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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), 196-208. 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), 431-450.  Google Scholar

[3]

V. Capasso and R. E. Wilson, Analysis of a reaction-diffusion system modeling manenvironment-manepidemics, SIAM J. Appl. Math., 57 (1997), 327-346. 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), 541-548. doi: 10.1137/0512047.  Google Scholar

[5]

K. P. Hadeler, Quiescent phases and stability, Linear Alge. Appl., 428 (2008), 1620-1627. 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, Springer, (2007), 7-23. 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) 473-499.  Google Scholar

[8]

T. Hillen, Transport equations with resting phases, European J. Appl. Math., 14 (2003), 613-636. doi: 10.1017/S0956792503005291.  Google Scholar

[9]

S. B. Hsu, On global stability of a predator-prey system, Math. Biosci., 39 (1978), 1-10. 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), 151-173.  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), 893-911. 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), 225-239. 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), 1-25.  Google Scholar

[14]

M. G. Neubert, P. Klepac and P. Driessche, Stabilizing dispersal delays in predator-prey metapopulation models, Theor. Popul. Biol., 61 (2002), 339-347. 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), 231-252. 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), 377-406. Google Scholar

[17]

M. L. Rosenzweig, Paradox of enrichment: Destablization of exploitation ecosystems in ecological time, Science, 171 (1971), 385-387. 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, 2. Springer-Verlag, New York, 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), 1944-1977. 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), 1029-1043. 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), 196-208. 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), 431-450.  Google Scholar

[3]

V. Capasso and R. E. Wilson, Analysis of a reaction-diffusion system modeling manenvironment-manepidemics, SIAM J. Appl. Math., 57 (1997), 327-346. 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), 541-548. doi: 10.1137/0512047.  Google Scholar

[5]

K. P. Hadeler, Quiescent phases and stability, Linear Alge. Appl., 428 (2008), 1620-1627. 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, Springer, (2007), 7-23. 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) 473-499.  Google Scholar

[8]

T. Hillen, Transport equations with resting phases, European J. Appl. Math., 14 (2003), 613-636. doi: 10.1017/S0956792503005291.  Google Scholar

[9]

S. B. Hsu, On global stability of a predator-prey system, Math. Biosci., 39 (1978), 1-10. 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), 151-173.  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), 893-911. 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), 225-239. 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), 1-25.  Google Scholar

[14]

M. G. Neubert, P. Klepac and P. Driessche, Stabilizing dispersal delays in predator-prey metapopulation models, Theor. Popul. Biol., 61 (2002), 339-347. 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), 231-252. 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), 377-406. Google Scholar

[17]

M. L. Rosenzweig, Paradox of enrichment: Destablization of exploitation ecosystems in ecological time, Science, 171 (1971), 385-387. 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, 2. Springer-Verlag, New York, 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), 1944-1977. 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), 1029-1043. doi: 10.1098/rspa.2006.1806.  Google Scholar

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