zbMATH — the first resource for mathematics

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

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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.