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Phase portraits of predator--prey systems with harvesting rates
1. | Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada, Canada |
References:
[1] |
A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maǐer, "Qualitative Theory of Second-Order Dynamical Systems," Halsted Press (A division of John Wiley & Sons), New York-Toronto, Ont., Israel Program for Scientific Translations, Jerusalem-London, 1973. |
[2] |
R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theoret. Biol., 139 (1989), 311-326.
doi: 10.1016/S0022-5193(89)80211-5. |
[3] |
J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.
doi: 10.2307/3866. |
[4] |
F. Berezovskaya, G. Karev and R. Arditi, Parameteric analysis of the ratio-dependent predator-prey model, J. Math. Biol., 43 (2001), 221-246.
doi: 10.1007/s002850000078. |
[5] |
R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.
doi: 10.1006/jmaa.2000.7343. |
[6] |
C. W. Clark, "Mathmatics Bioeconomics, The Optimal Management of Renewable Resources," Second edition, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1990. |
[7] |
C. Cosner, D. L. DeAngelis, J. S. Ault and D. B. Olson, Effects of spatial grouping on the functional response of predators, Theoret. Popul. Biol., 56 (1999), 65-75.
doi: 10.1006/tpbi.1999.1414. |
[8] |
D. L. DeAngelis, R. A. Goldstein and R. V. O'Neil, A model for trophic interaction, Ecology, 56 (1975), 881-892.
doi: 10.2307/1936298. |
[9] |
J. Dieudonné, "Foundations of Modern Analsis," Pure and Applied Mathematics, Vol. 10-I, Academic Press, New York-London, 1969. |
[10] |
D. T. Dimitrov and H. V. Kojouharov, Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis functional response, Applied Math. Comput., 162 (2005), 523-538. |
[11] |
M. Fan and K. Wang, Optimal harvesting policy for single population with periodic coefficients, Math. Biosci., 15 (1998), 165-177.
doi: 10.1016/S0025-5564(98)10024-X. |
[12] |
M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 295 (2004), 15-39.
doi: 10.1016/j.jmaa.2004.02.038. |
[13] |
S.-B. Hsu, T.-W. Hwang and Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol., 42 (2001), 489-506.
doi: 10.1007/s002850100079. |
[14] |
T.-W. Hwang, Global analysis of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 281 (2003), 395-401. |
[15] |
T.-W. Hwang, Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 290 (2004), 113-122.
doi: 10.1016/j.jmaa.2003.09.073. |
[16] |
Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.
doi: 10.1007/s002850050105. |
[17] |
S. Liu and E. Beretta, A stage-structured predator-prey model of Beddington-DeAngelis type, SIAM J. Appl. Math., 66 (2006), 1101-1129.
doi: 10.1137/050630003. |
[18] |
Z. Liu and R. Yuan, Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 296 (2004), 521-537.
doi: 10.1016/j.jmaa.2004.04.051. |
[19] |
L. Perko, "Differential Equations and Dynamical Systems," Second edition, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 1996. |
[20] |
G. T. Skalski and J. F. Gilliam, Functional responses with predator interference: Viable alternatives to the Holling type II model, Ecology, 82 (2001), 3083-3092.
doi: 10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2. |
[21] |
D. Xiao and L. S. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753.
doi: 10.1137/S0036139903428719. |
[22] |
D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43 (2001), 268-290.
doi: 10.1007/s002850100097. |
show all references
References:
[1] |
A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maǐer, "Qualitative Theory of Second-Order Dynamical Systems," Halsted Press (A division of John Wiley & Sons), New York-Toronto, Ont., Israel Program for Scientific Translations, Jerusalem-London, 1973. |
[2] |
R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theoret. Biol., 139 (1989), 311-326.
doi: 10.1016/S0022-5193(89)80211-5. |
[3] |
J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.
doi: 10.2307/3866. |
[4] |
F. Berezovskaya, G. Karev and R. Arditi, Parameteric analysis of the ratio-dependent predator-prey model, J. Math. Biol., 43 (2001), 221-246.
doi: 10.1007/s002850000078. |
[5] |
R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.
doi: 10.1006/jmaa.2000.7343. |
[6] |
C. W. Clark, "Mathmatics Bioeconomics, The Optimal Management of Renewable Resources," Second edition, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1990. |
[7] |
C. Cosner, D. L. DeAngelis, J. S. Ault and D. B. Olson, Effects of spatial grouping on the functional response of predators, Theoret. Popul. Biol., 56 (1999), 65-75.
doi: 10.1006/tpbi.1999.1414. |
[8] |
D. L. DeAngelis, R. A. Goldstein and R. V. O'Neil, A model for trophic interaction, Ecology, 56 (1975), 881-892.
doi: 10.2307/1936298. |
[9] |
J. Dieudonné, "Foundations of Modern Analsis," Pure and Applied Mathematics, Vol. 10-I, Academic Press, New York-London, 1969. |
[10] |
D. T. Dimitrov and H. V. Kojouharov, Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis functional response, Applied Math. Comput., 162 (2005), 523-538. |
[11] |
M. Fan and K. Wang, Optimal harvesting policy for single population with periodic coefficients, Math. Biosci., 15 (1998), 165-177.
doi: 10.1016/S0025-5564(98)10024-X. |
[12] |
M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 295 (2004), 15-39.
doi: 10.1016/j.jmaa.2004.02.038. |
[13] |
S.-B. Hsu, T.-W. Hwang and Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol., 42 (2001), 489-506.
doi: 10.1007/s002850100079. |
[14] |
T.-W. Hwang, Global analysis of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 281 (2003), 395-401. |
[15] |
T.-W. Hwang, Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 290 (2004), 113-122.
doi: 10.1016/j.jmaa.2003.09.073. |
[16] |
Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.
doi: 10.1007/s002850050105. |
[17] |
S. Liu and E. Beretta, A stage-structured predator-prey model of Beddington-DeAngelis type, SIAM J. Appl. Math., 66 (2006), 1101-1129.
doi: 10.1137/050630003. |
[18] |
Z. Liu and R. Yuan, Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 296 (2004), 521-537.
doi: 10.1016/j.jmaa.2004.04.051. |
[19] |
L. Perko, "Differential Equations and Dynamical Systems," Second edition, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 1996. |
[20] |
G. T. Skalski and J. F. Gilliam, Functional responses with predator interference: Viable alternatives to the Holling type II model, Ecology, 82 (2001), 3083-3092.
doi: 10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2. |
[21] |
D. Xiao and L. S. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753.
doi: 10.1137/S0036139903428719. |
[22] |
D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43 (2001), 268-290.
doi: 10.1007/s002850100097. |
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