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Dynamic behavior of a stochastic predator-prey system under regime switching
On stochastic multi-group Lotka-Volterra ecosystems with regime switching
School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China |
Focusing on stochastic dynamics involving continuous states as well as discrete events, this paper investigates dynamical behaviors of stochastic multi-group Lotka-Volterra model with regime switching. The contributions of the paper lie on: (a) giving the sufficient conditions of stochastic permanence for generic stochastic multi-group Lotka-Volterra model, which are much weaker than the existing results in the literature; (b) obtaining the stochastic strong permanence and ergodic property for the mutualistic systems; (c) establishing the almost surely asymptotic estimate of solutions. These can specify some realistic recurring phenomena and reveal the fact that regime switching can suppress the impermanence. A couple of examples and numerical simulations are given to illustrate our results.
References:
[1] |
J. Bao and J. Shao,
Permanence and extinction of regime-switching predator-prey models, SIAM J. Math. Anal., 48 (2016), 725-739.
doi: 10.1137/15M1024512. |
[2] |
M. Benaïm and C. Lobry,
Lotka-Volterra with randomly fluctuating environments or "how switching between beneficial environments can make survival harder", Ann. Appl. Probab., 26 (2016), 3754-3785.
doi: 10.1214/16-AAP1192. |
[3] |
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. |
[4] |
A. Friedman,
Stochastic Differential Equations and Applications, Dover Publications, Inc. , Mineola, NY, 2006. |
[5] |
X. He and K. Gopalsamy,
Persistence, attractivity, and delay in facultative mutualism, J. Math. Anal. Appl., 215 (1997), 154-173.
doi: 10.1006/jmaa.1997.5632. |
[6] |
Y. Hu, F. Wu and C. Huang,
Stochastic Lotka-Volterra models with multiple delays, J. Math. Anal. Appl., 375 (2011), 42-57.
doi: 10.1016/j.jmaa.2010.08.017. |
[7] |
G. E. Hutchinson,
The Paradox of the plankton, Amer. Nat., 95 (1961), 137-145.
doi: 10.1086/282171. |
[8] |
A. M. Il'in, R. Z. Khasminskii and G. Yin,
Asymptotic expansions of solutions of integro-differential equations for transition densities of singularly perturbed switching diffusions: Rapid switching, J. Math. Anal. Appl., 238 (1999), 516-539.
doi: 10.1006/jmaa.1998.6532. |
[9] |
R. Khasminskii,
Stochastic Stability of Differential Equations, 2nd edition, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-23280-0. |
[10] |
S. D. Lawley, J. C. Mattingly and M. C. Reed,
Sensitivity to switching rates in stochastically switched ODEs, Commun. Math. Sci., 12 (2014), 1343-1352.
doi: 10.4310/CMS.2014.v12.n7.a9. |
[11] |
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. |
[12] |
X. Li and G. Yin,
Logistic models with regime switching: Permanence and ergodicity, J. Math. Anal. Appl., 441 (2016), 593-611.
doi: 10.1016/j.jmaa.2016.04.016. |
[13] |
R. Liptser,
A strong law of large numbers for local martingale, Stochastics, 3 (1980), 217-228.
doi: 10.1080/17442508008833146. |
[14] |
H. Liu, X. Li and Q. Yang,
The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Systems Control Lett., 62 (2013), 805-810.
doi: 10.1016/j.sysconle.2013.06.002. |
[15] |
M. Liu,
Global asymptotic stability of stochastic Lotka-Volterra systems with infinite delays, IMA J. Appl. Math., 80 (2015), 1431-1453.
doi: 10.1093/imamat/hxv002. |
[16] |
A. J. Lotka,
Elements of Physical Biology, William and Wilkins, Baltimore, 1925. |
[17] |
Q. Luo and X. Mao,
tochastic population dynamics under regime switching Ⅱ, J. Math. Anal. Appl., 355 (2009), 577-593.
doi: 10.1016/j.jmaa.2009.02.010. |
[18] |
X. Mao,
Stochastic Differential Equations and Applications, 2nd edition, Horwwood Publishing, Chichester, 2008.
doi: 10.1533/9780857099402. |
[19] |
X. Mao and C. Yuan,
Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.
doi: 10.1142/p473. |
[20] |
X. Mao,
Stationary distribution of stochastic population systems, Systems Control Lett., 60 (2011), 398-405.
doi: 10.1016/j.sysconle.2011.02.013. |
[21] |
D. H. Nguyen and G. Yin,
Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 267 (2017), 1192-1225.
doi: 10.1016/j.jde.2016.10.005. |
[22] |
M. Slatkin,
The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249-256.
doi: 10.2307/1936370. |
[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] |
V. Volterra,
Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. Acad. Lincei., 2 (1926), 31-113.
|
[25] |
F. Wu and Y. Xu,
Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM J. Appl. Math., 70 (2009), 641-657.
doi: 10.1137/080719194. |
[26] |
G. Yin and Q. Zhang,
Continuous-Time Markov Chains and Applications: A Two-Time-Scale Approach, 2nd edition, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4346-9. |
[27] |
G. Yin and C. Zhu,
Hybrid Switching Diffusions Properties and Applications, Springer, New York, 2010.
doi: 10.1007/978-1-4419-1105-6. |
[28] |
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. |
[29] |
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. |
show all references
References:
[1] |
J. Bao and J. Shao,
Permanence and extinction of regime-switching predator-prey models, SIAM J. Math. Anal., 48 (2016), 725-739.
doi: 10.1137/15M1024512. |
[2] |
M. Benaïm and C. Lobry,
Lotka-Volterra with randomly fluctuating environments or "how switching between beneficial environments can make survival harder", Ann. Appl. Probab., 26 (2016), 3754-3785.
doi: 10.1214/16-AAP1192. |
[3] |
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. |
[4] |
A. Friedman,
Stochastic Differential Equations and Applications, Dover Publications, Inc. , Mineola, NY, 2006. |
[5] |
X. He and K. Gopalsamy,
Persistence, attractivity, and delay in facultative mutualism, J. Math. Anal. Appl., 215 (1997), 154-173.
doi: 10.1006/jmaa.1997.5632. |
[6] |
Y. Hu, F. Wu and C. Huang,
Stochastic Lotka-Volterra models with multiple delays, J. Math. Anal. Appl., 375 (2011), 42-57.
doi: 10.1016/j.jmaa.2010.08.017. |
[7] |
G. E. Hutchinson,
The Paradox of the plankton, Amer. Nat., 95 (1961), 137-145.
doi: 10.1086/282171. |
[8] |
A. M. Il'in, R. Z. Khasminskii and G. Yin,
Asymptotic expansions of solutions of integro-differential equations for transition densities of singularly perturbed switching diffusions: Rapid switching, J. Math. Anal. Appl., 238 (1999), 516-539.
doi: 10.1006/jmaa.1998.6532. |
[9] |
R. Khasminskii,
Stochastic Stability of Differential Equations, 2nd edition, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-23280-0. |
[10] |
S. D. Lawley, J. C. Mattingly and M. C. Reed,
Sensitivity to switching rates in stochastically switched ODEs, Commun. Math. Sci., 12 (2014), 1343-1352.
doi: 10.4310/CMS.2014.v12.n7.a9. |
[11] |
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. |
[12] |
X. Li and G. Yin,
Logistic models with regime switching: Permanence and ergodicity, J. Math. Anal. Appl., 441 (2016), 593-611.
doi: 10.1016/j.jmaa.2016.04.016. |
[13] |
R. Liptser,
A strong law of large numbers for local martingale, Stochastics, 3 (1980), 217-228.
doi: 10.1080/17442508008833146. |
[14] |
H. Liu, X. Li and Q. Yang,
The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Systems Control Lett., 62 (2013), 805-810.
doi: 10.1016/j.sysconle.2013.06.002. |
[15] |
M. Liu,
Global asymptotic stability of stochastic Lotka-Volterra systems with infinite delays, IMA J. Appl. Math., 80 (2015), 1431-1453.
doi: 10.1093/imamat/hxv002. |
[16] |
A. J. Lotka,
Elements of Physical Biology, William and Wilkins, Baltimore, 1925. |
[17] |
Q. Luo and X. Mao,
tochastic population dynamics under regime switching Ⅱ, J. Math. Anal. Appl., 355 (2009), 577-593.
doi: 10.1016/j.jmaa.2009.02.010. |
[18] |
X. Mao,
Stochastic Differential Equations and Applications, 2nd edition, Horwwood Publishing, Chichester, 2008.
doi: 10.1533/9780857099402. |
[19] |
X. Mao and C. Yuan,
Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.
doi: 10.1142/p473. |
[20] |
X. Mao,
Stationary distribution of stochastic population systems, Systems Control Lett., 60 (2011), 398-405.
doi: 10.1016/j.sysconle.2011.02.013. |
[21] |
D. H. Nguyen and G. Yin,
Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 267 (2017), 1192-1225.
doi: 10.1016/j.jde.2016.10.005. |
[22] |
M. Slatkin,
The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249-256.
doi: 10.2307/1936370. |
[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] |
V. Volterra,
Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. Acad. Lincei., 2 (1926), 31-113.
|
[25] |
F. Wu and Y. Xu,
Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM J. Appl. Math., 70 (2009), 641-657.
doi: 10.1137/080719194. |
[26] |
G. Yin and Q. Zhang,
Continuous-Time Markov Chains and Applications: A Two-Time-Scale Approach, 2nd edition, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4346-9. |
[27] |
G. Yin and C. Zhu,
Hybrid Switching Diffusions Properties and Applications, Springer, New York, 2010.
doi: 10.1007/978-1-4419-1105-6. |
[28] |
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. |
[29] |
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. |









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