Complexity of a Discrete-Time Predator-Prey Model Involving Prey Refuge Proportional to Predator

P. K. Santra, Hasan S. Panigoro, G. S. Mahapatra

Abstract


In this paper, a discrete-time predator-prey model involving prey refuge proportional to predator density is studied. It is assumed that the rate at which prey moves to the refuge is proportional to the predator density. The fixed points, their local stability, and the existence of Neimark-Sacker bifurcation are investigated. At last, the numerical simulations consisting of bifurcation diagrams, phase portraits, and time-series are given to support analytical findings. The occurrence of chaotic solutions are also presented by showing the Lyapunov exponent while some parameters are varied.

Keywords


Chaos; Neimark-Sacker Bifurcation; Predator-Prey; Refuge; Stability

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References


M. G. Neubert and M. Kot, “The subcritical collapse of predator populations in discretetime predator-prey models,” Mathematical Biosciences, vol. 110, no. 1, pp. 45–66, 1992, doi: https://doi.org/10.1016/0025-5564(92)90014-N.

J. Huang, S. Liu, S. Ruan, and D. Xiao, “Bifurcations in a discrete predator–prey model with nonmonotonic functional response,” Journal of Mathematical Analysis and Applications, vol. 464, no. 1, pp. 201–230, 2018, doi: https://doi.org/10.1016/j.jmaa.2018.03.074.

M. Gamez, I. L ´ opez, C. Rodr ´ ´ıguez, Z. Varga, and J. Garay, “Ecological monitoring in a discrete-time prey–predator model,” Journal of Theoretical Biology, vol. 429, pp. 52–60, 2017, doi: https://doi.org/10.1016/j.jtbi.2017.06.025.

A. Gkana and L. Zachilas, “Non-overlapping Generation Species: Complex Prey–Predator Interactions,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 16, no. 5, pp. 207–219, 2015, doi: https://doi.org/10.1515/ijnsns-2014-0121.

E. Sebastian, A. Victor, Preethi Gkana, and L. Zachilas, “The dynamics of a discrete-time ratio-dependent prey-predator model incorporating prey refuge and harvesting on prey,” International Journal of Applied Engineering Research, vol. 10, no. 55, pp. 2385–2388, 2015.

P. K. Santra, G. S. Mahapatra, and G. R. 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: https://doi.org/10.1155/2020/5309814.

H. S. Panigoro, E. Rahmi, N. Achmad, S. L. Mahmud, R. Resmawan, and A. R. Nuha, “A discrete-time fractional-order Rosenzweig-Macarthur predator-prey model involving prey refuge,” Communications in Mathematical Biology and Neuroscience, no. 1925, pp. 1—-19, 2021, doi: https://doi.org/10.28919/cmbn/6586.

P. Santra, “Discrete-time prey-predator model with q-logistic growth for prey incorporating square root functional response,” Jambura Journal of Biomathematics, vol. 1, no. 2, pp. 41–48, 2020, doi: https://doi.org/10.34312/jjbm.v1i2.7660.

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/10.34312/jjbm.v2i2.10444.

P. Santra, G. S. Mahapatra, and D. Pal, “Prey–predator nonlinear harvesting model with functional response incorporating prey refuge,” International Journal of Dynamics and Control, vol. 4, no. 3, pp. 293–302, 2016, doi: https://doi.org/10.1007/s40435-015-0198-6.

Q. Wang, Z. Liu, X. Zhang, and R. A. Cheke, “Incorporating prey refuge into a predator–prey system with imprecise parameter estimates,” Computational and Applied Mathematics, vol. 36, no. 2, pp. 1067–1084, 2017, doi: https://doi.org/10.1007/s40314-015-0282-8.

C. Banerjee and P. Das, “Impulsive Effect on Tri-Trophic Food Chain Model with Mixed Functional Responses under Seasonal Perturbations,” Differential Equations and Dynamical Systems, vol. 26, no. 1-3, pp. 157–176, 2018, doi: https://doi.org/10.1007/s12591-016-0328-4.

B. Sahoo and S. Poria, “Oscillatory Coexistence of Species in a Food Chain Model With General Holling Interactions,” Differential Equations and Dynamical Systems, vol. 22, no. 3, pp. 221–238, 2014, doi: https://doi.org/10.1007/s12591-013-0171-9.

H. S. Panigoro, A. Suryanto, W. M. Kusumahwinahyu, and I. Darti, “Dynamics of a fractional-order predator-prey model with infectious diseases in prey,” Communication in Biomathematical Sciences, vol. 2, no. 2, p. 105, 2019, doi: https://doi.org/10.5614/cbms.2019.2.2.4.

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.

L. K. Beay and M. Saija, “A stage-structure Rosenzweig-MacArthur model with effect of prey refuge,” Jambura Journal of Biomathematics, vol. 1, no. 1, pp. 1–7, 2020, doi: https://doi.org/10.34312/jjbm.v1i1.6891.

Y. Wu, F. Chen, and C. Du, “Dynamic behaviors of a nonautonomous predator–prey system with Holling type II schemes and a prey refuge,” Advances in Difference Equations, vol. 2021, no. 1, pp. 1–15, 2021, doi: https://doi.org/10.1186/s13662-021-03222-1.

Y. Tao, X. Wang, and X. Song, “Effect of prey refuge on a harvested predator–prey model with generalized functional response, ”Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 2, pp. 1052–1059, 2011, doi:https://doi.org/10.1016/j.cnsns.2010.05.026.

H. S. Panigoro, E. Rahmi, N. Achmad, and S. L. Mahmud, “The influence of additive Allee effect and periodic harvesting to the dynamics of Leslie-Gower predator-prey model,” Jambura Journal of Mathematics, vol. 2, no. 2, pp. 87–96, 2020, doi: https://doi.org/10.34312/jjom.v2i2.4566.

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.

G. Ruxton, “Short Term Refuge Use and Stability of Predator-Prey Models,” Theoretical Population Biology, vol. 47, no. 1, pp. 1–17, 1995, doi: https://doi.org/10.1006/tpbi.1995.1001.

R. Cressman and J. Garay, “A predator–prey refuge system: Evolutionary stability in ecological systems,” Theoretical Population Biology, vol. 76, no. 4, pp. 248–257, 2009, doi:https://doi.org/10.1016/j.tpb.2009.08.005.

U. Ufuktepe, B. Kulahcioglu, and O. Akman, “Stability analysis of a prey refuge predator–prey model with Allee effects,” Journal of Biosciences, vol. 44, no. 4, p. 85, 2019, doi:https://doi.org/10.1007/s12038-019-9911-5.

S. Kapc¸ak, S. Elaydi, and U. Ufuktepe, “Stability of a predator–prey model with refuge effect,” Journal of Difference Equations and Applications, vol. 22, no. 7, pp. 989–1004, 2016, doi:https://doi.org/10.1080/10236198.2016.1170823.

A. Gkana and L. Zachilas, “Incorporating prey refuge in a prey–predator model with a Holling type I functional response: random dynamics and population outbreaks,” Journal of Biological Physics, vol. 39, no. 4, pp. 587–606, 2013, doi: https://doi.org/10.1007/s10867-013-9319-7.

F. Chen, L. Chen, and X. Xie, “On a Leslie–Gower predator–prey model incorporating a prey refuge,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 2905–2908, 2009, doi:https://doi.org/10.1016/j.nonrwa.2008.09.009.

Z. Ma, S. Wang, W. Li, and Z. Li, “The effect of prey refuge in a patchy predator–prey system,” Mathematical Biosciences, vol. 243, no. 1, pp. 126–130, 2013, doi: https://doi.org/10.1016/j.mbs.2013.02.011.

G. S. Mahapatra and P. Santra, “Prey–predator model for optimal harvesting with functional response incorporating prey refuge,” International Journal of Biomathematics, vol. 09, no. 01, p.1650014, 2016, doi: https://doi.org/10.1142/S1793524516500145.

R. N. Fan, “A Predator-Prey Model Incorporating Prey Refuge and Allee Effect,” Applied Mechanics and Materials, vol. 713-715, pp. 1534–1539, 2015, doi: https://doi.org/10.4028/www.scientific.net/AMM.713-715.1534.




DOI: https://doi.org/10.34312/jjom.v4i1.11918



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