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Boundary layer separation of 2-D incompressible Dirichlet flows
Optimal harvesting for a stochastic N-dimensional competitive Lotka-Volterra model with jumps
1. | Department of Mathematics, Harbin Institute of Technology, Weihai 264209 |
2. | Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209 |
References:
[1] |
L. H. Alvarez, Optimal harvesting under stochastic fluctuations and critical depensation, Math Biosci., 152 (1998), 63-85.
doi: 10.1016/S0025-5564(98)10018-4. |
[2] |
V. S. Anishchenko, Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments, Springer-Verlag, New York, 2007. |
[3] |
D. Applebaum, Lévy Processes and Stochastics Calculus, Cambridge University Press, 2 edition, 2009.
doi: 10.1017/CBO9780511809781. |
[4] |
L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1974. |
[5] |
L. Arnold, Random Dynamical Systems, Springer, New York, 1998.
doi: 10.1007/978-3-662-12878-7. |
[6] |
J. Bao, X. Mao,G. Yin and C. Yuan, Competitive lotka-volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616.
doi: 10.1016/j.na.2011.06.043. |
[7] |
J. Bao and C. Yuan, Stochastic population dtnamics driven by lévy noise, J. Math. Anal. Appl., 391 (2012), 363-375.
doi: 10.1016/j.jmaa.2012.02.043. |
[8] |
I. Barbalat, Systemes d'équations différentielles d'oscillations non linéaires, Rev. Math. Pures. Appl., 4 (1959), 267-270. |
[9] |
J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465.
doi: 10.1126/science.197.4302.463. |
[10] |
J. X. Chen, C. H. Yu and L. Jin, Mathematical Analysis, Higher Education Press, Beijing, 2 edition, 2004. |
[11] |
C. Chiarella, X. He, D. Wang and M. Zheng, The stochastic bifurcation behaviour of speculative financial markets, Physica A., 387 (2008), 3837-3846.
doi: 10.1016/j.physa.2008.01.078. |
[12] |
C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewal Resources, John Wiley and Sons Inc., New York, 2 edition, 1990. |
[13] |
H. Crauel and M. Gundlach, Stochastic Dynamics, Springer-Verlag, New York, 1999.
doi: 10.1007/b97846. |
[14] |
T. C. Gard, Persistence in stochastic food web models, Bull. Math. Biol., 46 (1984), 357-370.
doi: 10.1007/BF02462011. |
[15] |
T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419.
doi: 10.1016/0362-546X(86)90111-2. |
[16] |
G. Hu and K. Wang, Stability in distribution of competitive lotka-volterra system with markovian switching, Appl. Math. Model., 35 (2011), 3189-3200.
doi: 10.1016/j.apm.2010.12.025. |
[17] |
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Amsterdam, North-Holland, 1981. |
[18] |
D. Jiang, C. Ji, X. Li and D. O'Regan, Analysis of autonomous lotka-volterra competition system with random perturbation, J. Math. Anal. Appl., 390 (2012), 582-595.
doi: 10.1016/j.jmaa.2011.12.049. |
[19] |
F. C. Klebaner, Introduction to Stochastic Calculus With Applications, Imperial College Press, London, 2005.
doi: 10.1142/p386. |
[20] |
H. Kunita, Itô's stochastic calculus: Its surprising power for applications, Stochastic Process. Appl., 120 (2010), 622-652.
doi: 10.1016/j.spa.2010.01.013. |
[21] |
W. Li, K. Wang and H. Su, Optimal harvesting policy for stochastic logistic population model, Appl. Math. Comput., 218 (2011), 157-162.
doi: 10.1016/j.amc.2011.05.079. |
[22] |
X. Li, D. Jiang and X. Mao, Population dynamical behavior of lotka-volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448.
doi: 10.1016/j.cam.2009.06.021. |
[23] |
X. Li and X. Mao, Population dynamical behavior of non-autonomous lotka-volterra competitive system with random perturbation, Discret. Contin. Dyn. S., 24 (2009), 523-545.
doi: 10.3934/dcds.2009.24.523. |
[24] |
R. S. Liptser, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228.
doi: 10.1080/17442508008833146. |
[25] |
A. J. Lotka, Elements of Physical Biology, William and Wilkins, Baltimore, 1925. |
[26] |
E. M. Lungu and B. Øksendal, Optimal harvesting from a population in a stochastic crowded environment, Math. Biosci., 145 (1997), 47-75.
doi: 10.1016/S0025-5564(97)00029-1. |
[27] |
X. Mao, Stochastic Differential Equations and Applications, Horwood, New York, 1997.
doi: 10.1533/9780857099402. |
[28] |
X. Mao, Stationary distribution of stochastic population systems, Syst. Control Letters, 60 (2011), 398-405.
doi: 10.1016/j.sysconle.2011.02.013. |
[29] |
X. Mao, G. Marion and E. Renshaw, Environmental brownian noise suppresses explosions in populations dynamics, Stochastic Process. Appl., 97 (2002), 95-110.
doi: 10.1016/S0304-4149(01)00126-0. |
[30] |
X. Mao, S. Sabanis and E. Renshaw, Asymptotic behavior of the stochastic lotka-volterra model, J. Math. Anal. Appl., 287 (2003), 141-156.
doi: 10.1016/S0022-247X(03)00539-0. |
[31] |
X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.
doi: 10.1142/p473. |
[32] |
R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 2001. |
[33] |
B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin, 6 edition, 2003.
doi: 10.1007/978-3-642-14394-6. |
[34] |
S. Peng and X. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Process. Appl., 116 (2006), 370-380.
doi: 10.1016/j.spa.2005.08.004. |
[35] |
D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996.
doi: 10.1017/CBO9780511662829. |
[36] |
M. A. Shah and U. Sharma, Optimal harvesting policies for a generalized gordon-schaefer model in randomly varying environment, Appl. Stochastic Models Bus. Ind., 19 (2003), 43-49.
doi: 10.1002/asmb.490. |
[37] |
R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005. |
[38] |
V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie d'animali conviventi, Mem. Acad. Lincei, 2 (1926), 31-113. |
[39] |
K. Wang, Stochastic Biomathematics Models, Science Press, Beijing, 2010. |
[40] |
C. Zhu and G. Yin, On competitive lotka-volterra model in random environments, J. Math. Anal. Appl., 357 (2009), 154-170.
doi: 10.1016/j.jmaa.2009.03.066. |
[41] |
C. Zhu and G. Yin, On hybrid competitive lotka-volterra ecosystems, Nonlinear Anal., 71 (2009), e1370-e1379.
doi: 10.1016/j.na.2009.01.166. |
[42] |
X. Zou and K. Wang, Numerical simulations and modeling for stochastic biological systems with jumps, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 1557-1568.
doi: 10.1016/j.cnsns.2013.09.010. |
show all references
References:
[1] |
L. H. Alvarez, Optimal harvesting under stochastic fluctuations and critical depensation, Math Biosci., 152 (1998), 63-85.
doi: 10.1016/S0025-5564(98)10018-4. |
[2] |
V. S. Anishchenko, Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments, Springer-Verlag, New York, 2007. |
[3] |
D. Applebaum, Lévy Processes and Stochastics Calculus, Cambridge University Press, 2 edition, 2009.
doi: 10.1017/CBO9780511809781. |
[4] |
L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1974. |
[5] |
L. Arnold, Random Dynamical Systems, Springer, New York, 1998.
doi: 10.1007/978-3-662-12878-7. |
[6] |
J. Bao, X. Mao,G. Yin and C. Yuan, Competitive lotka-volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616.
doi: 10.1016/j.na.2011.06.043. |
[7] |
J. Bao and C. Yuan, Stochastic population dtnamics driven by lévy noise, J. Math. Anal. Appl., 391 (2012), 363-375.
doi: 10.1016/j.jmaa.2012.02.043. |
[8] |
I. Barbalat, Systemes d'équations différentielles d'oscillations non linéaires, Rev. Math. Pures. Appl., 4 (1959), 267-270. |
[9] |
J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465.
doi: 10.1126/science.197.4302.463. |
[10] |
J. X. Chen, C. H. Yu and L. Jin, Mathematical Analysis, Higher Education Press, Beijing, 2 edition, 2004. |
[11] |
C. Chiarella, X. He, D. Wang and M. Zheng, The stochastic bifurcation behaviour of speculative financial markets, Physica A., 387 (2008), 3837-3846.
doi: 10.1016/j.physa.2008.01.078. |
[12] |
C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewal Resources, John Wiley and Sons Inc., New York, 2 edition, 1990. |
[13] |
H. Crauel and M. Gundlach, Stochastic Dynamics, Springer-Verlag, New York, 1999.
doi: 10.1007/b97846. |
[14] |
T. C. Gard, Persistence in stochastic food web models, Bull. Math. Biol., 46 (1984), 357-370.
doi: 10.1007/BF02462011. |
[15] |
T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419.
doi: 10.1016/0362-546X(86)90111-2. |
[16] |
G. Hu and K. Wang, Stability in distribution of competitive lotka-volterra system with markovian switching, Appl. Math. Model., 35 (2011), 3189-3200.
doi: 10.1016/j.apm.2010.12.025. |
[17] |
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Amsterdam, North-Holland, 1981. |
[18] |
D. Jiang, C. Ji, X. Li and D. O'Regan, Analysis of autonomous lotka-volterra competition system with random perturbation, J. Math. Anal. Appl., 390 (2012), 582-595.
doi: 10.1016/j.jmaa.2011.12.049. |
[19] |
F. C. Klebaner, Introduction to Stochastic Calculus With Applications, Imperial College Press, London, 2005.
doi: 10.1142/p386. |
[20] |
H. Kunita, Itô's stochastic calculus: Its surprising power for applications, Stochastic Process. Appl., 120 (2010), 622-652.
doi: 10.1016/j.spa.2010.01.013. |
[21] |
W. Li, K. Wang and H. Su, Optimal harvesting policy for stochastic logistic population model, Appl. Math. Comput., 218 (2011), 157-162.
doi: 10.1016/j.amc.2011.05.079. |
[22] |
X. Li, D. Jiang and X. Mao, Population dynamical behavior of lotka-volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448.
doi: 10.1016/j.cam.2009.06.021. |
[23] |
X. Li and X. Mao, Population dynamical behavior of non-autonomous lotka-volterra competitive system with random perturbation, Discret. Contin. Dyn. S., 24 (2009), 523-545.
doi: 10.3934/dcds.2009.24.523. |
[24] |
R. S. Liptser, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228.
doi: 10.1080/17442508008833146. |
[25] |
A. J. Lotka, Elements of Physical Biology, William and Wilkins, Baltimore, 1925. |
[26] |
E. M. Lungu and B. Øksendal, Optimal harvesting from a population in a stochastic crowded environment, Math. Biosci., 145 (1997), 47-75.
doi: 10.1016/S0025-5564(97)00029-1. |
[27] |
X. Mao, Stochastic Differential Equations and Applications, Horwood, New York, 1997.
doi: 10.1533/9780857099402. |
[28] |
X. Mao, Stationary distribution of stochastic population systems, Syst. Control Letters, 60 (2011), 398-405.
doi: 10.1016/j.sysconle.2011.02.013. |
[29] |
X. Mao, G. Marion and E. Renshaw, Environmental brownian noise suppresses explosions in populations dynamics, Stochastic Process. Appl., 97 (2002), 95-110.
doi: 10.1016/S0304-4149(01)00126-0. |
[30] |
X. Mao, S. Sabanis and E. Renshaw, Asymptotic behavior of the stochastic lotka-volterra model, J. Math. Anal. Appl., 287 (2003), 141-156.
doi: 10.1016/S0022-247X(03)00539-0. |
[31] |
X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.
doi: 10.1142/p473. |
[32] |
R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 2001. |
[33] |
B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin, 6 edition, 2003.
doi: 10.1007/978-3-642-14394-6. |
[34] |
S. Peng and X. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Process. Appl., 116 (2006), 370-380.
doi: 10.1016/j.spa.2005.08.004. |
[35] |
D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996.
doi: 10.1017/CBO9780511662829. |
[36] |
M. A. Shah and U. Sharma, Optimal harvesting policies for a generalized gordon-schaefer model in randomly varying environment, Appl. Stochastic Models Bus. Ind., 19 (2003), 43-49.
doi: 10.1002/asmb.490. |
[37] |
R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005. |
[38] |
V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie d'animali conviventi, Mem. Acad. Lincei, 2 (1926), 31-113. |
[39] |
K. Wang, Stochastic Biomathematics Models, Science Press, Beijing, 2010. |
[40] |
C. Zhu and G. Yin, On competitive lotka-volterra model in random environments, J. Math. Anal. Appl., 357 (2009), 154-170.
doi: 10.1016/j.jmaa.2009.03.066. |
[41] |
C. Zhu and G. Yin, On hybrid competitive lotka-volterra ecosystems, Nonlinear Anal., 71 (2009), e1370-e1379.
doi: 10.1016/j.na.2009.01.166. |
[42] |
X. Zou and K. Wang, Numerical simulations and modeling for stochastic biological systems with jumps, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 1557-1568.
doi: 10.1016/j.cnsns.2013.09.010. |
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