Stability and Hopf Bifurcation in Three-Dimensional Predator-Prey Models with Allee Effect

Yıl 2019, Cilt: 23 Sayı: 5, 1005 - 1011, 01.10.2019

İlknur Kuşbeyzi Aybar

https://doi.org/10.16984/saufenbilder.537485

Öz

In this
study, we perform the stability and Hopf bifurcation analysis for two
population models with Allee effect. The population models within the scope of
this study are the one prey-two predator model with Allee growth in the prey
and the two prey-one predator model with Allee growth in the preys. Our
procedure for investigating each model is as follows. First, we investigate the
singular points where the system is stable. We provide the necessary parameter
conditions for the system to be stable at the singular points. Then, we look
for Hopf bifurcation at each singular point where a family of limit cycles
cycle or oscillate. We provide the parameter conditions for Hopf bifurcation to
occur. We apply the algebraic invariants method to fully examine the system. We
investigate the algebraic properties of the system by finding all algebraic
invariants of degree two and three. We give the conditions for the system to
have a first integral.

Anahtar Kelimeler

predator-prey model, stability, Hopf bifurcation

Kaynakça

• A. A. Berryman, “The origins and evolution of predator-prey theory,” Ecology, vol. 73, no. 5, pp. 1530 – 1535, 1992.
• H. W. Hethcote, W. D. Wang, L. T. Han and M. Zhien, “A predator-prey model with infected prey,” Theoretical Population Biology, vol. 66, no. 3, pp. 259—268, 2004.
• I. Kusbeyzi Aybar, O. O. Aybar, M. Dukaric and B. Fercec, “Dynamical analysis of a two prey-one predator system with quadratic self interaction,” Applied Mathematics and Computation, vol. 333, pp. 118—132, 2018.
• X. C. Huang, Y. M. Wang and L. M. Zhu, “One and three limit cycles in a cubic predator-prey system,” Mathematical Methods in the Applied Sciences, vol. 30, no. 5, pp. 501—511, 2007.
• I. Kusbeyzi, O. O. Aybar and A. Hacinliyan, “Stability and bifurcation in two species predator-prey models,” Nonlinear Analysis: Real World Applications, vol. 12, pp. 377—387, 2011.
• X. N. Liu and L. S. Chen, “Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator,” Chaos Solitons and Fractals, vol. 16, no. 2, pp. 311—320, 2003.
• M. A. Aziz-Alaoui and M. D. Okiye, “Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes,” Applied Mathematics Letters, vol. 16, no. 7, pp. 1069—1075, 2003.
• R. S. Cantrell and C. Cosner, “A predator-prey model with infected prey,” Journal of Mathematical Analysis and Applications, vol. 257, no. 1, pp. 206—222, 2001.
• M. Fan and Y. Kuang, “Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 15—39, 2004.
• A. J. Lotka, Elements of Physical Biology, Williams and Wilkins Company, Baltimore, 1925.
• V. Volterra, Variations and fluctuations of a number of individuals in animal species living together, in: Animal Ecology, Chapman, Mcgraw-Hill, New York, 1931.
• E. Venturino, “Epidemics in predator-prey models: disease in the predators,” IMA Journal of Mathematics Applied in Medicine and Biology, vol. 19, no. 3, pp. 185—205, 2002.
• G. T. Skalski and J. F. Gilliam, “Functional responses with predator interference: viable alternatives to the Holling Type II model,” Ecology, vol. 82, no. 11, pp. 3083—3092, 2001.
• C. M. Taylor and A. Hastings, “Allee effects in biological invasions,” Ecology Letters, vol. 8, no. 8, pp. 895-908, 2005.
• N. Knowlton, “Thresholds and Multiple Stable States in Coral-Reef Community Dynamics,” Integrative and Comparative Biology, vol. 32, no. 6, pp. 674—682, 1992.
• J. A. Hutchings, and J. D. Reynolds, “Marine fish population collapses: Consequences for recovery and extinction risk,” Bioscience, vol. 54, no. 4, pp. 297—309, 2004.
• R. A. Myers, N. J. Barrowman, J. A. Hutchings, and A. A. Rosenberg, “Population dynamics of exploited fish stocks at low population levels,” Science, vol. 269, no. 5227, pp. 1106—1108, 1995.
• W.C. Allee, Animal Aggregations: A Study in General Sociology. University of Chicago Press, Chicago, 1931.
• P. A. Stephens, W. J. Sutherland and R. P. Freckleton, “What is the Allee effect?,” Oikos, vol. 87, no. 1, pp. 185—190, 1999.
• F. Courchamp, “Inverse density dependence and the Allee effect,” Trends in Ecology and Evolution, vol. 14, no. 10, 405—410, 1999.
• M. Kot, M. A. Lewis and P. vandenDriessche, “Dispersal data and the spread of invading organisms,” Ecology, vol. 77, no. 7, pp. 2027—2042, 1996.
• F. Courchamp, J. Berec and J. Gascoigne, Allee effects in ecology and conservation, Oxford University Press, New York, 2008.
• V. G. Romanovski and D.S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhauser, Boston-Basel-Berlin, 2009.
• L. Liu, O. O. Aybar, V. G. Romanovski and W. Zhang, “Identifying weak foci and centers in the Maxwell–Bloch system,” Journal of Mathematical Analysis and Applications, vol. 430, no. 1, pp. 549—571, 2015.
• I. Kusbeyzi Aybar, “Stability and Hopf bifurcation of 3D predator-prey models with Allee effect via computational algebra”, International Conference of Mathematical Sciences (ICMS 2018), 31 July–06 August 2018, Maltepe University, Istanbul, Turkey