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Bifurcations and dynamics of a predator-prey model with double Allee effects and time delays. (English) Zbl 1404.34092

MSC:
34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D25 Population dynamics (general)
34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
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[1] Allee, W. C., Animal Aggregations: A Study in General Sociology, (1931), University of Chicago Press, Chicago
[2] Berec, L.; Angulo, E.; Courchamp, F., Multiple allee effects and population management, Ecol. Model., 22, 185-191, (2006)
[3] Bertness, M. D.; Leonard, G. H., The role of positive interactions in communities: lessons from intertidal habitats, Ecology, 78, 1976-1989, (1997)
[4] Callaway, R. M.; Walker, L. R., Competition and facilitation: A synthetic approach to interactions in plant communities, Ecology, 78, 1958-1965, (1997)
[5] Celik, C.; Merdan, H.; Duman, O.; Akin, O., Allee effects on population dynamics with delay, Chaos Solit. Fract., 37, 65-74, (2008) · Zbl 1137.92364
[6] Chen, Y.; Yu, J.; Sun, C., Stability and Hopf bifurcation analysis in a three-level food chain system with delay, Chaos Solit. Fract., 31, 683-694, (2007) · Zbl 1146.34051
[7] Cooke, K. L.; Grossman, Z., Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86, 592-627, (1982) · Zbl 0492.34064
[8] Correigh, M. G., Habitat selection reduces extinction of populations subject to allee effects, Theor. Popul. Biol., 64, 1-10, (2003) · Zbl 1100.92041
[9] Courchamp, F.; Brock, T. C.; Grenfell, B., Inverse density dependence and the allee effect, Trends Ecol. Evol., 14, 405-410, (1999)
[10] Feng, P.; Kang, Y., Dynamics of a modied Leslie-gower model with double allee effects, Nonlin. Dyn., 80, 1051-1062, (2015) · Zbl 1345.92115
[11] Freedman, H. I.; So, J.; Waltman, P., Coexistence in a model of competition in the chemostat incorporating discrete time delays, SIAM J. Appl. Math., 49, 859-870, (1989) · Zbl 0676.92013
[12] Gopalsamy, K., Stability and Oscillation in Delay Differential Equation of Population Dynamics, (1992), Kluwer Academic, Dordrecht · Zbl 0752.34039
[13] Groom, M., Allee effects limit population viability of an annual plant, Am. Nat., 151, 487-496, (1998)
[14] Hacker, S. D.; Gaines, S. D., Some implications of direct positive interactions for species diversity, Ecology, 78, 1990-2003, (1997)
[15] Hurwitz, A., Ueber die bedingungen, unter welchen eine gleichung nur wurzeln mit negativen reellen theilen besitzt, Math. Ann., 46, 273-284, (1895) · JFM 26.0119.03
[16] Kang, Y.; Yakubu, A. A., Weak allee effects and species coexistence, Nonlin. Anal.: Real World Appl., 12, 3329-3345, (2011) · Zbl 1231.39008
[17] Kooten, T. V.; Roos, A. M.; Persson, L., Bistability and an allee effect as emergent consequences of stage-specific predation, J. Theoret. Biol., 203, 67-74, (2005)
[18] Kuang, Y., Delay Differential Equation with Applications in Population Dynamics, (1993), Academic Press, NY
[19] Li, Y. F.; Xie, D. L.; Cui, J. A., Complex dynamics of a predator-prey model with impulsive state feedback control, Appl. Math. Comput., 230, 395-405, (2014) · Zbl 1410.37076
[20] Liu, Z.; Yuan, R., Stability and bifurcation in a delayed predator-prey system with beddington-deangelis functional response, J. Math. Anal. Appl., 296, 521-537, (2004) · Zbl 1051.34060
[21] Liu, X.; Dai, B., Dynamics of a predator-prey model with double allee effects and impulse, Nonlin. Dyn., 88, 685-701, (2017) · Zbl 1373.92105
[22] MacDonald, N., Biological Delay Systems: Linear Stability Theory, (1989), Cambridge University Press, Cambridge · Zbl 0669.92001
[23] Negi, K.; Gakkhar, S., Dynamics in a beddington-deangelis prey-predator system with impulsive harvesting, Ecol. Model., 206, 421-430, (2007)
[24] Pal, P. J.; Saha, T.; Sen, M.; Banerjee, M., A delayed predator-prey model with strong allee effect in prey population growth, Nonlin. Dyn., 68, 23-42, (2012) · Zbl 1242.92060
[25] Routh, E. J., A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion, (1877), Macmillan
[26] Ruan, S.; Wei, J., On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, Math. Med. Biol., 18, 41-52, (2001) · Zbl 0982.92008
[27] Sen, M.; Banerjee, M.; Morozov, A., Bifurcation analysis of a ratio-dependent prey-predator model with the allee effect, Ecol. Complex., 11, 12-27, (2012)
[28] Shen, J.; Liu, Z.; Zheng, W.; Xu, F.; Chen, L., Oscillatory dynamics in a simple gene regulatory network mediated by small rnas, Physica A, 388, 2995-3000, (2009)
[29] Stephens, P. A.; Sutherland, W. J., Consequences of the allee effect for behaviour, ecology and conservation, Trends Ecol. Evol., 14, 401-405, (1999)
[30] Taylor, C. M.; Hastings, A., Allee effects in biological invasions, Ecol. Lett., 8, 895, (2005)
[31] Terry, A. J., Prey resurgence from mortality events in predator-prey models, Nonlin. Anal.: Real World Appl., 14, 2180-2203, (2013) · Zbl 1323.92191
[32] Wang, M. H.; Kot, M., Speeds of invasion in a model with strong or weak allee effects, Math. Biosci., 171, 83-97, (2001) · Zbl 0978.92033
[33] Wang, W.; Zhang, Y.; Liu, C., Analysis of a discrete-time predator-prey system with allee effect, Ecol. Complex., 8, 81-85, (2011)
[34] Xu, R.; Gan, Q.; Ma, Z., Stability and bifurcation analysis on a ratio-dependent predator-prey model with time delay, J. Comput. Appl. Math., 230, 187-203, (2009) · Zbl 1186.34122
[35] Yan, J.; Zhao, A.; Yan, W., Existence and global attractivity of periodic solution for an impulsive delay differential equation with allee effect, J. Math. Anal. Appl., 309, 489-504, (2005) · Zbl 1086.34066
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