-
Previous Article
Stability of an efficient Navier-Stokes solver with Navier boundary condition
- DCDS-B Home
- This Issue
-
Next Article
Periodic solutions of a non-divergent diffusion equation with nonlinear sources
Qualitative analysis of a diffusive prey-predator model with trophic interactions of three levels
1. | Department of Mathematics, Southeast University, Nanjing 210096, China |
2. | Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076, Singapore |
3. | Natural Science Research Center, Harbin Institute of Technology, Harbin 150080, China |
References:
[1] |
P. A. Abrams and L. R. Ginzburg, The nature of predation: Prey dependent, ratio dependent or neither?, Trends in Ecology & Evolution, 15 (2000), 337-341.
doi: 10.1016/S0169-5347(00)01908-X. |
[2] |
C. O. Alves and D. G. de Figueiredo, Nonvariational elliptic systems. Current developments in partial differential equations, Discrete Contin. Dyn. Syst., 8 (2002), 289-302. |
[3] |
J. Blat and K. J. Brown, Global bifurcation on positive solutions in some systems of elliptic equations, SIAM. J. Math. Anal., 17 (1986), 1339-1353.
doi: 10.1137/0517094. |
[4] |
G. Buffoni, M. P. Cassinari and M. Groppi, Modelling of predator-prey trophic interactions. Part II: Three trophic levels, J. Math. Biol., 54 (2007), 623-644. |
[5] |
G. Buffoni, M. P. Cassinari, M. Groppi and M. Serluca, Modelling of predator-prey trophic interactions. Part I: Two trophic levels, J. Math. Biol., 50 (2005), 713-732. |
[6] |
A. Casal, J. C. Eilbede and J. López-Gómez, Existence and uniqueness of coexitence states for a predator-prey model with diffusion, Differential Integral Equations, 7 (1994), 411-439. |
[7] |
W.-Y. Chen and M.-X. Wang, Positive steady states of a competitor-competitor-mutualist model, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 53-57.
doi: 10.1007/s10255-004-0148-0. |
[8] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[9] |
E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.
doi: 10.1016/0022-247X(83)90098-7. |
[10] |
E. N. Dancer and Y. H. Du, Effects of certain degeneracies in the predator-prey model, SIAM. J. Math. Anal., 34 (2002), 292-314.
doi: 10.1137/S0036141001387598. |
[11] |
M. Delgado, J. López-Gómez and A. Suárez, On the symbiotic Lotka-Volterra model with diffusion and transport effects, J. Differential Equations, 160 (2000), 175-262. |
[12] |
Y. H. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475.
doi: 10.1090/S0002-9947-97-01842-4. |
[13] |
Y. H. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations, 144 (1998), 390-440. |
[14] |
C. F. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure Appl. Math., 47 (1994), 1571-1594.
doi: 10.1002/cpa.3160471203. |
[15] |
T. Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. |
[16] |
W. Ko and K. Ryu, Coexistence states of a predator-prey system with non-monotonic functional response, Nonlinear Anal. RWA, 8 (2007), 769-786.
doi: 10.1016/j.nonrwa.2006.03.003. |
[17] |
L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166.
doi: 10.1090/S0002-9947-1988-0920151-1. |
[18] |
J. López-Gómez and C. Mora-Corral, Minimal complexity of semi-bounded components in bifurcation theory, Nonlinear Anal., 58 (2004), 749-777.
doi: 10.1016/j.na.2004.04.011. |
[19] |
C. V. Pao, "Nonlinear Parabolic and Elliptic Equations," Plenum Press, New York, 1992. |
[20] |
R. Peng and M. X. Wang, On multiplicity and stability of positive solutions of a diffusive prey-predator model, J. Math. Anal. Appl., 316 (2006), 256-268.
doi: 10.1016/j.jmaa.2005.04.033. |
[21] |
W. H. Ruan and W. Feng, On the fixed point index and multiple steady states of reaction-diffusion systems, Differential Integral Equations, 8 (1995), 371-391. |
[22] |
K. Ryu and I. Ahn, Positive solutions for ratio-dependent predator-prey interaction systems, J. Differential Equations, 218 (2005), 117-135. |
[23] |
Y. M. Svirezhev and D. O. Logofet, "Stability of Biological Communities," Translated from the Russian by Alexey Voinov, "MIR," Moscow, 1983. |
[24] |
M. X. Wang, "Nonlinear Partial Differential Equations of Parabolic Type," (Chinese), Science Press, Beijing, 1993. |
[25] |
M. X. Wang and Q. Wu, Positive solutions of a prey-predator model with predator saturation and competition, J. Math. Anal. Appl., 345 (2008), 708-718.
doi: 10.1016/j.jmaa.2008.04.054. |
[26] |
Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM. J. Math. Anal., 21 (1990), 327-345.
doi: 10.1137/0521018. |
show all references
References:
[1] |
P. A. Abrams and L. R. Ginzburg, The nature of predation: Prey dependent, ratio dependent or neither?, Trends in Ecology & Evolution, 15 (2000), 337-341.
doi: 10.1016/S0169-5347(00)01908-X. |
[2] |
C. O. Alves and D. G. de Figueiredo, Nonvariational elliptic systems. Current developments in partial differential equations, Discrete Contin. Dyn. Syst., 8 (2002), 289-302. |
[3] |
J. Blat and K. J. Brown, Global bifurcation on positive solutions in some systems of elliptic equations, SIAM. J. Math. Anal., 17 (1986), 1339-1353.
doi: 10.1137/0517094. |
[4] |
G. Buffoni, M. P. Cassinari and M. Groppi, Modelling of predator-prey trophic interactions. Part II: Three trophic levels, J. Math. Biol., 54 (2007), 623-644. |
[5] |
G. Buffoni, M. P. Cassinari, M. Groppi and M. Serluca, Modelling of predator-prey trophic interactions. Part I: Two trophic levels, J. Math. Biol., 50 (2005), 713-732. |
[6] |
A. Casal, J. C. Eilbede and J. López-Gómez, Existence and uniqueness of coexitence states for a predator-prey model with diffusion, Differential Integral Equations, 7 (1994), 411-439. |
[7] |
W.-Y. Chen and M.-X. Wang, Positive steady states of a competitor-competitor-mutualist model, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 53-57.
doi: 10.1007/s10255-004-0148-0. |
[8] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[9] |
E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.
doi: 10.1016/0022-247X(83)90098-7. |
[10] |
E. N. Dancer and Y. H. Du, Effects of certain degeneracies in the predator-prey model, SIAM. J. Math. Anal., 34 (2002), 292-314.
doi: 10.1137/S0036141001387598. |
[11] |
M. Delgado, J. López-Gómez and A. Suárez, On the symbiotic Lotka-Volterra model with diffusion and transport effects, J. Differential Equations, 160 (2000), 175-262. |
[12] |
Y. H. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475.
doi: 10.1090/S0002-9947-97-01842-4. |
[13] |
Y. H. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations, 144 (1998), 390-440. |
[14] |
C. F. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure Appl. Math., 47 (1994), 1571-1594.
doi: 10.1002/cpa.3160471203. |
[15] |
T. Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. |
[16] |
W. Ko and K. Ryu, Coexistence states of a predator-prey system with non-monotonic functional response, Nonlinear Anal. RWA, 8 (2007), 769-786.
doi: 10.1016/j.nonrwa.2006.03.003. |
[17] |
L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166.
doi: 10.1090/S0002-9947-1988-0920151-1. |
[18] |
J. López-Gómez and C. Mora-Corral, Minimal complexity of semi-bounded components in bifurcation theory, Nonlinear Anal., 58 (2004), 749-777.
doi: 10.1016/j.na.2004.04.011. |
[19] |
C. V. Pao, "Nonlinear Parabolic and Elliptic Equations," Plenum Press, New York, 1992. |
[20] |
R. Peng and M. X. Wang, On multiplicity and stability of positive solutions of a diffusive prey-predator model, J. Math. Anal. Appl., 316 (2006), 256-268.
doi: 10.1016/j.jmaa.2005.04.033. |
[21] |
W. H. Ruan and W. Feng, On the fixed point index and multiple steady states of reaction-diffusion systems, Differential Integral Equations, 8 (1995), 371-391. |
[22] |
K. Ryu and I. Ahn, Positive solutions for ratio-dependent predator-prey interaction systems, J. Differential Equations, 218 (2005), 117-135. |
[23] |
Y. M. Svirezhev and D. O. Logofet, "Stability of Biological Communities," Translated from the Russian by Alexey Voinov, "MIR," Moscow, 1983. |
[24] |
M. X. Wang, "Nonlinear Partial Differential Equations of Parabolic Type," (Chinese), Science Press, Beijing, 1993. |
[25] |
M. X. Wang and Q. Wu, Positive solutions of a prey-predator model with predator saturation and competition, J. Math. Anal. Appl., 345 (2008), 708-718.
doi: 10.1016/j.jmaa.2008.04.054. |
[26] |
Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM. J. Math. Anal., 21 (1990), 327-345.
doi: 10.1137/0521018. |
[1] |
Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045 |
[2] |
Xinfu Chen, Yuanwei Qi, Mingxin Wang. Steady states of a strongly coupled prey-predator model. Conference Publications, 2005, 2005 (Special) : 173-180. doi: 10.3934/proc.2005.2005.173 |
[3] |
Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 |
[4] |
Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875 |
[5] |
Shanshan Chen. Nonexistence of nonconstant positive steady states of a diffusive predator-prey model. Communications on Pure and Applied Analysis, 2018, 17 (2) : 477-485. doi: 10.3934/cpaa.2018026 |
[6] |
Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189 |
[7] |
Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597 |
[8] |
Isam Al-Darabsah, Xianhua Tang, Yuan Yuan. A prey-predator model with migrations and delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 737-761. doi: 10.3934/dcdsb.2016.21.737 |
[9] |
R. P. Gupta, Peeyush Chandra, Malay Banerjee. Dynamical complexity of a prey-predator model with nonlinear predator harvesting. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 423-443. doi: 10.3934/dcdsb.2015.20.423 |
[10] |
Malay Banerjee, Nayana Mukherjee, Vitaly Volpert. Prey-predator model with nonlocal and global consumption in the prey dynamics. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2109-2120. doi: 10.3934/dcdss.2020180 |
[11] |
Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3087-3107. doi: 10.3934/dcdsb.2017209 |
[12] |
Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536 |
[13] |
Mingxin Wang, Peter Y. H. Pang. Qualitative analysis of a diffusive variable-territory prey-predator model. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1061-1072. doi: 10.3934/dcds.2009.23.1061 |
[14] |
J. Gani, R. J. Swift. Prey-predator models with infected prey and predators. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5059-5066. doi: 10.3934/dcds.2013.33.5059 |
[15] |
Komi Messan, Yun Kang. A two patch prey-predator model with multiple foraging strategies in predator: Applications to insects. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 947-976. doi: 10.3934/dcdsb.2017048 |
[16] |
Yun Kang, Sourav Kumar Sasmal, Komi Messan. A two-patch prey-predator model with predator dispersal driven by the predation strength. Mathematical Biosciences & Engineering, 2017, 14 (4) : 843-880. doi: 10.3934/mbe.2017046 |
[17] |
Siyu Liu, Haomin Huang, Mingxin Wang. A free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1649-1670. doi: 10.3934/dcdsb.2019245 |
[18] |
Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172 |
[19] |
Pankaj Kumar, Shiv Raj. Modelling and analysis of prey-predator model involving predation of mature prey using delay differential equations. Numerical Algebra, Control and Optimization, 2021 doi: 10.3934/naco.2021035 |
[20] |
Guoqiang Ren, Bin Liu. Global existence and convergence to steady states for a predator-prey model with both predator- and prey-taxis. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 759-779. doi: 10.3934/dcds.2021136 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]