step size. The model's behavior is also affected by the prey population's conservation rate. The model suggests that excessive conservation may reduce predator populations, causing food shortages. Thus, predator-prey dynamics management must account for prey conservation rateReferences
Ahmed, R. (2020). 'Complex dynamics of a fractional-order predator-prey interaction with harvesting.' Open Journal of Discrete Applied Mathematics, 3, 24–32. Retrieved from: [link]
Ahmed, R., Ahmad, A., & Ali, N. (2022). 'Stability analysis and neimark-sacker bifurcation of a nonstandard finite difference scheme for lotka-volterra prey-predator model.' Communications in Mathematical Biology and Neuroscience, 2022.
Ahmed, R., Akhtar, S., Farooq, U., & Ali, S. (2023). 'Stability, bifurcation, and chaos control of predator-prey system with additive allee effect.' Communications in Mathematical Biology and Neuroscience, 2023.
Ahmed, R., & Yazdani, M. S. (2022). 'Complex dynamics of a discrete-time model with prey refuge and holling type-ii functional response.' Journal of Mathematical and Computational Science.
Ajraldi, V., Pittavino, M., & Venturino, E. (2011). 'Modeling herd behavior in population systems.' Nonlinear Analysis: Real World Applications, 12, 2319–2338.
Akhtar, S., Ahmed, R., Batool, M., Shah, N., & Chung, J. (2021). 'Stability, bifurcation and chaos control of a discretized leslie prey-predator model.' Chaos, Solitons & Fractals, 152.
Alsakaji, H. J., Kundu, S., & Rihan, F. A. (2021). 'Delay differential model of one-predator two-prey system with monod-haldane and holling type ii functional responses.' Applied Mathematics and Computation, 397, 125919.
AlSharawi, Z., Pal, S., Pal, N., & Chattopadhyay, J. (2020). 'A discrete-time model with non-monotonic functional response and strong allee effect in prey.' Journal of Difference Equations and Applications, 26, 404–431.
Arancibia-Ibarra, C., Aguirre, P., Flores, J. and van Heijster, P. (2021), 'Bifurcation analysis of a predator-prey model with predator intraspecific interactions and ratio-dependent functional response', Applied Mathematics and Computation 402, 126152.
Baydemir, P., Merdan, H., Karaoglu, E. and Sucu, G. (2020), 'Complex dynamics of a discrete-time prey-predator system with leslie type: Stability, bifurcation analyses and chaos', International Journal of Bifurcation and Chaos 30, 2050149.
Beddington, J. R. (1975), 'Mutual interference between parasites or predators and its effect on searching efficiency', The Journal of Animal Ecology 44, 331–340.
Braza, P. A. (2012), 'Predator-prey dynamics with square root functional responses', Nonlinear Analysis: Real World Applications 13, 1837–1843.
Chen, X. and Zhang, X. (2021), 'Dynamics of the predator-prey model with the sigmoid functional response', Studies in Applied Mathematics 147, 300–318.
Crowley, P. H. and Martin, E. K. (1989), 'Functional responses and interference within and between year classes of a dragonfly population', Journal of the North American Benthological Society 8, 211–221.
DeAngelis, D. L., Goldstein, R. A. and O’Neill, R. V. (1975), 'A model for tropic interaction', Ecology 56, 881–892.
Elettreby, M. F., Khawagi, A. and Nabil, T. (2019), 'Dynamics of a discrete prey-predator model with mixed functional response', International Journal of Bifurcation and Chaos 29, 1950199.
Guckenheimer, J. and Holmes, P. (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Vol. 42, Springer New York.
Holling, C. S. (1959), 'Some characteristics of simple types of predation and parasitism', The Canadian Entomologist 91, 385–398.
Hong, B. and Zhang, C. (2023), 'Neimark-Sacker bifurcation of a discrete-time predator-prey model with prey refuge effect', Mathematics 11, 1399.
J., L. A. C. (2012), Regularity and complexity in dynamical systems, Springer.
Khabyah, A. A., Ahmed, R., Akram, M. S. and Akhtar, S. (2023), 'Stability, bifurcation, and chaos control in a discrete predator-prey model with strong Allee effect', AIMS Mathematics 8, 8060–8081.
Khan, A. Q., Ahmad, I., Alayachi, H. S., Noorani, M. S. M. and Khaliq, A. (2020), 'Discrete-time predator-prey model with flip bifurcation and chaos control', Mathematical Biosciences and Engineering 17, 5944–5960.
Liu, W. and Cai, D. (2019), 'Bifurcation, chaos analysis and control in a discrete-time predator-prey system', Advances in Difference Equations 2019, 11.
Luo, X. S., Chen, G., Wang, B. H. and Fang, J. Q. (2003), 'Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems', Chaos, Solitons & Fractals 18, 775–783.
Ma, Z., Li, W., Zhao, Y., Wang, W., Zhang, H. and Li, Z. (2009), 'Effects of prey refuges on a predator-prey model with a class of functional responses: The role of refuges', Mathematical Biosciences 218, 73–79.
Mortuja, M. G., Chaube, M. K. and Kumar, S. (2021), 'Dynamic analysis of a predator-prey system with nonlinear prey harvesting and square root functional response', Chaos, Solitons & Fractals 148, 111071.
Mukherjee, D. (2016), '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 31, 277–287.
Naik, P. A., Eskandari, Z., Yavuz, M. and Zu, J. (2022), 'Complex dynamics of a discrete-time Bazykin-Berezovskaya prey-predator model with a strong Allee effect', Journal of Computational and Applied Mathematics 413, 114401.
Pal, D., Santra, P. and Mahapatra, G. S. (2017), 'Predator-prey dynamical behavior and stability analysis with square root functional response', International Journal of Applied and Computational Mathematics 3, 1833–1845.
Panja, P. (2021), 'Combine effects of square root functional response and prey refuge on predator-prey dynamics', International Journal of Modelling and Simulation 41, 426–433.
Rana, S. M. S. (2019), 'Dynamics and chaos control in a discrete-time ratio-dependent Holling-Tanner model', Journal of the Egyptian Mathematical Society 27, 48.
Rana, S. M. S. and Kulsum, U. (2017), 'Bifurcation analysis and chaos control in a discrete-time predator-prey system of Leslie type with simplified Holling type IV functional response', Discrete Dynamics in Nature and Society 2017, 1–11.
Rayungsari, M., Suryanto, A., Kusumawinahyu, W. M. and Darti, I. (2022), 'Dynamical analysis of a predator-prey model incorporating predator cannibalism and refuge', Axioms 11, 116.
Salman, S., Yousef, A. and Elsadany, A. (2016), 'Stability, bifurcation analysis and chaos control of a discrete predator-prey system with square root functional response', Chaos, Solitons & Fractals 93, 20–31.
Shu, Q. and Xie, J. (2022), 'Stability and bifurcation analysis of discrete predator-prey model with nonlinear prey harvesting and prey refuge', Mathematical Methods in the Applied Sciences 45, 3589–3604.
Tassaddiq, A., Shabbir, M. S., Din, Q. and Naaz, H. (2022), 'Discretization, bifurcation, and control for a class of predator-prey interactions', Fractal and Fractional 6, 31.
Wang, X. and Smit, A. (2023), 'Studying the fear effect in a predator-prey system with apparent competition', Discrete and Continuous Dynamical Systems - B 28, 1393.
Wiggins, S. and Golubitsky, M. (2003), Introduction to Applied Nonlinear Dynamical Systems and Chaos, Vol. 2, Springer-Verlag.
Wu, Y., Chen, F. and Du, C. (2021), 'Dynamic behaviors of a nonautonomous predator-prey system with Holling type II schemes and a prey refuge', Advances in Difference Equations 2021, 62.
Xie, B. and Zhang, N. (2022), 'Influence of fear effect on a Holling type III prey-predator system with the prey refuge', AIMS Mathematics 7, 1811–1830.
Zhao, M., Li, C. and Wang, J. [2017], ‘Complex dynamic behaviors of a discrete-time predator-prey system’, Journal of Applied Analysis & Computation 7, 478–500.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC-By) that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
This work is licensed under a Creative Commons Attribution License CC BY
