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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 |
References:
[1] |
L. Bilinsky and K. P. Hadeler, Quiescence stabiizes predator prey relations,, J. Biol. Dyna., 3 (2009), 196.
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.
|
[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. |
[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. |
[5] |
K. P. Hadeler, Quiescent phases and stability,, Linear Alge. Appl., 428 (2008), 1620.
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, (2007), 7.
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.
|
[8] |
T. Hillen, Transport equations with resting phases,, European J. Appl. Math., 14 (2003), 613.
doi: 10.1017/S0956792503005291. |
[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. |
[10] |
S. B. Hsu, A survey of constructing lyapunov functions for mathematical models in population biology,, Taiwanese Journal of Mathematics, 9 (2005), 151.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[2] |
V. Capasso and K. Kunisch, A reaction-diffusion system arising in modelling manenvironment diseases,, Quart. Appl. Math., 46 (1988), 431.
|
[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. |
[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. |
[5] |
K. P. Hadeler, Quiescent phases and stability,, Linear Alge. Appl., 428 (2008), 1620.
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, (2007), 7.
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.
|
[8] |
T. Hillen, Transport equations with resting phases,, European J. Appl. Math., 14 (2003), 613.
doi: 10.1017/S0956792503005291. |
[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. |
[10] |
S. B. Hsu, A survey of constructing lyapunov functions for mathematical models in population biology,, Taiwanese Journal of Mathematics, 9 (2005), 151.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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