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Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response
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A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced
Codimension 3 B-T bifurcations in an epidemic model with a nonlinear incidence
1. | School of Mathematical Sciences and LMAM, Peking University, Beijing, 100871 |
2. | Faculty of Science, Air Force Engineering University, Xi'an 710051 |
3. | School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China |
References:
[1] |
M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96.
doi: 10.1016/j.mbs.2004.01.003. |
[2] |
F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemics, Springer-Verlag, New York, 2000. |
[3] |
L. Cai, G. Chen and D. Xiao, Multiparametric bifurcations of an epidemiological model with strong Allee effect, J. Math. Biol., 67 (2013), 185-215.
doi: 10.1007/s00285-012-0546-5. |
[4] |
S.-N. Chow, C. Li and D. Wang, Normal Forms and Bifurcations of Planar Vector Fields, Cambridge University Press, 1994.
doi: 10.1017/CBO9780511665639. |
[5] |
C. Christopher and C. Li, Limit Cycles of Differential Equations, Birkhäuser Verlag, 2007. |
[6] |
J. Cui,, X. Mu and H. Wan, Saturation recovery leads to multiple endemic equilibria and backward bifurcation, J. Theoret. Biol., 254 (2008), 275-283.
doi: 10.1016/j.jtbi.2008.05.015. |
[7] |
F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter family of vector feilds on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergod. Theor. & Dyn. Sys., 7 (1987), 375-413.
doi: 10.1017/S0143385700004119. |
[8] |
H. W. Hethcote, Mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[9] |
J. Huang, Y. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2101-2121.
doi: 10.3934/dcdsb.2013.18.2101. |
[10] |
C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.
doi: 10.1016/j.jde.2012.10.003. |
[11] |
J. Li, Y. Zhou, J. Wu and Z. Ma, Complex dynamics of a simple epidemic model with a nonlinear incidence, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 161-173.
doi: 10.3934/dcdsb.2007.8.161. |
[12] |
W. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological model with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.
doi: 10.1007/BF00277162. |
[13] |
W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.
doi: 10.1007/BF00276956. |
[14] |
Z. Ma and J. Li, Dynamical Modeling and Analysis of Epidemics, World Scientific, Singapore, 2009.
doi: 10.1142/9789812797506. |
[15] |
S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163.
doi: 10.1016/S0022-0396(02)00089-X. |
[16] |
Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in an SIRS model with nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639.
doi: 10.1137/070700966. |
[17] |
H. Zhu, S. A. Campbell and G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63 (2002), 636-682.
doi: 10.1137/S0036139901397285. |
show all references
References:
[1] |
M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96.
doi: 10.1016/j.mbs.2004.01.003. |
[2] |
F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemics, Springer-Verlag, New York, 2000. |
[3] |
L. Cai, G. Chen and D. Xiao, Multiparametric bifurcations of an epidemiological model with strong Allee effect, J. Math. Biol., 67 (2013), 185-215.
doi: 10.1007/s00285-012-0546-5. |
[4] |
S.-N. Chow, C. Li and D. Wang, Normal Forms and Bifurcations of Planar Vector Fields, Cambridge University Press, 1994.
doi: 10.1017/CBO9780511665639. |
[5] |
C. Christopher and C. Li, Limit Cycles of Differential Equations, Birkhäuser Verlag, 2007. |
[6] |
J. Cui,, X. Mu and H. Wan, Saturation recovery leads to multiple endemic equilibria and backward bifurcation, J. Theoret. Biol., 254 (2008), 275-283.
doi: 10.1016/j.jtbi.2008.05.015. |
[7] |
F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter family of vector feilds on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergod. Theor. & Dyn. Sys., 7 (1987), 375-413.
doi: 10.1017/S0143385700004119. |
[8] |
H. W. Hethcote, Mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[9] |
J. Huang, Y. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2101-2121.
doi: 10.3934/dcdsb.2013.18.2101. |
[10] |
C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.
doi: 10.1016/j.jde.2012.10.003. |
[11] |
J. Li, Y. Zhou, J. Wu and Z. Ma, Complex dynamics of a simple epidemic model with a nonlinear incidence, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 161-173.
doi: 10.3934/dcdsb.2007.8.161. |
[12] |
W. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological model with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.
doi: 10.1007/BF00277162. |
[13] |
W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.
doi: 10.1007/BF00276956. |
[14] |
Z. Ma and J. Li, Dynamical Modeling and Analysis of Epidemics, World Scientific, Singapore, 2009.
doi: 10.1142/9789812797506. |
[15] |
S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163.
doi: 10.1016/S0022-0396(02)00089-X. |
[16] |
Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in an SIRS model with nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639.
doi: 10.1137/070700966. |
[17] |
H. Zhu, S. A. Campbell and G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63 (2002), 636-682.
doi: 10.1137/S0036139901397285. |
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