# American Institute of Mathematical Sciences

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

## Phase portraits of predator--prey systems with harvesting rates

 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.
Citation: K. Q. Lan, C. R. Zhu. Phase portraits of predator--prey systems with harvesting rates. Discrete and Continuous Dynamical Systems, 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), 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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##### 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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