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.
Share and Cite
ISRP Style
Hafizul Molla, Sahabuddin Sarwardi, Mohammad Sajid, Predator-prey dynamics with Allee effect on predator species subject to intra-specific competition and nonlinear prey refuge, Journal of Mathematics and Computer Science, 25 (2022), no. 2, 150--165
AMA Style
Molla Hafizul, Sarwardi Sahabuddin, Sajid Mohammad, Predator-prey dynamics with Allee effect on predator species subject to intra-specific competition and nonlinear prey refuge. J Math Comput SCI-JM. (2022); 25(2):150--165
Chicago/Turabian Style
Molla, Hafizul, Sarwardi, Sahabuddin, Sajid, Mohammad. "Predator-prey dynamics with Allee effect on predator species subject to intra-specific competition and nonlinear prey refuge." Journal of Mathematics and Computer Science, 25, no. 2 (2022): 150--165
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)
-
[2]
W. C. Allee, The social life of animals, William Heinemann, London (1938)
-
[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
-
[4]
A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific Publishing Co., River Edge (1998)
-
[5]
L. Berec, E. Angulo, F. Courchamp, Multiple Allee effects and population management, Trends Ecology Evol., 22 (2007), 185--191
-
[6]
G. Birkhoff, G. C. Rota, Ordinary differential equations, Wiley, New York (1975)
-
[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
-
[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
-
[9]
F. Courchamp, T. Clutton-Brock, B. Grenfell, Inverse density dependence and the Allee effect, Trends Ecology Evol., 14 (1999), 405--410
-
[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
-
[11]
J. K. Hale, Analytic theory of differential equations, Appl. Math. Sci., 1971 (1971), 9--22
-
[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
-
[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
-
[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
-
[15]
T. K. Kar, Stability analysis of a prey-predator model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Simul., 10 (2005), 681--691
-
[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
-
[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
-
[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
-
[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
-
[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
-
[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
-
[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
-
[23]
J. D. Murray, Mathematical biology, Springer-Verlag, Berlin (1989)
-
[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
-
[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
-
[26]
L. Perko, Differential equations and dynamical systems, Springer-Verlag, New York (2001)
-
[27]
C. Rebelo, C. Soresina, Persistence in seasonally varying predator-prey systems with Allee effect, arXiv, 2019 (2019), 26 pages
-
[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
-
[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
-
[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
-
[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
-
[32]
S. Sarwardi, P. K. Mandal, S. Ray, Analysis of a competitive prey-predator system with a prey refuge, Biosystems, 110 (2012), 133--148
-
[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
-
[34]
P. A. Stephens, W. J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends Ecol. Evol., 14 (1999), 401--405
-
[35]
M. Teixeira Alves, F. M. Hilker, Hunting cooperation and Allee effects in predators, J. Theoret. Biol., 419 (2017), 13--22
-
[36]
A. J. Terry, Predator--prey models with component Allee effect for predator reproduction, J. Math. Biol., 71 (2015), 1325--1352
-
[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
-
[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
-
[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
-
[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
-
[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
-
[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
-
[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
-
[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
-
[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
-
[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