Predator-prey dynamics with Allee effect on predator species subject to intra-specific competition and nonlinear prey refuge

Volume 25, Issue 2, pp 150--165 http://dx.doi.org/10.22436/jmcs.025.02.04

Publication Date: May 26, 2021 Submission Date: December 05, 2020 Revision Date: February 22, 2021 Accteptance Date: April 20, 2021

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Authors

Hafizul Molla - Department of Mathematics , Manbhum Mahavidyalaya, Purulia - 723 131, West Bengal, India. Sahabuddin Sarwardi - Department of Mathematics \(\&\) Statistics, Aliah University, IIA/27, New Town, Kolkata- 700 160, West Bengal, India. Mohammad Sajid - Department of Mechanical Engineering, College of Engineering, Qassim University, Buraidah-51452, Al-Qassim, P.O. Box: 6677, Kingdom of Saudi Arabia.

Abstract

A modified version of our previously analyzed prey-predator refuge model is presented in this article by introducing Allee effect on the predator species and mutual interference among the predators. Possible number of coexistence equilibrium points are investigated with the help of prey and predator nullcline. The local stability and Hopf-bifurcation conditions are established around the coexistence equilibria. We have also discussed the nature of Hopf-bifurcation around the unique coexistence equilibrium point of the system as well. Finally, a comprehensive numerical simulation is carried out to justify our obtained analytical findings.

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Keywords

• Ecological model
• refuge
• stability
• Hopf-bifurcation
• nature of Hopf-bifurcation
• numerical simulations

MSC

•  92D25
•  92D40
•  34D20
•  37G10
•  37G15
•  34L16

References

• [1] W. C. Allee, Animal aggregations, A study in general sociology, Univ. Chicago Press, Chicago (1931)
  • View Article
  • Google Scholar


• [2] W. C. Allee, The social life of animals, William Heinemann, London (1938)
  • View Article
  • Google Scholar


• [3] J. Banerjee, S. K. Sasmal, R. K. Layek, Supercritical and subcritical Hopf-bifurcations in a two-delayed prey--predator system with density-dependent mortality of predator and strong Allee effect in prey, Biosystems, 180 (2019), 19--37
  • View Article
  • Google Scholar


• [4] A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific Publishing Co., River Edge (1998)
  • View Article
  • MathSciNet
  • Google Scholar


• [5] L. Berec, E. Angulo, F. Courchamp, Multiple Allee effects and population management, Trends Ecology Evol., 22 (2007), 185--191
  • View Article
  • Google Scholar


• [6] G. Birkhoff, G. C. Rota, Ordinary differential equations, Wiley, New York (1975)
  • Google Scholar


• [7] Y. L. Cai, C. D. Zhao, W. M. Wang, J. F. Wang, Dynamics of Leslie-Gower predator-prey Model With Additive Allee Effect, Appl. Math. Model., 39 (2015), 2092--2106
  • View Article
  • MathSciNet
  • Google Scholar


• [8] M. I. S. Costa, L. dos Anjos, Multiple hydra effect in a predator--prey model with Allee effect and mutual interference in the predator, Ecolog. Model., 373 (2018), 22--24
  • View Article
  • Google Scholar


• [9] F. Courchamp, T. Clutton-Brock, B. Grenfell, Inverse density dependence and the Allee effect, Trends Ecology Evol., 14 (1999), 405--410
  • View Article
  • Google Scholar


• [10] E. Gonzalez-Olivares, J. Cabrera-Villegas, F. Cordova-Lepe, A. Rojas-Palma, Competition among Predators and Allee Effect on Prey, Their Influence on a Gause-Type Predation Model, Math. Probl. Eng., 2019 (2019), 19 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [11] J. K. Hale, Analytic theory of differential equations, Appl. Math. Sci., 1971 (1971), 9--22
  • Google Scholar


• [12] C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European Pine Sawfly1, Canad. Entomolog., 91 (1959), 293--320
  • View Article
  • Google Scholar


• [13] C. S. Holling, The functional response of predator to prey density and its role in mimicry and population regulations, Memoirs Entomolog. Soc. Canada, 97 (1965), 5--60
  • View Article
  • Google Scholar


• [14] S. Isik, A study of stability and bifurcation analysis in discrete-time predator-prey system involving the Allee effect, Int. J. Biomath., 12 (2019), 15 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [15] T. K. Kar, Stability analysis of a prey-predator model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Simul., 10 (2005), 681--691
  • View Article
  • MathSciNet
  • Google Scholar


• [16] T. Li, X. Huang, X. Xie, Stability of a stage-structured predator-prey model with Allee effect and harvesting, Commun. Math. Biol. Neurosci., 2019 (2019), 11 pages
  • View Article
  • Google Scholar


• [17] M. Manarul Haque, S. Sarwardi, Dynamics of a harvested prey–predator model with prey refuge dependent on both species, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 16 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [18] N. Min, M. X. Wang, Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey, Discrete Contin. Dyn. Syst., 39 (2019), 1071--1099
  • View Article
  • Google Scholar


• [19] H. Molla, M. S. Rahman, S. Sarwardi, Dynamics of a Predator-Prey Model with Holling Type II Functional Response Incorporating a Prey Refuge Depending on Both the Species, Int. J. Nonlinear Sci. Numer. Simul., 20 (2019), 1--16
  • MathSciNet
  • Google Scholar


• [20] H. Molla, M. S. Rahman, S. Sarwardi, Dynamical study of a prey-predator model incorporating nonlinear prey refuge and additive Allee effect acting on prey species, Model. Earth Syst. Environ., 20 (2020), 1--17
  • View Article
  • Google Scholar


• [21] H. Molla, M. S. Rahman, S. Sarwardi, Incorporating Prey Refuge in a Prey-Predator Model with Beddington-DeAngelis Type Functional Response: A Comparative Study on Intra-Specific Competition, Discontin., Nonlinear. Complex., 9 (2020), 395--419
  • View Article


• [22] D. Mukherjee, The effect of refuge and immigration in a predator–prey system in the presence of a competitor for the prey, Nonlinear Anal. Real World Appl., 31 (2016), 277--287
  • View Article
  • MathSciNet
  • Google Scholar


• [23] J. D. Murray, Mathematical biology, Springer-Verlag, Berlin (1989)
  • View Article
  • MathSciNet
  • Google Scholar


• [24] P. J. Pal, P. K. Mandal, Bifurcation analysis of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and strong Allee effect, Math. Comput. Simulation, 97 (2014), 123--146
  • View Article
  • MathSciNet
  • Google Scholar


• [25] P. J. Pal, T. Saha, M. Sen, M. Banerjee, A delayed predator--prey model with strong Allee effect in prey population growth, Nonlinear Dynam., 68 (2012), 23--42
  • View Article
  • MathSciNet
  • Google Scholar


• [26] L. Perko, Differential equations and dynamical systems, Springer-Verlag, New York (2001)
  • View Article
  • MathSciNet
  • Google Scholar


• [27] C. Rebelo, C. Soresina, Persistence in seasonally varying predator-prey systems with Allee effect, arXiv, 2019 (2019), 26 pages
  • View Article
  • Google Scholar


• [28] S. Saha, A. Maiti, G. P. Samanta, A Michaelis-Menten predator-prey model with strong Allee effect and disease in prey incorporating prey refuge, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 21 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [29] S. Sarwardi, M. M. Haque, S. Hossain, Analysis of Bogdanov-Takens bifurcations in a spatiotemporal harvested-predator and prey system with Beddington-DeAngelis type response function, Nonlinear Dynam., 100 (2020), 1755--1778
  • View Article
  • Google Scholar


• [30] S. Sarwardi, M. Haque, P. K. Mandal, Ratio--dependent predator-prey model of interacting population with delay effect, Nonlinear Dynam., 69 (2012), 817--836
  • View Article
  • MathSciNet
  • Google Scholar


• [31] S. Sarwardi, M. Haque, P. K. Mandal, Persistence and global stability of Bazykin predator-prey model with Beddington-DeAngelis response function, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 189--209
  • View Article
  • MathSciNet
  • Google Scholar


• [32] S. Sarwardi, P. K. Mandal, S. Ray, Analysis of a competitive prey-predator system with a prey refuge, Biosystems, 110 (2012), 133--148
  • View Article
  • Google Scholar


• [33] M. Sen, M. Banerjee, Y. Takeuchi, Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model, Math. Biosci. Eng., 15 (2018), 883--904
  • View Article
  • MathSciNet
  • Google Scholar


• [34] P. A. Stephens, W. J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends Ecol. Evol., 14 (1999), 401--405
  • View Article
  • Google Scholar


• [35] M. Teixeira Alves, F. M. Hilker, Hunting cooperation and Allee effects in predators, J. Theoret. Biol., 419 (2017), 13--22
  • View Article
  • MathSciNet
  • Google Scholar


• [36] A. J. Terry, Predator--prey models with component Allee effect for predator reproduction, J. Math. Biol., 71 (2015), 1325--1352
  • View Article
  • MathSciNet
  • Google Scholar


• [37] U. Ufuktepe, B. Kulahcioglu, O. Akman, Stability analysis of a prey refuge predator--prey model with Allee effects, J. Biosci., 44 (2019), 1--9
  • View Article
  • Google Scholar


• [38] M. Verma, A. K. Misra, Modeling the effect of prey refuge on a ratio-dependent predator--prey system with the Allee effect, Bull. Math. Biol., 80 (2018), 626--656
  • View Article
  • MathSciNet
  • Google Scholar


• [39] X. Q. Wang, Y. L. Cai, H. H. Ma, Dynamics of a Diffusive Predator-Prey Model with Allee Effect on Predator, Discrete Dyn. Nat. Soc., 2013 (2013), 10 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [40] J. F. Wang, J. P. Shi, J. J. Wei, Predator prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291--331
  • View Article
  • MathSciNet
  • Google Scholar


• [41] Z. W. Xiao, Z. Li, Stability and Bifurcation in a Stage-structured Predator-prey Model with Allee Effect and Time Delay, IAENG Int. J. Appl. Math., 49 (2019), 6--13
  • View Article
  • MathSciNet
  • Google Scholar


• [42] Z. W. Xiao, X. D. Xie, Y. Xue, Stability and bifurcation in a Holling type II predator--prey model with Allee effect and time delay, Adv. Difference Equ., 2018 (2018), 21 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [43] Y. Ye, H. Liu, Y.-M. Wei, M. Ma, K. Zhang, Dynamic Study of a Predator-Prey Model with Weak Allee Effect and Delay, Adv. Math. Phys., 2019 (2019), 15 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [44] T. T. Yu, Y. Tian, H. J. Guo, X. Y. Song, Dynamical analysis of an integrated pest management predator--prey model with weak Allee effect, J. Biol. Dyn., 13 (2019), 218--244
  • View Article
  • MathSciNet
  • Google Scholar


• [45] L. M. Zhang, C. F. Zhang, Z. R. He, Codimension-one and codimension-two bifurcations of a discrete predator–prey system with strong Allee effect, Math. Comput. Simulation, 162 (2019), 155--178
  • View Article
  • MathSciNet
  • Google Scholar


• [46] J. Zu, Global qualitative analysis of a predator prey system with Allee effect on the prey species, Math. Comput. Simulation, 94 (2013), 33--54
  • View Article
  • MathSciNet
  • Google Scholar