American Institute of Mathematical Sciences

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March  2012, 32(3): 901-933. doi: 10.3934/dcds.2012.32.901

Phase portraits of predator--prey systems with harvesting rates

K. Q. Lan 1, and  C. R. Zhu 1,

1. 

Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada, Canada

Received  October 2010 Revised  April 2011 Published  October 2011

We investigate positive equilibria and phase portraits of predator-prey systems with constant harvesting rates arising in ecology. These systems are generalizations of the well--known predator--prey systems with Beddington--DeAngelis functional responses. We seek the ranges of the five parameters involved for which the equilibria of the systems to be positive and obtain all the positive equilibria of the systems. We prove that these positive equilibria are saddles, topological saddles, nodes, saddle-nodes, foci, centers, or cusps by providing suitable ranges of the five parameters. These results show how the harvesting rate and another parameter used in the Beddington--DeAngelis functional responses affect the dynamical behaviors of these systems. In particular, if the harvesting rate is larger than $1/4$ or the parameter just mentioned is too large, then the mutual extinction occurs.
Keywords: Predator--prey system, positive equilibria, harvesting rate, phase portraits., Beddington--DeAngelis functional response.
Mathematics Subject Classification: Primary: 34C20; Secondary: 34C23, 34K18, 37N25, 92D40, 92B0.
Citation: K. Q. Lan, C. R. Zhu. Phase portraits of predator--prey systems with harvesting rates. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 901-933. doi: 10.3934/dcds.2012.32.901
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), (1973).   Google Scholar

[2]

R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence,, J. Theoret. Biol., 139 (1989), 311.  doi: 10.1016/S0022-5193(89)80211-5.  Google Scholar

[3]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency,, J. Animal Ecol., 44 (1975), 331.  doi: 10.2307/3866.  Google Scholar

[4]

F. Berezovskaya, G. Karev and R. Arditi, Parameteric analysis of the ratio-dependent predator-prey model,, J. Math. Biol., 43 (2001), 221.  doi: 10.1007/s002850000078.  Google Scholar

[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.  doi: 10.1006/jmaa.2000.7343.  Google Scholar

[6]

C. W. Clark, "Mathmatics Bioeconomics, The Optimal Management of Renewable Resources,", Second edition, (1990).   Google Scholar

[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.  doi: 10.1006/tpbi.1999.1414.  Google Scholar

[8]

D. L. DeAngelis, R. A. Goldstein and R. V. O'Neil, A model for trophic interaction,, Ecology, 56 (1975), 881.  doi: 10.2307/1936298.  Google Scholar

[9]

J. Dieudonné, "Foundations of Modern Analsis,", Pure and Applied Mathematics, (1969).   Google Scholar

[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.   Google Scholar

[11]

M. Fan and K. Wang, Optimal harvesting policy for single population with periodic coefficients,, Math. Biosci., 15 (1998), 165.  doi: 10.1016/S0025-5564(98)10024-X.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2004.02.038.  Google Scholar

[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.  doi: 10.1007/s002850100079.  Google Scholar

[14]

T.-W. Hwang, Global analysis of the predator-prey system with Beddington-DeAngelis functional response,, J. Math. Anal. Appl., 281 (2003), 395.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2003.09.073.  Google Scholar

[16]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system,, J. Math. Biol., 36 (1998), 389.  doi: 10.1007/s002850050105.  Google Scholar

[17]

S. Liu and E. Beretta, A stage-structured predator-prey model of Beddington-DeAngelis type,, SIAM J. Appl. Math., 66 (2006), 1101.  doi: 10.1137/050630003.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2004.04.051.  Google Scholar

[19]

L. Perko, "Differential Equations and Dynamical Systems,", Second edition, 7 (1996).   Google Scholar

[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.  doi: 10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2.  Google Scholar

[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.  doi: 10.1137/S0036139903428719.  Google Scholar

[22]

D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system,, J. Math. Biol., 43 (2001), 268.  doi: 10.1007/s002850100097.  Google Scholar

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), (1973).   Google Scholar

[2]

R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence,, J. Theoret. Biol., 139 (1989), 311.  doi: 10.1016/S0022-5193(89)80211-5.  Google Scholar

[3]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency,, J. Animal Ecol., 44 (1975), 331.  doi: 10.2307/3866.  Google Scholar

[4]

F. Berezovskaya, G. Karev and R. Arditi, Parameteric analysis of the ratio-dependent predator-prey model,, J. Math. Biol., 43 (2001), 221.  doi: 10.1007/s002850000078.  Google Scholar

[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.  doi: 10.1006/jmaa.2000.7343.  Google Scholar

[6]

C. W. Clark, "Mathmatics Bioeconomics, The Optimal Management of Renewable Resources,", Second edition, (1990).   Google Scholar

[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.  doi: 10.1006/tpbi.1999.1414.  Google Scholar

[8]

D. L. DeAngelis, R. A. Goldstein and R. V. O'Neil, A model for trophic interaction,, Ecology, 56 (1975), 881.  doi: 10.2307/1936298.  Google Scholar

[9]

J. Dieudonné, "Foundations of Modern Analsis,", Pure and Applied Mathematics, (1969).   Google Scholar

[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.   Google Scholar

[11]

M. Fan and K. Wang, Optimal harvesting policy for single population with periodic coefficients,, Math. Biosci., 15 (1998), 165.  doi: 10.1016/S0025-5564(98)10024-X.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2004.02.038.  Google Scholar

[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.  doi: 10.1007/s002850100079.  Google Scholar

[14]

T.-W. Hwang, Global analysis of the predator-prey system with Beddington-DeAngelis functional response,, J. Math. Anal. Appl., 281 (2003), 395.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2003.09.073.  Google Scholar

[16]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system,, J. Math. Biol., 36 (1998), 389.  doi: 10.1007/s002850050105.  Google Scholar

[17]

S. Liu and E. Beretta, A stage-structured predator-prey model of Beddington-DeAngelis type,, SIAM J. Appl. Math., 66 (2006), 1101.  doi: 10.1137/050630003.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2004.04.051.  Google Scholar

[19]

L. Perko, "Differential Equations and Dynamical Systems,", Second edition, 7 (1996).   Google Scholar

[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.  doi: 10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2.  Google Scholar

[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.  doi: 10.1137/S0036139903428719.  Google Scholar

[22]

D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system,, J. Math. Biol., 43 (2001), 268.  doi: 10.1007/s002850100097.  Google Scholar

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