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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), 196-208.
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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