-
Previous Article
Global stability of a five-dimensional model with immune responses and delay
- DCDS-B Home
- This Issue
-
Next Article
Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay
Stability and Hopf bifurcations for a delayed diffusion system in population dynamics
1. | Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China |
2. | School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000 |
References:
[1] |
S. Busenberg and W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differential Equations, 124 (1996), 80-107. |
[2] |
S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," Grundlehren der Mathematischen Wissenschaften, 251, Springer-Verlag, New York-Berlin, 1982. |
[3] |
G. E. Hutchinson, Circular causal systems in ecology, Ann. New York Acad. Sci., 50 (1948), 221-246.
doi: 10.1111/j.1749-6632.1948.tb39854.x. |
[4] |
L. C. Evans, "Partial Differential Equations," 2nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. |
[5] |
T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463. |
[6] |
J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. |
[7] |
J. Hale, "Theory of Functional Differential Equations," 2nd edition, Applied Mathematical Sciences, 3, Springer-Verlag, New York-Heidelberg, 1977. |
[8] |
W. Huang, Global Dynamics for a Reaction-Diffusion Equation with Time Delay, J. Differential Equations, 143 (1998), 293-326. |
[9] |
H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004. |
[10] |
Y. Kuang and H. L. Smith, Global stability in diffusive delay Lotka-Volterra systems, Differential Integral Equations, 4 (1991), 117-128. |
[11] |
Y. Kuang and H. L. Smith, Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks, J. Austral. Math. Soc. Ser. B, 34 (1993), 471-494.
doi: 10.1017/S0334270000009036. |
[12] |
W.-T. Li, G. Lin and S. Ruan, Existence of traveling wave solutions in delayed reaction-diffusion systems with application to diffusion-competition system, Nonlinearity, 19 (2006), 1253-1273.
doi: 10.1088/0951-7715/19/6/003. |
[13] |
W.-T. Li, X.-P. Yan and C.-H. Zhang, Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions, Chaos, Solitons and Fractals, 38 (2008), 227-237.
doi: 10.1016/j.chaos.2006.11.015. |
[14] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. |
[15] |
K. Ryu and I. Ahn, Positive coexistence of steady states for competitive interacting system with self-diffusion pressures, Bull. Korean Math. Soc., 38 (2001), 643-655. |
[16] |
S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays, Quart. Appl. Math., 59 (2001), 159-173. |
[17] |
Y. Su, J. Wei and J. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, J. Differential Equations, 247 (2009), 1156-1184. |
[18] |
C. Travis and G. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418.
doi: 10.1090/S0002-9947-1974-0382808-3. |
[19] |
J. Wu, "Theory and Applications of Partial Functional-Differential Equations," Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996. |
[20] |
X.-P. Yan and W.-T. Li, Stability and Hopf bifurcation for a delayed cooperative system with diffusion effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 441-453.
doi: 10.1142/S0218127408020434. |
[21] |
X.-P. Yan and W.-T. Li, Stability of bifurcated periodic solutions in a delayed competition system with diffusion effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 857-871.
doi: 10.1142/S0218127409023342. |
[22] |
X.-P. Yan and C.-H. Zhang, Direction of Hopf bifurcation in a delayed Lotka-Volterra competition diffusion system, Nonlinear Anal. Real World Appl., 10 (2009), 2758-2773.
doi: 10.1016/j.nonrwa.2008.08.004. |
[23] |
X.-P. Yan and C.-H. Zhang, Asymptotic stability of positive equilibrium solution for a delayed prey-predator diffusion system, Appl. Math. Model., 34 (2010), 184-199.
doi: 10.1016/j.apm.2009.03.040. |
[24] |
Y. Yang and J. W.-H. So, Dynamics for the diffusive Nicholson's blowflies equation, in "Dynamical Systems and Differential Equations," Vol. II (eds. W. Chen and S. Hu) (Springfield, MO, 1996), Discrete Contin. Dynam. Systems, 1998, Added Volume II, 333-352. |
[25] |
Q. Ye and Z. Li, "An Introduction to Reaction-Diffusion Equations," (in Chinese), Science Press, Beijing, 1990. |
[26] |
K. Yosida, "Functional Analysis," Reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. |
[27] |
L. Zhou, Y. Tang and S. Hussein, Stability and Hopf bifurcation for a delay competition diffusion system, Chaos Solitons Fractals, 14 (2002), 1201-1225.
doi: 10.1016/S0960-0779(02)00068-1. |
show all references
References:
[1] |
S. Busenberg and W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differential Equations, 124 (1996), 80-107. |
[2] |
S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," Grundlehren der Mathematischen Wissenschaften, 251, Springer-Verlag, New York-Berlin, 1982. |
[3] |
G. E. Hutchinson, Circular causal systems in ecology, Ann. New York Acad. Sci., 50 (1948), 221-246.
doi: 10.1111/j.1749-6632.1948.tb39854.x. |
[4] |
L. C. Evans, "Partial Differential Equations," 2nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. |
[5] |
T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463. |
[6] |
J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. |
[7] |
J. Hale, "Theory of Functional Differential Equations," 2nd edition, Applied Mathematical Sciences, 3, Springer-Verlag, New York-Heidelberg, 1977. |
[8] |
W. Huang, Global Dynamics for a Reaction-Diffusion Equation with Time Delay, J. Differential Equations, 143 (1998), 293-326. |
[9] |
H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004. |
[10] |
Y. Kuang and H. L. Smith, Global stability in diffusive delay Lotka-Volterra systems, Differential Integral Equations, 4 (1991), 117-128. |
[11] |
Y. Kuang and H. L. Smith, Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks, J. Austral. Math. Soc. Ser. B, 34 (1993), 471-494.
doi: 10.1017/S0334270000009036. |
[12] |
W.-T. Li, G. Lin and S. Ruan, Existence of traveling wave solutions in delayed reaction-diffusion systems with application to diffusion-competition system, Nonlinearity, 19 (2006), 1253-1273.
doi: 10.1088/0951-7715/19/6/003. |
[13] |
W.-T. Li, X.-P. Yan and C.-H. Zhang, Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions, Chaos, Solitons and Fractals, 38 (2008), 227-237.
doi: 10.1016/j.chaos.2006.11.015. |
[14] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. |
[15] |
K. Ryu and I. Ahn, Positive coexistence of steady states for competitive interacting system with self-diffusion pressures, Bull. Korean Math. Soc., 38 (2001), 643-655. |
[16] |
S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays, Quart. Appl. Math., 59 (2001), 159-173. |
[17] |
Y. Su, J. Wei and J. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, J. Differential Equations, 247 (2009), 1156-1184. |
[18] |
C. Travis and G. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418.
doi: 10.1090/S0002-9947-1974-0382808-3. |
[19] |
J. Wu, "Theory and Applications of Partial Functional-Differential Equations," Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996. |
[20] |
X.-P. Yan and W.-T. Li, Stability and Hopf bifurcation for a delayed cooperative system with diffusion effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 441-453.
doi: 10.1142/S0218127408020434. |
[21] |
X.-P. Yan and W.-T. Li, Stability of bifurcated periodic solutions in a delayed competition system with diffusion effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 857-871.
doi: 10.1142/S0218127409023342. |
[22] |
X.-P. Yan and C.-H. Zhang, Direction of Hopf bifurcation in a delayed Lotka-Volterra competition diffusion system, Nonlinear Anal. Real World Appl., 10 (2009), 2758-2773.
doi: 10.1016/j.nonrwa.2008.08.004. |
[23] |
X.-P. Yan and C.-H. Zhang, Asymptotic stability of positive equilibrium solution for a delayed prey-predator diffusion system, Appl. Math. Model., 34 (2010), 184-199.
doi: 10.1016/j.apm.2009.03.040. |
[24] |
Y. Yang and J. W.-H. So, Dynamics for the diffusive Nicholson's blowflies equation, in "Dynamical Systems and Differential Equations," Vol. II (eds. W. Chen and S. Hu) (Springfield, MO, 1996), Discrete Contin. Dynam. Systems, 1998, Added Volume II, 333-352. |
[25] |
Q. Ye and Z. Li, "An Introduction to Reaction-Diffusion Equations," (in Chinese), Science Press, Beijing, 1990. |
[26] |
K. Yosida, "Functional Analysis," Reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. |
[27] |
L. Zhou, Y. Tang and S. Hussein, Stability and Hopf bifurcation for a delay competition diffusion system, Chaos Solitons Fractals, 14 (2002), 1201-1225.
doi: 10.1016/S0960-0779(02)00068-1. |
[1] |
Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6185-6205. doi: 10.3934/dcdsb.2021013 |
[2] |
Qi Wang. On steady state of some Lotka-Volterra competition-diffusion-advection model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 859-875. doi: 10.3934/dcdsb.2019193 |
[3] |
Xiaoli Liu, Dongmei Xiao. Bifurcations in a discrete time Lotka-Volterra predator-prey system. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 559-572. doi: 10.3934/dcdsb.2006.6.559 |
[4] |
Fuke Wu, Yangzi Hu. Stochastic Lotka-Volterra system with unbounded distributed delay. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 275-288. doi: 10.3934/dcdsb.2010.14.275 |
[5] |
Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197 |
[6] |
Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083 |
[7] |
Fang Li, Liping Wang, Yang Wang. On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 669-686. doi: 10.3934/dcdsb.2011.15.669 |
[8] |
Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135 |
[9] |
S. Nakaoka, Y. Saito, Y. Takeuchi. Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system. Mathematical Biosciences & Engineering, 2006, 3 (1) : 173-187. doi: 10.3934/mbe.2006.3.173 |
[10] |
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 |
[11] |
Yuzo Hosono. Traveling waves for the Lotka-Volterra predator-prey system without diffusion of the predator. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 161-171. doi: 10.3934/dcdsb.2015.20.161 |
[12] |
Qi Wang. Some global dynamics of a Lotka-Volterra competition-diffusion-advection system. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3245-3255. doi: 10.3934/cpaa.2020142 |
[13] |
Li-Jun Du, Wan-Tong Li, Jia-Bing Wang. Invasion entire solutions in a time periodic Lotka-Volterra competition system with diffusion. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1187-1213. doi: 10.3934/mbe.2017061 |
[14] |
Qian Guo, Xiaoqing He, Wei-Ming Ni. Global dynamics of a general Lotka-Volterra competition-diffusion system in heterogeneous environments. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6547-6573. doi: 10.3934/dcds.2020290 |
[15] |
Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147 |
[16] |
Li Ma, Shangjiang Guo. Bifurcation and stability of a two-species diffusive Lotka-Volterra model. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1205-1232. doi: 10.3934/cpaa.2020056 |
[17] |
Lih-Ing W. Roeger, Razvan Gelca. Dynamically consistent discrete-time Lotka-Volterra competition models. Conference Publications, 2009, 2009 (Special) : 650-658. doi: 10.3934/proc.2009.2009.650 |
[18] |
Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031 |
[19] |
Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035 |
[20] |
Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]