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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

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.
Citation: Jinfeng Wang, Hongxia Fan. Dynamics in a Rosenzweig-Macarthur predator-prey system with quiescence. Discrete and 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.

[2]

V. Capasso and K. Kunisch, A reaction-diffusion system arising in modelling manenvironment diseases, Quart. Appl. Math., 46 (1988), 431-450.

[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.

[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.

[5]

K. P. Hadeler, Quiescent phases and stability, Linear Alge. Appl., 428 (2008), 1620-1627. doi: 10.1016/j.laa.2007.10.008.

[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.

[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.

[8]

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

[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.

[10]

S. B. Hsu, A survey of constructing lyapunov functions for mathematical models in population biology, Taiwanese Journal of Mathematics, 9 (2005), 151-173.

[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.

[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.

[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.

[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.

[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.

[16]

D. H. Robert, Spatial heterogeneity, indirect interaction and the coexistence of prey species, The American Naturalist, 124 (1984), 377-406.

[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.

[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.

[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.

[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.

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.

[2]

V. Capasso and K. Kunisch, A reaction-diffusion system arising in modelling manenvironment diseases, Quart. Appl. Math., 46 (1988), 431-450.

[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.

[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.

[5]

K. P. Hadeler, Quiescent phases and stability, Linear Alge. Appl., 428 (2008), 1620-1627. doi: 10.1016/j.laa.2007.10.008.

[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.

[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.

[8]

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

[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.

[10]

S. B. Hsu, A survey of constructing lyapunov functions for mathematical models in population biology, Taiwanese Journal of Mathematics, 9 (2005), 151-173.

[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.

[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.

[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.

[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.

[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.

[16]

D. H. Robert, Spatial heterogeneity, indirect interaction and the coexistence of prey species, The American Naturalist, 124 (1984), 377-406.

[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.

[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.

[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.

[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.

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