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Kolmogorov-type systems with regime-switching jump diffusion perturbations
1. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China |
2. | Department of Mathematics, Wayne State University, Detroit, Michigan 48202 |
3. | Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia |
References:
[1] |
W. J. Anderson, Continuous-Time Markov Chains,, Springer-Verlag, (1991).
doi: 10.1007/978-1-4612-3038-0. |
[2] |
D. Applebaum, Levy Processes and Stochastic Calculus,, $2^{nd}$ Edition, (2009).
doi: 10.1017/CBO9780511809781. |
[3] |
D. Applebaum and M. Siakalli, Asymptotic stability properties of stochastic differential equations driven by Lévy noise,, Journal of Applied Probability, 46 (2009), 1116. Google Scholar |
[4] |
A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model,, Journal of Mathematical Analysis and Applications, 292 (2004), 364. Google Scholar |
[5] |
A. Bahar and X. Mao, Stochastic delay population dynamcis,, International Journal of pure and applied mathematics, 11 (2004), 377.
|
[6] |
J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps,, Nonlinear Analysis: Theory, 74 (2011), 6601.
doi: 10.1016/j.na.2011.06.043. |
[7] |
J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise,, Journal of Mathematical Analysis and Applications, 391 (2012), 363.
doi: 10.1016/j.jmaa.2012.02.043. |
[8] |
N. H. Dang, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise,, Journal of Differential Equations, 257 (2014), 2078.
doi: 10.1016/j.jde.2014.05.029. |
[9] |
A. Dembo and O. Zeitouni, Large Deviation Techniques and Applications,, $2^{nd}$, (1998).
doi: 10.1007/978-1-4612-5320-4. |
[10] |
J. D. Deuschel and D. W. Stroock, Large Deviations,, Academic Press, (1989).
|
[11] |
M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, I,, Communications on Pure and Applied Mathematics, 28 (1975), 1.
|
[12] |
M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, II,, Communications on Pure and Applied Mathematics, 28 (1975), 279.
|
[13] |
M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, III,, Communications on Pure and Applied Mathematics, 29 (1976), 389.
doi: 10.1002/cpa.3160290405. |
[14] |
M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, IV,, Communications on Pure and Applied Mathematics, 36 (1983), 183.
doi: 10.1002/cpa.3160360204. |
[15] |
N. H. Du and N. H. Dang, Dynamics of Kolmogorov systems of competitive type under the telegraph noise,, Journal of Differential Equations, 250 (2011), 386.
doi: 10.1016/j.jde.2010.08.023. |
[16] |
N. H. Du, D. H. Nguyen and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models,, to appear in Journal of Applied Probability., ().
doi: 10.1017/jpr.2015.18. |
[17] |
T. Faria and J. J. Oliveira, Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks,, Journal of Differential Equations, 244 (2008), 1049.
doi: 10.1016/j.jde.2007.12.005. |
[18] |
B. M. Gary, A functional equation characterizing monomial functions used in permanence theory for ecological differential equation,, Universitatis Iagellonicae acta mathematica, 42 (2004), 69.
|
[19] |
T. C. Gard, Persistence in stochastic food web models,, Bulletin of Mathematical Biology, 46 (1984), 357.
doi: 10.1007/BF02462011. |
[20] |
T. C. Gard, Stability for multispecies population models in random environments,, Nonlinear Analysis, 10 (1986), 1411.
doi: 10.1016/0362-546X(86)90111-2. |
[21] |
T. C. Gard, Introduction to Stochastic Differential Equations,, Dekker, (1988).
|
[22] |
K. Gopalsamy, Global asymptotic stability in a periodic Lotka-Volterra system,, The Journal of the Australian Mathematical Society. Series B, 27 (1985), 66.
doi: 10.1017/S0334270000004768. |
[23] |
X. Han, Z. Teng and D. Xiao, Persistence and average persistence of a nonautonomous Kolmogorov system,, Chaos Solitons Fractals, 30 (2006), 748.
doi: 10.1016/j.chaos.2006.04.026. |
[24] |
O. Kallenberg, Foundations of Modern Probability,, $2^{nd}$ Edition, (2002).
|
[25] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic press, (1993).
|
[26] |
Y. Li and Y. Kuang, Periodic solutions of periodic delay Lotka-Volterra equations and systems,, Journal of Mathematical Analysis and Applications, 255 (2001), 260.
doi: 10.1006/jmaa.2000.7248. |
[27] |
M. Liu and K. Wang, Stochastic Lotka-Volterra systems with Lévy noise,, Journal of Mathematical Analysis and Applications, 410 (2014), 750.
doi: 10.1016/j.jmaa.2013.07.078. |
[28] |
Q. Luo and X. Mao, Stochastic population dynamics under regime switching,, Journal of Mathematical Analysis and applications, 334 (2007), 69.
doi: 10.1016/j.jmaa.2006.12.032. |
[29] |
Q. Luo and X. Mao, Stochastic population dynamics under regime switching II,, Journal of Mathematical Analysis and Applications, 355 (2009), 577.
doi: 10.1016/j.jmaa.2009.02.010. |
[30] |
E. Lungu and B. Øksendal, Optimal harvesting from a population in a stochastic crowded environment,, Mathematical Biosciences, 145 (1997), 47.
doi: 10.1016/S0025-5564(97)00029-1. |
[31] |
E. Lungu and B. Øksendal, Optimal Harvesting from Interacting Populations in a Stochastic Environment,, Bernoulli, 7 (2001), 527.
doi: 10.2307/3318500. |
[32] |
X. Mao, G. Marion and E. Renshaw, Environmental noise supresses explosion in population dynamics,, Stochastic Processes and their Applications, 97 (2002), 95.
doi: 10.1016/S0304-4149(01)00126-0. |
[33] |
X. Mao, Delay population dynamics and environmental noise,, Stochastics and Dynamics, 5 (2005), 149.
doi: 10.1142/S021949370500133X. |
[34] |
X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, (2006).
doi: 10.1142/p473. |
[35] |
X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ Edition, (2008).
doi: 10.1533/9780857099402. |
[36] |
J. D. Murray, Mathematical Biology, I. an Introduction,, $3^{rd}$ Edition, (2002).
|
[37] |
H. D. Nguyen, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise,, Journal of Differential Equations, 257 (2014), 2078.
doi: 10.1016/j.jde.2014.05.029. |
[38] |
S. Pang, F. Deng and X. Mao, Asymptotic properties of stochastic population dynamics,, Dynamics of Continuous Discrete and Impulsive Systems Series A, 15 (2008), 603.
|
[39] |
P. E. Protter, Stochastic Integration and Differential Equations,, $2^{nd}$ Edition, (2004).
|
[40] |
M. Slatkin, The dynamics of a population in a Markovian environment,, Ecology, 59 (1978), 249.
doi: 10.2307/1936370. |
[41] |
Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems,, World Scientific, (1996).
doi: 10.1142/9789812830548. |
[42] |
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,, Journal of Mathematical Analysis and applications, 323 (2006), 938.
doi: 10.1016/j.jmaa.2005.11.009. |
[43] |
B. Tang and Y. Kuang, Permanence in Kolmogorov-type systems of nonautonomous functional differential equations,, Journal of Mathematical Analysis and Applications, 197 (1996), 427.
doi: 10.1006/jmaa.1996.0030. |
[44] |
Z. Teng, The almost periodic Kolmogorov competitive sysems,, Nonlinear Analysis, 42 (2000), 1221.
doi: 10.1016/S0362-546X(99)00149-2. |
[45] |
F. Wu and S. Hu, Stochastic functional Kolmogorov-type population dynamics,, Journal of Mathematical analysis and applications, 347 (2008), 534.
doi: 10.1016/j.jmaa.2008.06.038. |
[46] |
F. Wu and S. Hu, Suppression and stabilisation of noise,, International Journal of Control, 82 (2009), 2150.
doi: 10.1080/00207170902968108. |
[47] |
F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay,, SIAM Journal on Applied Mathematics, 70 (2009), 641.
doi: 10.1137/080719194. |
[48] |
F. Wu, S. Hu and Y. Liu, Positive solution and its asymptotic behaviour of stochastic functional Kolmogorov-type system,, Journal of Mathematical Analysis and Applications, 364 (2010), 104.
doi: 10.1016/j.jmaa.2009.10.072. |
[49] |
F. Wu and G. Yin, Environmental noise impact on regularity and extinction of population systems with infinite delay,, Journal of Mathematical Analysis and Applications, 396 (2012), 772.
doi: 10.1016/j.jmaa.2012.07.017. |
[50] |
G. Yin and F. Xi, Stablity of regime-switching jump diffusions,, SIAM Journal on Control and Optimization, 48 (2010), 4525.
doi: 10.1137/080738301. |
[51] |
G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications,, Springer, (2010).
doi: 10.1007/978-1-4419-1105-6. |
[52] |
G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems,, SIAM Journal on Applied Mathematics, 72 (2012), 1361.
doi: 10.1137/110851171. |
[53] |
C. Zhu and G. Yin, On hybrid competitive Lotka-Volterra ecosystems,, Nonlinear Analysis: Theory, 71 (2009).
doi: 10.1016/j.na.2009.01.166. |
[54] |
C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments,, Journal of Mathematical Analysis and Applications, 357 (2009), 154.
doi: 10.1016/j.jmaa.2009.03.066. |
[55] |
X. Zong, F. Wu, G. Yin and Z. Jin, Almost sure and $p$th-moment stability and stabilization of regime-switching jump diffusion systems,, SIAM Journal on Control and Optimization, 52 (2014), 2595.
doi: 10.1137/14095251X. |
show all references
References:
[1] |
W. J. Anderson, Continuous-Time Markov Chains,, Springer-Verlag, (1991).
doi: 10.1007/978-1-4612-3038-0. |
[2] |
D. Applebaum, Levy Processes and Stochastic Calculus,, $2^{nd}$ Edition, (2009).
doi: 10.1017/CBO9780511809781. |
[3] |
D. Applebaum and M. Siakalli, Asymptotic stability properties of stochastic differential equations driven by Lévy noise,, Journal of Applied Probability, 46 (2009), 1116. Google Scholar |
[4] |
A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model,, Journal of Mathematical Analysis and Applications, 292 (2004), 364. Google Scholar |
[5] |
A. Bahar and X. Mao, Stochastic delay population dynamcis,, International Journal of pure and applied mathematics, 11 (2004), 377.
|
[6] |
J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps,, Nonlinear Analysis: Theory, 74 (2011), 6601.
doi: 10.1016/j.na.2011.06.043. |
[7] |
J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise,, Journal of Mathematical Analysis and Applications, 391 (2012), 363.
doi: 10.1016/j.jmaa.2012.02.043. |
[8] |
N. H. Dang, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise,, Journal of Differential Equations, 257 (2014), 2078.
doi: 10.1016/j.jde.2014.05.029. |
[9] |
A. Dembo and O. Zeitouni, Large Deviation Techniques and Applications,, $2^{nd}$, (1998).
doi: 10.1007/978-1-4612-5320-4. |
[10] |
J. D. Deuschel and D. W. Stroock, Large Deviations,, Academic Press, (1989).
|
[11] |
M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, I,, Communications on Pure and Applied Mathematics, 28 (1975), 1.
|
[12] |
M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, II,, Communications on Pure and Applied Mathematics, 28 (1975), 279.
|
[13] |
M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, III,, Communications on Pure and Applied Mathematics, 29 (1976), 389.
doi: 10.1002/cpa.3160290405. |
[14] |
M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, IV,, Communications on Pure and Applied Mathematics, 36 (1983), 183.
doi: 10.1002/cpa.3160360204. |
[15] |
N. H. Du and N. H. Dang, Dynamics of Kolmogorov systems of competitive type under the telegraph noise,, Journal of Differential Equations, 250 (2011), 386.
doi: 10.1016/j.jde.2010.08.023. |
[16] |
N. H. Du, D. H. Nguyen and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models,, to appear in Journal of Applied Probability., ().
doi: 10.1017/jpr.2015.18. |
[17] |
T. Faria and J. J. Oliveira, Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks,, Journal of Differential Equations, 244 (2008), 1049.
doi: 10.1016/j.jde.2007.12.005. |
[18] |
B. M. Gary, A functional equation characterizing monomial functions used in permanence theory for ecological differential equation,, Universitatis Iagellonicae acta mathematica, 42 (2004), 69.
|
[19] |
T. C. Gard, Persistence in stochastic food web models,, Bulletin of Mathematical Biology, 46 (1984), 357.
doi: 10.1007/BF02462011. |
[20] |
T. C. Gard, Stability for multispecies population models in random environments,, Nonlinear Analysis, 10 (1986), 1411.
doi: 10.1016/0362-546X(86)90111-2. |
[21] |
T. C. Gard, Introduction to Stochastic Differential Equations,, Dekker, (1988).
|
[22] |
K. Gopalsamy, Global asymptotic stability in a periodic Lotka-Volterra system,, The Journal of the Australian Mathematical Society. Series B, 27 (1985), 66.
doi: 10.1017/S0334270000004768. |
[23] |
X. Han, Z. Teng and D. Xiao, Persistence and average persistence of a nonautonomous Kolmogorov system,, Chaos Solitons Fractals, 30 (2006), 748.
doi: 10.1016/j.chaos.2006.04.026. |
[24] |
O. Kallenberg, Foundations of Modern Probability,, $2^{nd}$ Edition, (2002).
|
[25] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic press, (1993).
|
[26] |
Y. Li and Y. Kuang, Periodic solutions of periodic delay Lotka-Volterra equations and systems,, Journal of Mathematical Analysis and Applications, 255 (2001), 260.
doi: 10.1006/jmaa.2000.7248. |
[27] |
M. Liu and K. Wang, Stochastic Lotka-Volterra systems with Lévy noise,, Journal of Mathematical Analysis and Applications, 410 (2014), 750.
doi: 10.1016/j.jmaa.2013.07.078. |
[28] |
Q. Luo and X. Mao, Stochastic population dynamics under regime switching,, Journal of Mathematical Analysis and applications, 334 (2007), 69.
doi: 10.1016/j.jmaa.2006.12.032. |
[29] |
Q. Luo and X. Mao, Stochastic population dynamics under regime switching II,, Journal of Mathematical Analysis and Applications, 355 (2009), 577.
doi: 10.1016/j.jmaa.2009.02.010. |
[30] |
E. Lungu and B. Øksendal, Optimal harvesting from a population in a stochastic crowded environment,, Mathematical Biosciences, 145 (1997), 47.
doi: 10.1016/S0025-5564(97)00029-1. |
[31] |
E. Lungu and B. Øksendal, Optimal Harvesting from Interacting Populations in a Stochastic Environment,, Bernoulli, 7 (2001), 527.
doi: 10.2307/3318500. |
[32] |
X. Mao, G. Marion and E. Renshaw, Environmental noise supresses explosion in population dynamics,, Stochastic Processes and their Applications, 97 (2002), 95.
doi: 10.1016/S0304-4149(01)00126-0. |
[33] |
X. Mao, Delay population dynamics and environmental noise,, Stochastics and Dynamics, 5 (2005), 149.
doi: 10.1142/S021949370500133X. |
[34] |
X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, (2006).
doi: 10.1142/p473. |
[35] |
X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ Edition, (2008).
doi: 10.1533/9780857099402. |
[36] |
J. D. Murray, Mathematical Biology, I. an Introduction,, $3^{rd}$ Edition, (2002).
|
[37] |
H. D. Nguyen, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise,, Journal of Differential Equations, 257 (2014), 2078.
doi: 10.1016/j.jde.2014.05.029. |
[38] |
S. Pang, F. Deng and X. Mao, Asymptotic properties of stochastic population dynamics,, Dynamics of Continuous Discrete and Impulsive Systems Series A, 15 (2008), 603.
|
[39] |
P. E. Protter, Stochastic Integration and Differential Equations,, $2^{nd}$ Edition, (2004).
|
[40] |
M. Slatkin, The dynamics of a population in a Markovian environment,, Ecology, 59 (1978), 249.
doi: 10.2307/1936370. |
[41] |
Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems,, World Scientific, (1996).
doi: 10.1142/9789812830548. |
[42] |
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,, Journal of Mathematical Analysis and applications, 323 (2006), 938.
doi: 10.1016/j.jmaa.2005.11.009. |
[43] |
B. Tang and Y. Kuang, Permanence in Kolmogorov-type systems of nonautonomous functional differential equations,, Journal of Mathematical Analysis and Applications, 197 (1996), 427.
doi: 10.1006/jmaa.1996.0030. |
[44] |
Z. Teng, The almost periodic Kolmogorov competitive sysems,, Nonlinear Analysis, 42 (2000), 1221.
doi: 10.1016/S0362-546X(99)00149-2. |
[45] |
F. Wu and S. Hu, Stochastic functional Kolmogorov-type population dynamics,, Journal of Mathematical analysis and applications, 347 (2008), 534.
doi: 10.1016/j.jmaa.2008.06.038. |
[46] |
F. Wu and S. Hu, Suppression and stabilisation of noise,, International Journal of Control, 82 (2009), 2150.
doi: 10.1080/00207170902968108. |
[47] |
F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay,, SIAM Journal on Applied Mathematics, 70 (2009), 641.
doi: 10.1137/080719194. |
[48] |
F. Wu, S. Hu and Y. Liu, Positive solution and its asymptotic behaviour of stochastic functional Kolmogorov-type system,, Journal of Mathematical Analysis and Applications, 364 (2010), 104.
doi: 10.1016/j.jmaa.2009.10.072. |
[49] |
F. Wu and G. Yin, Environmental noise impact on regularity and extinction of population systems with infinite delay,, Journal of Mathematical Analysis and Applications, 396 (2012), 772.
doi: 10.1016/j.jmaa.2012.07.017. |
[50] |
G. Yin and F. Xi, Stablity of regime-switching jump diffusions,, SIAM Journal on Control and Optimization, 48 (2010), 4525.
doi: 10.1137/080738301. |
[51] |
G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications,, Springer, (2010).
doi: 10.1007/978-1-4419-1105-6. |
[52] |
G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems,, SIAM Journal on Applied Mathematics, 72 (2012), 1361.
doi: 10.1137/110851171. |
[53] |
C. Zhu and G. Yin, On hybrid competitive Lotka-Volterra ecosystems,, Nonlinear Analysis: Theory, 71 (2009).
doi: 10.1016/j.na.2009.01.166. |
[54] |
C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments,, Journal of Mathematical Analysis and Applications, 357 (2009), 154.
doi: 10.1016/j.jmaa.2009.03.066. |
[55] |
X. Zong, F. Wu, G. Yin and Z. Jin, Almost sure and $p$th-moment stability and stabilization of regime-switching jump diffusion systems,, SIAM Journal on Control and Optimization, 52 (2014), 2595.
doi: 10.1137/14095251X. |
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