American Institute of Mathematical Sciences

Advanced
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

[1]

Haiyin Li, Yasuhiro Takeuchi. Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1117-1134. doi: 10.3934/dcdsb.2015.20.1117
[2]

Xiao He, Sining Zheng. Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4641-4657. doi: 10.3934/dcdsb.2020117
[3]

Seong Lee, Inkyung Ahn. Diffusive predator-prey models with stage structure on prey and beddington-deangelis functional responses. Communications on Pure & Applied Analysis, 2017, 16 (2) : 427-442. doi: 10.3934/cpaa.2017022
[4]

Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035
[5]

Zengji Du, Xiao Chen, Zhaosheng Feng. Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1203-1214. doi: 10.3934/dcdss.2014.7.1203
[6]

C. R. Zhu, K. Q. Lan. Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 289-306. doi: 10.3934/dcdsb.2010.14.289
[7]

Walid Abid, Radouane Yafia, M.A. Aziz-Alaoui, Habib Bouhafa, Azgal Abichou. Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type. Evolution Equations & Control Theory, 2015, 4 (2) : 115-129. doi: 10.3934/eect.2015.4.115
[8]

Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597
[9]

Jinliang Wang, Jiying Lang, Xianning Liu. Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3215-3233. doi: 10.3934/dcdsb.2015.20.3215
[10]

Renji Han, Binxiang Dai, Lin Wang. Delay induced spatiotemporal patterns in a diffusive intraguild predation model with Beddington-DeAngelis functional response. Mathematical Biosciences & Engineering, 2018, 15 (3) : 595-627. doi: 10.3934/mbe.2018027
[11]

Sze-Bi Hsu, Shigui Ruan, Ting-Hui Yang. On the dynamics of two-consumers-one-resource competing systems with Beddington-DeAngelis functional response. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2331-2353. doi: 10.3934/dcdsb.2013.18.2331
[12]

Shanshan Chen, Junping Shi, Junjie Wei. The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response. Communications on Pure & Applied Analysis, 2013, 12 (1) : 481-501. doi: 10.3934/cpaa.2013.12.481
[13]

Xin Jiang, Zhikun She, Shigui Ruan. Global dynamics of a predator-prey system with density-dependent mortality and ratio-dependent functional response. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020041
[14]

Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875
[15]

Gianni Gilioli, Sara Pasquali, Fabrizio Ruggeri. Nonlinear functional response parameter estimation in a stochastic predator-prey model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 75-96. doi: 10.3934/mbe.2012.9.75
[16]

Haiying Jing, Zhaoyu Yang. The impact of state feedback control on a predator-prey model with functional response. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 607-614. doi: 10.3934/dcdsb.2004.4.607
[17]

Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214
[18]

Wan-Tong Li, Yong-Hong Fan. Periodic solutions in a delayed predator-prey models with nonmonotonic functional response. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 175-185. doi: 10.3934/dcdsb.2007.8.175
[19]

Prabir Panja, Soovoojeet Jana, Shyamal kumar Mondal. Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020033
[20]

Yongli Cai, Malay Banerjee, Yun Kang, Weiming Wang. Spatiotemporal complexity in a predator--prey model with weak Allee effects. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1247-1274. doi: 10.3934/mbe.2014.11.1247

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (86)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences