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The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response
1. | Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, China, China |
2. | Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187 |
References:
[1] |
S. Chen, J. Shi and J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system, Int. J. Bifurcation Chaos., 22 (2012), 1250061.
doi: 10.1142/S0218127412500617. |
[2] |
T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238.
doi: 10.1090/S0002-9947-00-02280-7. |
[3] |
T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.
doi: 10.1006/jmaa.2000.7182. |
[4] |
T. Faria and L. T. Magalh, Normal forms for retarded functional-differential equations with parameters and applications to Hopf bifurcation, J. Differential Equations., 122 (1995), 181-200.
doi: 10.1006/jdeq.1995.1144. |
[5] |
T. Faria and L. T. Magalh, Normal forms for retarded functional-differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations., 122 (1995), 201-224.
doi: 10.1006/jdeq.1995.1145. |
[6] |
B. C. Goodwin, "Temporal Organzization in Cells," Academic Press, London and New York, 1963. |
[7] |
J. K. Hale, "Theory of Functinal Differentail Equations," Second edition. Applied Mathematical Sciences, 3. Springer-Verlag, New York-Heidelberg, 1977. |
[8] |
B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, "Theory and Application of Hopf Bifurcation," Cambridge University Press, Cambridge-New York, 1981. |
[9] |
C. S. Holling, Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91 (1959), 385-398. |
[10] |
G. Hu and W. Li, Hopf bifurcation analysis for a delayed predator-prey system with diffusion effects, Nonl. Anal. Real World Appl., 11 (2010), 819-826.
doi: 10.1016/j.nonrwa.2009.01.027. |
[11] |
G. E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246. |
[12] |
W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type-II functional response incorporating a prey refuge, J. Differential Equations., 231 (2006), 534-550.
doi: 10.1016/j.jde.2006.08.001. |
[13] |
X. Lin, J. W.-H. So and J. Wu, Center manifolds for partial differential equations with delay, Proc. Roy. Soc. Edinburgh., 122 A (1992), 237-254.
doi: 10.1017/S0308210500021090. |
[14] |
R. M. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900-902. |
[15] |
A. B. Medvinsky, S. V. Petrovskii, I. A. Tikhonova, H. Malchow and B.-L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev., 44 (2002), 900-902.
doi: 10.1137/S0036144502404442. |
[16] |
R. Peng and J. Shi, Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differential Equations., 247 (2009), 866-886.
doi: 10.1016/j.jde.2009.03.008. |
[17] |
M. L. Rosenzweig and R. MacArthur, Graphical representation and stability conditions of predator-prey interaction, Amer. Natur., 97 (1963), 209-223. |
[18] |
S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874. |
[19] |
H. Thieme and X. Zhao, A non-local delayed and diffusive predator-prey model, Nonl. Anal. Real World Appl., 2 (2001), 145-160.
doi: 10.1016/S0362-546X(00)00112-7. |
[20] |
P. K. C. Wang, Asymptotic stability of a time-delayed diffusion system, J. Appl. Mech. Ser., E 30 (1963), 500-504.
doi: 10.1115/1.3636609. |
[21] |
Y. Wang, Asymptotic behavior of solutions for a class of predator-prey reaction-diffusion systems with time delays, J. Math. Anal. Appl., 328 (2007), 137-150.
doi: 10.1016/j.jmaa.2006.05.02. |
[22] |
J. Wu, "Theory and Applications of Partial Functional-Differential Equations," Applied Mathematical Sciences, 119. Springer, New York, 1996. |
[23] |
D. Xiao and S. Ruan, Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response, J. Differential Equations., 2001 (2009), 494-510.
doi: 10.1006/jdeq.2000.3982. |
[24] |
X. Yan, Stability and Hopf bifurcation for a delayed prey-predator system with diffusion effects, Appl. Math. Comput., 192 (2007), 137-150.
doi: 10.1016/j.amc.2007.03.033. |
[25] |
X. Yan and C. Zhang, Asymptotic stability of positive equilibrium solution for a delayed prey-predator diffusion system, Applied Mathematical Modelling, 34 (2010), 184-199.
doi: 10.1016/j.apm.2009.03.040. |
[26] |
F. Yi, J. Wei and J. 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. |
[27] |
J. Zhang, W. Li and X. Yan, Multiple bifurcations in a delayed predator-prey diffusion system with a functional response, Nonl. Anal. Real World Appl., 11 (2010), 2708-2725.
doi: 10.1016/j.nonrwa.2009.09.019. |
show all references
References:
[1] |
S. Chen, J. Shi and J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system, Int. J. Bifurcation Chaos., 22 (2012), 1250061.
doi: 10.1142/S0218127412500617. |
[2] |
T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238.
doi: 10.1090/S0002-9947-00-02280-7. |
[3] |
T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.
doi: 10.1006/jmaa.2000.7182. |
[4] |
T. Faria and L. T. Magalh, Normal forms for retarded functional-differential equations with parameters and applications to Hopf bifurcation, J. Differential Equations., 122 (1995), 181-200.
doi: 10.1006/jdeq.1995.1144. |
[5] |
T. Faria and L. T. Magalh, Normal forms for retarded functional-differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations., 122 (1995), 201-224.
doi: 10.1006/jdeq.1995.1145. |
[6] |
B. C. Goodwin, "Temporal Organzization in Cells," Academic Press, London and New York, 1963. |
[7] |
J. K. Hale, "Theory of Functinal Differentail Equations," Second edition. Applied Mathematical Sciences, 3. Springer-Verlag, New York-Heidelberg, 1977. |
[8] |
B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, "Theory and Application of Hopf Bifurcation," Cambridge University Press, Cambridge-New York, 1981. |
[9] |
C. S. Holling, Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91 (1959), 385-398. |
[10] |
G. Hu and W. Li, Hopf bifurcation analysis for a delayed predator-prey system with diffusion effects, Nonl. Anal. Real World Appl., 11 (2010), 819-826.
doi: 10.1016/j.nonrwa.2009.01.027. |
[11] |
G. E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246. |
[12] |
W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type-II functional response incorporating a prey refuge, J. Differential Equations., 231 (2006), 534-550.
doi: 10.1016/j.jde.2006.08.001. |
[13] |
X. Lin, J. W.-H. So and J. Wu, Center manifolds for partial differential equations with delay, Proc. Roy. Soc. Edinburgh., 122 A (1992), 237-254.
doi: 10.1017/S0308210500021090. |
[14] |
R. M. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900-902. |
[15] |
A. B. Medvinsky, S. V. Petrovskii, I. A. Tikhonova, H. Malchow and B.-L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev., 44 (2002), 900-902.
doi: 10.1137/S0036144502404442. |
[16] |
R. Peng and J. Shi, Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differential Equations., 247 (2009), 866-886.
doi: 10.1016/j.jde.2009.03.008. |
[17] |
M. L. Rosenzweig and R. MacArthur, Graphical representation and stability conditions of predator-prey interaction, Amer. Natur., 97 (1963), 209-223. |
[18] |
S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874. |
[19] |
H. Thieme and X. Zhao, A non-local delayed and diffusive predator-prey model, Nonl. Anal. Real World Appl., 2 (2001), 145-160.
doi: 10.1016/S0362-546X(00)00112-7. |
[20] |
P. K. C. Wang, Asymptotic stability of a time-delayed diffusion system, J. Appl. Mech. Ser., E 30 (1963), 500-504.
doi: 10.1115/1.3636609. |
[21] |
Y. Wang, Asymptotic behavior of solutions for a class of predator-prey reaction-diffusion systems with time delays, J. Math. Anal. Appl., 328 (2007), 137-150.
doi: 10.1016/j.jmaa.2006.05.02. |
[22] |
J. Wu, "Theory and Applications of Partial Functional-Differential Equations," Applied Mathematical Sciences, 119. Springer, New York, 1996. |
[23] |
D. Xiao and S. Ruan, Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response, J. Differential Equations., 2001 (2009), 494-510.
doi: 10.1006/jdeq.2000.3982. |
[24] |
X. Yan, Stability and Hopf bifurcation for a delayed prey-predator system with diffusion effects, Appl. Math. Comput., 192 (2007), 137-150.
doi: 10.1016/j.amc.2007.03.033. |
[25] |
X. Yan and C. Zhang, Asymptotic stability of positive equilibrium solution for a delayed prey-predator diffusion system, Applied Mathematical Modelling, 34 (2010), 184-199.
doi: 10.1016/j.apm.2009.03.040. |
[26] |
F. Yi, J. Wei and J. 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. |
[27] |
J. Zhang, W. Li and X. Yan, Multiple bifurcations in a delayed predator-prey diffusion system with a functional response, Nonl. Anal. Real World Appl., 11 (2010), 2708-2725.
doi: 10.1016/j.nonrwa.2009.09.019. |
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