-
Previous Article
Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions
- CPAA Home
- This Issue
-
Next Article
The basis property of generalized Jacobian elliptic functions
Asymptotic behavior of Kolmogorov systems with predator-prey type in random environment
1. | Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam |
2. | Department of Mathematics, Wayne State University, Detroit, MI 48202, United States |
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, Heidelberg, New York, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
P. Auger, N. H. Du and N. T. Hieu, Evolution of Lotka-Volterra predator-prey systems under telegraph noise, Math. Biosci. Eng., 6 (2009), 683-700.
doi: 10.3934/mbe.2009.6.683. |
[3] |
A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific, Singapore, 1998.
doi: 10.1142/9789812798725. |
[4] |
A. Bobrowski, T. Lipniacki, K. Pichor and R. Rudnicki, Asymptotic behavior of distributions of mRNA and protein levels in a model of stochastic gene expression, J. Math. Anal. Appl., 333 (2007), 753-769.
doi: 10.1016/j.jmaa.2006.11.043. |
[5] |
Z. Brzeniak, M. Capifiski and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probab. Theory Relat. Fields, 95 (1993), 87-102
doi: 10.1007/BF01197339. |
[6] |
N. H. Dang, N. H. Du and T. V. Ton, Asymptotic behavior of predator-prey systems perturbed by white noise, Acta Appl. Math., 115 (2011), 351-370.
doi: 10.1007/s10440-011-9628-4. |
[7] |
N. H. Du and N. H. Dang, Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J. Differential Equations, 250 (2011), 386-409.
doi: 10.1016/j.jde.2010.08.023. |
[8] |
N. H. Du, R. Kon, K. Sato and Y. Takeuchi, Dynamical behavior of Lotka - Volterra competition systems: Non autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math, 170 (2004), 399-422.
doi: 10.1016/j.cam.2004.02.001. |
[9] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[10] |
I. I Gihman and A. V. Skorohod, The Theory of Stochastic Processes, Springer-Verlag, Berlin, Heidelberg, New York, 1979. |
[11] |
T. W. Hwang, Global analysis of the predator-prey system with Beddington DeAngelis functional response, J. Math. Anal. Appl., 281 (2003), 395-401. |
[12] |
T. W. Hwang, Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 290 (2004), 113-122.
doi: 10.1016/j.jmaa.2003.09.073. |
[13] |
C. Ji and D. Jiang, Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 381 (2011), 441-453.
doi: 10.1016/j.jmaa.2011.02.037. |
[14] |
C. Ji, D. Jiang and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498.
doi: 10.1016/j.jmaa.2009.05.039. |
[15] |
Q. Luo and X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84 .
doi: 10.1016/j.jmaa.2006.12.032. |
[16] |
Q. Luo and X. Mao, Stochastic population dynamics under regime switching. II, J. Math. Anal. Appl., 355 (2009), 577-593.
doi: 10.1016/j.jmaa.2009.02.010. |
[17] |
X. Mao, Attraction, stability and boundedness for stochastic differential delay equations, Nonlinear Analysis: Theory, Methods and Applications, 47 (2001), 4795-4806.
doi: 10.1016/S0362-546X(01)00591-0. |
[18] |
X. Mao, S. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156.
doi: 10.1016/S0022-247X(03)00539-0. |
[19] |
L. Michael, Conservative Markov processes on a topological space, Israel J. Math., 8 (1970), 165-186. |
[20] |
J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, Heidelberg, 2002.
doi: 10.1007/b98869. |
[21] |
K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000), 668-685.
doi: 10.1006/jmaa.2000.6968. |
[22] |
R. Rudnicki, K. Pichór and M. Tyran-Kaminska, Markov semigroups and their applications, in Dynamics of Dissipation (P. Garbaczewski and R. Olkiewicz Eds), Lecture Note in Physics, Vol 587, Springer, Berlin, 2002, 215-238. |
[23] |
Y. Takeuchi, N. H. Du, N. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957.
doi: 10.1016/j.jmaa.2005.11.009. |
[24] |
C. Yuan and X. Mao, Attraction and stochastic asymptotic stability and boundedness of stochastic functional differential equations with respect to semimartingales, Stochastic Analysis and Applications, 24 (2006), 1169-1184.
doi: 10.1080/07362990600958937. |
[25] |
C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments, Journal of Mathematical Analysis and Applications, 357 (2009), 154-170.
doi: 10.1016/j.jmaa.2009.03.066. |
show all references
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, Heidelberg, New York, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
P. Auger, N. H. Du and N. T. Hieu, Evolution of Lotka-Volterra predator-prey systems under telegraph noise, Math. Biosci. Eng., 6 (2009), 683-700.
doi: 10.3934/mbe.2009.6.683. |
[3] |
A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific, Singapore, 1998.
doi: 10.1142/9789812798725. |
[4] |
A. Bobrowski, T. Lipniacki, K. Pichor and R. Rudnicki, Asymptotic behavior of distributions of mRNA and protein levels in a model of stochastic gene expression, J. Math. Anal. Appl., 333 (2007), 753-769.
doi: 10.1016/j.jmaa.2006.11.043. |
[5] |
Z. Brzeniak, M. Capifiski and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probab. Theory Relat. Fields, 95 (1993), 87-102
doi: 10.1007/BF01197339. |
[6] |
N. H. Dang, N. H. Du and T. V. Ton, Asymptotic behavior of predator-prey systems perturbed by white noise, Acta Appl. Math., 115 (2011), 351-370.
doi: 10.1007/s10440-011-9628-4. |
[7] |
N. H. Du and N. H. Dang, Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J. Differential Equations, 250 (2011), 386-409.
doi: 10.1016/j.jde.2010.08.023. |
[8] |
N. H. Du, R. Kon, K. Sato and Y. Takeuchi, Dynamical behavior of Lotka - Volterra competition systems: Non autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math, 170 (2004), 399-422.
doi: 10.1016/j.cam.2004.02.001. |
[9] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[10] |
I. I Gihman and A. V. Skorohod, The Theory of Stochastic Processes, Springer-Verlag, Berlin, Heidelberg, New York, 1979. |
[11] |
T. W. Hwang, Global analysis of the predator-prey system with Beddington DeAngelis functional response, J. Math. Anal. Appl., 281 (2003), 395-401. |
[12] |
T. W. Hwang, Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 290 (2004), 113-122.
doi: 10.1016/j.jmaa.2003.09.073. |
[13] |
C. Ji and D. Jiang, Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 381 (2011), 441-453.
doi: 10.1016/j.jmaa.2011.02.037. |
[14] |
C. Ji, D. Jiang and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498.
doi: 10.1016/j.jmaa.2009.05.039. |
[15] |
Q. Luo and X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84 .
doi: 10.1016/j.jmaa.2006.12.032. |
[16] |
Q. Luo and X. Mao, Stochastic population dynamics under regime switching. II, J. Math. Anal. Appl., 355 (2009), 577-593.
doi: 10.1016/j.jmaa.2009.02.010. |
[17] |
X. Mao, Attraction, stability and boundedness for stochastic differential delay equations, Nonlinear Analysis: Theory, Methods and Applications, 47 (2001), 4795-4806.
doi: 10.1016/S0362-546X(01)00591-0. |
[18] |
X. Mao, S. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156.
doi: 10.1016/S0022-247X(03)00539-0. |
[19] |
L. Michael, Conservative Markov processes on a topological space, Israel J. Math., 8 (1970), 165-186. |
[20] |
J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, Heidelberg, 2002.
doi: 10.1007/b98869. |
[21] |
K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000), 668-685.
doi: 10.1006/jmaa.2000.6968. |
[22] |
R. Rudnicki, K. Pichór and M. Tyran-Kaminska, Markov semigroups and their applications, in Dynamics of Dissipation (P. Garbaczewski and R. Olkiewicz Eds), Lecture Note in Physics, Vol 587, Springer, Berlin, 2002, 215-238. |
[23] |
Y. Takeuchi, N. H. Du, N. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957.
doi: 10.1016/j.jmaa.2005.11.009. |
[24] |
C. Yuan and X. Mao, Attraction and stochastic asymptotic stability and boundedness of stochastic functional differential equations with respect to semimartingales, Stochastic Analysis and Applications, 24 (2006), 1169-1184.
doi: 10.1080/07362990600958937. |
[25] |
C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments, Journal of Mathematical Analysis and Applications, 357 (2009), 154-170.
doi: 10.1016/j.jmaa.2009.03.066. |
[1] |
P. Auger, N. H. Du, N. T. Hieu. Evolution of Lotka-Volterra predator-prey systems under telegraph noise. Mathematical Biosciences & Engineering, 2009, 6 (4) : 683-700. doi: 10.3934/mbe.2009.6.683 |
[2] |
Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507 |
[3] |
Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124 |
[4] |
Hongxiao Hu, Liguang Xu, Kai Wang. A comparison of deterministic and stochastic predator-prey models with disease in the predator. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2837-2863. doi: 10.3934/dcdsb.2018289 |
[5] |
Henri Berestycki, Alessandro Zilio. Predator-prey models with competition, Part Ⅲ: Classification of stationary solutions. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7141-7162. doi: 10.3934/dcds.2019299 |
[6] |
Fei Xu, Ross Cressman, Vlastimil Křivan. Evolution of mobility in predator-prey systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3397-3432. doi: 10.3934/dcdsb.2014.19.3397 |
[7] |
Christian Kuehn, Thilo Gross. Nonlocal generalized models of predator-prey systems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 693-720. doi: 10.3934/dcdsb.2013.18.693 |
[8] |
Guanqi Liu, Yuwen Wang. Stochastic spatiotemporal diffusive predator-prey systems. Communications on Pure and Applied Analysis, 2018, 17 (1) : 67-84. doi: 10.3934/cpaa.2018005 |
[9] |
Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002 |
[10] |
Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3253-3269. doi: 10.3934/dcdss.2020130 |
[11] |
Xiang-Sheng Wang, Haiyan Wang, Jianhong Wu. Traveling waves of diffusive predator-prey systems: Disease outbreak propagation. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3303-3324. doi: 10.3934/dcds.2012.32.3303 |
[12] |
Patrick D. Leenheer, David Angeli, Eduardo D. Sontag. On Predator-Prey Systems and Small-Gain Theorems. Mathematical Biosciences & Engineering, 2005, 2 (1) : 25-42. doi: 10.3934/mbe.2005.2.25 |
[13] |
Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162 |
[14] |
Kimun Ryu, Wonlyul Ko. On dynamics and stationary pattern formations of a diffusive predator-prey system with hunting cooperation. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022015 |
[15] |
Shanshan Chen, Junping Shi, Junjie Wei. The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response. Communications on Pure and Applied Analysis, 2013, 12 (1) : 481-501. doi: 10.3934/cpaa.2013.12.481 |
[16] |
Ming Liu, Dongpo Hu, Fanwei Meng. Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3197-3222. doi: 10.3934/dcdss.2020259 |
[17] |
Peter A. Braza. Predator-Prey Dynamics with Disease in the Prey. Mathematical Biosciences & Engineering, 2005, 2 (4) : 703-717. doi: 10.3934/mbe.2005.2.703 |
[18] |
Ke Wang, Qi Wang, Feng Yu. Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 505-543. doi: 10.3934/dcds.2017021 |
[19] |
Zhifu Xie. Turing instability in a coupled predator-prey model with different Holling type functional responses. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1621-1628. doi: 10.3934/dcdss.2011.4.1621 |
[20] |
Kexin Wang. Influence of feedback controls on the global stability of a stochastic predator-prey model with Holling type Ⅱ response and infinite delays. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1699-1714. doi: 10.3934/dcdsb.2019247 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]