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November  2017, 22(9): 3409-3420. doi: 10.3934/dcdsb.2017172

Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Mingxin Wang

Received  March 2016 Revised  May 2017 Published  July 2017

Fund Project: This work was supported by NSFC Grant 11371113.

This paper is devoted to study the dynamical properties of a Leslie-Gower prey-predator system with strong Allee effect in prey. We first gives some estimates, and then study the dynamical properties of solutions. In particular, we mainly investigate the unstable and stable manifolds of the positive equilibrium when the system has only one positive equilibrium.

Citation: Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172
References:
[1]

W. C. Allee, Animal Aggregations, A Study in General Sociology, The University of Chicago Press, 1931.

[2]

A. A. Berryman, The orgins and evolution of predator-prey theory, Ecology, 73 (1992), 1530-1535.  doi: 10.2307/1940005.

[3]

D. S. BoukalM. W. Sabelis and L. Berec, How predator functional responses and allee effects in prey affect the paradox of enrichment and population collapses, Theoretical Population Biology, 72 (2007), 136-147.  doi: 10.1016/j.tpb.2006.12.003.

[4]

F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001.

[5]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X.

[6]

A. M. KramerB. DennisA. M. Liebhold and J. M. Drake, The evidence for allee effects, Population Ecology, 51 (2009), 341-354.  doi: 10.1007/s10144-009-0152-6.

[7]

P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234.  doi: 10.1093/biomet/47.3-4.219.

[8]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 2001.

[9]

E. C. Pielou, An Introduction to Mathematical Ecology, Wiley Interscience. John Wiley and Sons, New York, 1969.

[10]

P. A. Stephens and W. J. Sutherland, Consequences of the allee effect for behaviour, ecology and conservation, Trends in Ecology and Evolution, 14 (1999), 401-405.  doi: 10.1016/S0169-5347(99)01684-5.

[11]

G. A. K. Van VoornL. HemerikM. P. Boer and B. W. Kooi, Heteroclinic orbits indicate overexploitation in predator--prey systems with a strong allee effect, Mathematical Biosciences, 209 (2007), 451-469.  doi: 10.1016/j.mbs.2007.02.006.

[12]

J. F. WangJ. P. Shi and J. J. Wei, Predator-prey system with strong allee effect in prey, Journal of Mathematical Biology, 62 (2011), 291-331.  doi: 10.1007/s00285-010-0332-1.

[13]

M. H. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Math. Biosci., 171 (2001), 83-97.  doi: 10.1016/S0025-5564(01)00048-7.

show all references

References:
[1]

W. C. Allee, Animal Aggregations, A Study in General Sociology, The University of Chicago Press, 1931.

[2]

A. A. Berryman, The orgins and evolution of predator-prey theory, Ecology, 73 (1992), 1530-1535.  doi: 10.2307/1940005.

[3]

D. S. BoukalM. W. Sabelis and L. Berec, How predator functional responses and allee effects in prey affect the paradox of enrichment and population collapses, Theoretical Population Biology, 72 (2007), 136-147.  doi: 10.1016/j.tpb.2006.12.003.

[4]

F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001.

[5]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X.

[6]

A. M. KramerB. DennisA. M. Liebhold and J. M. Drake, The evidence for allee effects, Population Ecology, 51 (2009), 341-354.  doi: 10.1007/s10144-009-0152-6.

[7]

P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234.  doi: 10.1093/biomet/47.3-4.219.

[8]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 2001.

[9]

E. C. Pielou, An Introduction to Mathematical Ecology, Wiley Interscience. John Wiley and Sons, New York, 1969.

[10]

P. A. Stephens and W. J. Sutherland, Consequences of the allee effect for behaviour, ecology and conservation, Trends in Ecology and Evolution, 14 (1999), 401-405.  doi: 10.1016/S0169-5347(99)01684-5.

[11]

G. A. K. Van VoornL. HemerikM. P. Boer and B. W. Kooi, Heteroclinic orbits indicate overexploitation in predator--prey systems with a strong allee effect, Mathematical Biosciences, 209 (2007), 451-469.  doi: 10.1016/j.mbs.2007.02.006.

[12]

J. F. WangJ. P. Shi and J. J. Wei, Predator-prey system with strong allee effect in prey, Journal of Mathematical Biology, 62 (2011), 291-331.  doi: 10.1007/s00285-010-0332-1.

[13]

M. H. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Math. Biosci., 171 (2001), 83-97.  doi: 10.1016/S0025-5564(01)00048-7.

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