American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On the uniqueness of weak solution for the 2-D Ericksen--Leslie system
  • DCDS-B Home
  • This Issue
  • Next Article
    Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions
May  2016, 21(3): 909-918. doi: 10.3934/dcdsb.2016.21.909

Dynamics in a Rosenzweig-Macarthur predator-prey system with quiescence

Jinfeng Wang 1, and  Hongxia Fan 2,

1. 

School of Mathematical Sciences, Harbin Normal University, Harbin, Heilongjiang, 150025, China
2. 

Department of Basic Science, Harbin University of Commerce, Harbin, Heilongjiang, 150001, China

Received  July 2015 Revised  September 2015 Published  January 2016

A system of four coupled ordinary differential equations is considered, which are coupled through migration of both prey and predator model with logistic type growth. Combined effect of quiescence provides a more realistic way of modeling the complex dynamical behavior. The global stability and Hopf bifurcation solutions are investigated.
Keywords: quiescence, Predator-prey, four coupled, stability, Hopf bifurcation..
Mathematics Subject Classification: Primary: 34D20; Secondary: 37G1.
Citation: Jinfeng Wang, Hongxia Fan. Dynamics in a Rosenzweig-Macarthur predator-prey system with quiescence. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 909-918. doi: 10.3934/dcdsb.2016.21.909
References:
[1]

L. Bilinsky and K. P. Hadeler, Quiescence stabiizes predator prey relations,, J. Biol. Dyna., 3 (2009), 196. doi: 10.1080/17513750802590707. Google Scholar

[2]

V. Capasso and K. Kunisch, A reaction-diffusion system arising in modelling manenvironment diseases,, Quart. Appl. Math., 46 (1988), 431. Google Scholar

[3]

V. Capasso and R. E. Wilson, Analysis of a reaction-diffusion system modeling manenvironment-manepidemics,, SIAM J. Appl. Math., 57 (1997), 327. doi: 10.1137/S0036139995284681. Google Scholar

[4]

K. S. Cheng, Uniqueness of a limit cycle for a predator-prey system,, SIAM J. Math. Anal., 12 (1981), 541. doi: 10.1137/0512047. Google Scholar

[5]

K. P. Hadeler, Quiescent phases and stability,, Linear Alge. Appl., 428 (2008), 1620. doi: 10.1016/j.laa.2007.10.008. Google Scholar

[6]

K. P. Hadeler and T. Hillen, Coupled dynamics and quiescent phases, math everywhere deterministic and stochastic modelling in biomedicine,, Economy and Industry, (2007), 7. doi: 10.1007/978-3-540-44446-6_2. Google Scholar

[7]

K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment,, Canad. Appl. Math. Quart., 10 (2002), 473. Google Scholar

[8]

T. Hillen, Transport equations with resting phases,, European J. Appl. Math., 14 (2003), 613. doi: 10.1017/S0956792503005291. Google Scholar

[9]

S. B. Hsu, On global stability of a predator-prey system,, Math. Biosci., 39 (1978), 1. doi: 10.1016/0025-5564(78)90025-1. Google Scholar

[10]

S. B. Hsu, A survey of constructing lyapunov functions for mathematical models in population biology,, Taiwanese Journal of Mathematics, 9 (2005), 151. Google Scholar

[11]

S. B. Hsu and J. P. Shi, Relaxation oscillator profile of limit cycle in predator-prey system,, Discrete Continuous Dynam. Systems - B, 11 (2009), 893. doi: 10.3934/dcdsb.2009.11.893. Google Scholar

[12]

W. Jager, S. Kroker and B. Tang, Quiescence and transient growth dynamics in chemostat models,, Math. Biosci., 119 (1994), 225. doi: 10.1016/0025-5564(94)90077-9. Google Scholar

[13]

M. A. Lewis and G. Schmitz, Biological invasion of an organism with separate mobile and stationary states: modeling and analysis,, Forma, 11 (1996), 1. Google Scholar

[14]

M. G. Neubert, P. Klepac and P. Driessche, Stabilizing dispersal delays in predator-prey metapopulation models,, Theor. Popul. Biol., 61 (2002), 339. doi: 10.1006/tpbi.2002.1578. Google Scholar

[15]

T. Malik and H. L. Smith, A resource-based model of microbial quiescence,, J. Math. Biol., 53 (2006), 231. doi: 10.1007/s00285-006-0003-4. Google Scholar

[16]

D. H. Robert, Spatial heterogeneity, indirect interaction and the coexistence of prey species,, The American Naturalist, 124 (1984), 377. Google Scholar

[17]

M. L. Rosenzweig, Paradox of enrichment: Destablization of exploitation ecosystems in ecological time,, Science, 171 (1971), 385. doi: 10.1126/science.171.3969.385. Google Scholar

[18]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,, Second edition. Texts in Applied Mathematics, (2003). doi: 10.1007/978-1-4757-4067-7. Google Scholar

[19]

F. Q. Yi, J. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system,, J. Differential Equations, 246 (2009), 1944. doi: 10.1016/j.jde.2008.10.024. Google Scholar

[20]

K. F. Zhang and X. Q. Zhao, Asymptotic behaviour of a reaction-diffusion model with a quiescent stage,, Proc. R. Soc. A, 463 (2007), 1029. doi: 10.1098/rspa.2006.1806. Google Scholar

show all references

References:
[1]

L. Bilinsky and K. P. Hadeler, Quiescence stabiizes predator prey relations,, J. Biol. Dyna., 3 (2009), 196. doi: 10.1080/17513750802590707. Google Scholar

[2]

V. Capasso and K. Kunisch, A reaction-diffusion system arising in modelling manenvironment diseases,, Quart. Appl. Math., 46 (1988), 431. Google Scholar

[3]

V. Capasso and R. E. Wilson, Analysis of a reaction-diffusion system modeling manenvironment-manepidemics,, SIAM J. Appl. Math., 57 (1997), 327. doi: 10.1137/S0036139995284681. Google Scholar

[4]

K. S. Cheng, Uniqueness of a limit cycle for a predator-prey system,, SIAM J. Math. Anal., 12 (1981), 541. doi: 10.1137/0512047. Google Scholar

[5]

K. P. Hadeler, Quiescent phases and stability,, Linear Alge. Appl., 428 (2008), 1620. doi: 10.1016/j.laa.2007.10.008. Google Scholar

[6]

K. P. Hadeler and T. Hillen, Coupled dynamics and quiescent phases, math everywhere deterministic and stochastic modelling in biomedicine,, Economy and Industry, (2007), 7. doi: 10.1007/978-3-540-44446-6_2. Google Scholar

[7]

K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment,, Canad. Appl. Math. Quart., 10 (2002), 473. Google Scholar

[8]

T. Hillen, Transport equations with resting phases,, European J. Appl. Math., 14 (2003), 613. doi: 10.1017/S0956792503005291. Google Scholar

[9]

S. B. Hsu, On global stability of a predator-prey system,, Math. Biosci., 39 (1978), 1. doi: 10.1016/0025-5564(78)90025-1. Google Scholar

[10]

S. B. Hsu, A survey of constructing lyapunov functions for mathematical models in population biology,, Taiwanese Journal of Mathematics, 9 (2005), 151. Google Scholar

[11]

S. B. Hsu and J. P. Shi, Relaxation oscillator profile of limit cycle in predator-prey system,, Discrete Continuous Dynam. Systems - B, 11 (2009), 893. doi: 10.3934/dcdsb.2009.11.893. Google Scholar

[12]

W. Jager, S. Kroker and B. Tang, Quiescence and transient growth dynamics in chemostat models,, Math. Biosci., 119 (1994), 225. doi: 10.1016/0025-5564(94)90077-9. Google Scholar

[13]

M. A. Lewis and G. Schmitz, Biological invasion of an organism with separate mobile and stationary states: modeling and analysis,, Forma, 11 (1996), 1. Google Scholar

[14]

M. G. Neubert, P. Klepac and P. Driessche, Stabilizing dispersal delays in predator-prey metapopulation models,, Theor. Popul. Biol., 61 (2002), 339. doi: 10.1006/tpbi.2002.1578. Google Scholar

[15]

T. Malik and H. L. Smith, A resource-based model of microbial quiescence,, J. Math. Biol., 53 (2006), 231. doi: 10.1007/s00285-006-0003-4. Google Scholar

[16]

D. H. Robert, Spatial heterogeneity, indirect interaction and the coexistence of prey species,, The American Naturalist, 124 (1984), 377. Google Scholar

[17]

M. L. Rosenzweig, Paradox of enrichment: Destablization of exploitation ecosystems in ecological time,, Science, 171 (1971), 385. doi: 10.1126/science.171.3969.385. Google Scholar

[18]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,, Second edition. Texts in Applied Mathematics, (2003). doi: 10.1007/978-1-4757-4067-7. Google Scholar

[19]

F. Q. Yi, J. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system,, J. Differential Equations, 246 (2009), 1944. doi: 10.1016/j.jde.2008.10.024. Google Scholar

[20]

K. F. Zhang and X. Q. Zhao, Asymptotic behaviour of a reaction-diffusion model with a quiescent stage,, Proc. R. Soc. A, 463 (2007), 1029. doi: 10.1098/rspa.2006.1806. Google Scholar

[1]

Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979
[2]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507
[3]

Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002
[4]

Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031
[5]

Leonid Braverman, Elena Braverman. Stability analysis and bifurcations in a diffusive predator-prey system. Conference Publications, 2009, 2009 (Special) : 92-100. doi: 10.3934/proc.2009.2009.92
[6]

Sílvia Cuadrado. Stability of equilibria of a predator-prey model of phenotype evolution. Mathematical Biosciences & Engineering, 2009, 6 (4) : 701-718. doi: 10.3934/mbe.2009.6.701
[7]

Zhong Li, Maoan Han, Fengde Chen. Global stability of a predator-prey system with stage structure and mutual interference. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 173-187. doi: 10.3934/dcdsb.2014.19.173
[8]

Wei Feng, Michael T. Cowen, Xin Lu. Coexistence and asymptotic stability in stage-structured predator-prey models. Mathematical Biosciences & Engineering, 2014, 11 (4) : 823-839. doi: 10.3934/mbe.2014.11.823
[9]

Antoni Leon Dawidowicz, Anna Poskrobko. Stability problem for the age-dependent predator-prey model. Evolution Equations & Control Theory, 2018, 7 (1) : 79-93. doi: 10.3934/eect.2018005
[10]

Wei Feng, Jody Hinson. Stability and pattern in two-patch predator-prey population dynamics. Conference Publications, 2005, 2005 (Special) : 268-279. doi: 10.3934/proc.2005.2005.268
[11]

Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019214
[12]

Michael Y. Li, Xihui Lin, Hao Wang. Global Hopf branches and multiple limit cycles in a delayed Lotka-Volterra predator-prey model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 747-760. doi: 10.3934/dcdsb.2014.19.747
[13]

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
[14]

Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041
[15]

Jicai Huang, Yijun Gong, Shigui Ruan. Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2101-2121. doi: 10.3934/dcdsb.2013.18.2101
[16]

Qing Zhu, Huaqin Peng, Xiaoxiao Zheng, Huafeng Xiao. Bifurcation analysis of a stage-structured predator-prey model with prey refuge. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2195-2209. doi: 10.3934/dcdss.2019141
[17]

Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045
[18]

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
[19]

Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065
[20]

Zhifu Xie. Turing instability in a coupled predator-prey model with different Holling type functional responses. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1621-1628. doi: 10.3934/dcdss.2011.4.1621

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences