Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model

Moitri Sen; Malay Banerjee; Yasuhiro Takeuchi; Moitri Sen; Malay Banerjee; Yasuhiro Takeuchi

doi:10.3934/mbe.2018040

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 4: 883-904. doi: 10.3934/mbe.2018040

Previous Article Next Article

Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model

1.
Department of Mathematics, National Institute of Technology, Patna, Bihar-800005, India
2.
Department of Mathematics & Statistics, Indian Institute of Technology Kanpur, Uttar Pradesh-208016, India
3.
Department of Physics and Mathematics, Aoyama Gakuin University, Kanagawa, Japan

Received: 26 June 2017 Accepted: 25 November 2017 Published: 01 August 2018
MSC : Primary: 34D20, 34F10; Secondary: 92D25

One of the important ecological challenges is to capture the complex dynamics and understand the underlying regulating ecological factors. Allee effect is one of the important factors in ecology and taking it into account can cause significant changes to the system dynamics. In this work we consider a two prey-one predator model where the growth of both the prey population is subjected to Allee effect, and the predator is generalist as it survives on both the prey populations. We analyze the role of Allee effect on the dynamics of the system, knowing the dynamics of the model without Allee effect. Interestingly we have observed through a comprehensive bifurcation study that incorporation of Allee effect enriches the local as well as the global dynamics of the system. Specially after a certain threshold value of the Allee effect, it has a very significant effect on the chaotic dynamics of the system. In course of the bifurcation analysis we have explored all possible bifurcations such as the existence of transcritical bifurcation, saddle-node bifurcation, Hopf-bifurcation, Bogdanov-Takens bifurcation and Bautin bifurcation and period-doubling route to chaos respectively.
- Prey-predator,
- Allee effect,
- stability,
- bifurcation,
- chaos
Citation: Moitri Sen, Malay Banerjee, Yasuhiro Takeuchi. Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model[J]. Mathematical Biosciences and Engineering, 2018, 15(4): 883-904. doi: 10.3934/mbe.2018040

Related Papers:

Abstract

One of the important ecological challenges is to capture the complex dynamics and understand the underlying regulating ecological factors. Allee effect is one of the important factors in ecology and taking it into account can cause significant changes to the system dynamics. In this work we consider a two prey-one predator model where the growth of both the prey population is subjected to Allee effect, and the predator is generalist as it survives on both the prey populations. We analyze the role of Allee effect on the dynamics of the system, knowing the dynamics of the model without Allee effect. Interestingly we have observed through a comprehensive bifurcation study that incorporation of Allee effect enriches the local as well as the global dynamics of the system. Specially after a certain threshold value of the Allee effect, it has a very significant effect on the chaotic dynamics of the system. In course of the bifurcation analysis we have explored all possible bifurcations such as the existence of transcritical bifurcation, saddle-node bifurcation, Hopf-bifurcation, Bogdanov-Takens bifurcation and Bautin bifurcation and period-doubling route to chaos respectively.

References

[1]	[ P. Aguirre,E. González-Olivares,E. Sáez, Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM Journal on Applied Mathematics, 69 (2009): 1244-1262.
[2]	[ P. Aguirre,E. González-Olivares,E. Sáez, Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Analysis: Real World Applications, 10 (2009): 1401-1416.
[3]	[ W. C. Allee, Animal Aggregations: A study in general sociology, The Quarterly Review of Biology, 2 (1927): 367-398.
[4]	[ L. Berec,E. Angulo,F. Courchamp, Multiple Allee effects and population management, Trends in Ecology & Evolution, 22 (2007): 185-191.
[5]	[ F. Berezovskaya,S. Wirkus,B. Song,C. Castillo-Chavez, Dynamics of population communities with prey migrations and Allee effects: a bifurcation approach, Mathematical Medicine and Biology, 28 (2011): 129-152.
[6]	[ E. D. Conway,J. A. Smoller, Global analysis of a system of predator-prey equations, SIAM Journal on Applied Mathematics, 46 (1986): 630-642.
[7]	[ F. Courchamp,T. Clutton-Brock,B. Grenfell, Inverse density dependence and the Allee effect, Trends in Ecology & Evolution, 14 (1999): 405-410.
[8]	[ B. Dennis, Allee effects: Population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3 (1989): 481-538.
[9]	[ Y. C. Lai and R. L. Winslow, Geometric properties of the chaotic saddle responsible for supertransients in spatiotemporal chaotic systems Physical Review Letters, 74 (1995), p5208.
[10]	[ M. A. Lewis,P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993): 141-158.
[11]	[ A. J. Lotka, A Natural Population Norm I & Ⅱ, 1913.
[12]	[ A. Morozov,S. Petrovskii,B.-L. Li, Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect, Journal of Theoretical Biology, 238 (2006): 18-35.
[13]	[ A. Y. Morozov,M. Banerjee,S. V. Petrovskii, Long-term transients and complex dynamics of a stage-structured population with time delay and the Allee effect, Journal of Theoretical Biology, 396 (2016): 116-124.
[14]	[ M. Sen,M. Banerjee,A. Morozov, Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect, Ecological Complexity, 11 (2012): 12-27.
[15]	[ M. Sen,M. Banerjee, Rich global dynamics in a prey-predator model with Allee effect and density dependent death rate of predator, International Journal of Bifurcation and Chaos, 25 (2015): 17pp.
[16]	[ P. A. Stephens,W. J. Sutherland, Consequences of the allee effect for behaviour, ecology and conservation, Trends in Ecology & Evolution, 14 (1999): 401-405.
[17]	[ Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems, World Scientific, 1996.
[18]	[ Y. Takeuchi,N. Adachi, Existence and bifurcation of stable equilibrium in two-prey, one-predator communities, Bulletin of Mathematical Biology, 45 (1983): 877-900.
[19]	[ V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 118 (1926): 558-560.
[20]	[ G. Wang,X. G. Liang,F. Z. Wang, The competitive dynamics of populations subject to an Allee effect, Ecological Modelling, 124 (1999): 183-192.
[21]	[ J. Zu,M. Mimura, The impact of Allee effect on a predator-prey system with Holling type ii functional response, Applied Mathematics and Computation, 217 (2010): 3542-3556.

Reader Comments

Your name:*

Email:*
© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)