Figure 1.
Dynamical behaviors of model (46) with $ c(t) = 0.85+0.75\sin t-\int_{\mathbb{Z}}h(t,z)\pi(dz) $ : (a) $ h = 0.4 $ ; (b) $ h = 2 $ . Other parameters are given by (47)
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October
2021, 26(10): 5641-5660.
doi: 10.3934/dcdsb.2020371
Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises
School of Mathematics and Statistics, Yangtze Normal University, Chongqing, 408100, China |
This paper considers a stochastic single-species model with Lévy noises and time periodic coefficients. By Lyapunov functions and stochastic estimates, the threshold conditions between the time-average persistence in probability and extinction for the model are derived where Lévy noises play an important role in persistence and extinction of populations. It is shown that the time-average persistence in probability of the model implies the existence and uniqueness of positive periodic solution and the existence and uniqueness of periodic measure of the model. An example and its numerical simulations are given to verify the effectiveness of the theoretical results.
Mathematics Subject Classification:
Primary: 92B05, 60G51, 60G57, 60H35.
Citation:
Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete and Continuous Dynamical Systems - B,
2021, 26
(10)
: 5641-5660.
doi: 10.3934/dcdsb.2020371
![]()
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A stochastic SIRI epidemic model with Lévy noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2415-2431.
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R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman Hall/CRC, 2004. |
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J. Cyr, P. Nguyen and R. Temam,
Stochastic one layer shallow water equations with Lévy noise, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3765-3818.
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J. M. Cushing,
Periodic time-dependent predator-prey system, SIAM J. Appl. Math., 32 (1977), 82-95.
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M. Deng, Dynamics of a stochastic population model with Allee effect and Lévy jumps, Physica A, 531 (2019), 121745, 11 pp.
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J. Duan, An Introduction to Stochastic Dynamics, Cambridge University Press, New York, 2015.
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S. N. Evans, P. L. Ralph, S. J. Schreiber and A. Sen,
Stochastic population growth in spatially heterogeneous environments, J. Math. Biol., 66 (2013), 423-476.
doi: 10.1007/s00285-012-0514-0. |
| [12] |
C. Feng and H. Zhao,
Random periodic processes, periodic measures and ergodicity, J. Differ. Equations, 269 (2020), 7382-7428.
doi: 10.1016/j.jde.2020.05.034. |
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T. G. Hallam and Z. E. Ma,
Persistence in population models with demographic fluctuations, J. Math. Biol., 24 (1986), 327-339.
doi: 10.1007/BF00275641. |
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P. R. Halmos, Measure Theory, Springer-Verlag, New York, 1970.
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A. Hening and D. H. Nguyen,
Stochastic Lotka-Volterra food chains, J. Math. Biol., 77 (2018), 135-163.
doi: 10.1007/s00285-017-1192-8. |
| [16] |
G. Hu and Y. Li,
Asymptotic behaviors of stochastic periodic differential equations with Markovian switching, Appl. Math. Compt., 264 (2015), 403-416.
doi: 10.1016/j.amc.2015.04.033. |
| [17] |
D. Jiang and N. Shi,
A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164-172.
doi: 10.1016/j.jmaa.2004.08.027. |
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D. Jiang, N. Shi and X. Li,
Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008), 588-597.
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Permanence and extinction for a single-species systems with jump-diffusion, J. Math. Anal. Appl., 430 (2015), 438-464.
doi: 10.1016/j.jmaa.2015.04.050. |
| [21] |
D. Li and D. Xu,
Periodic solutions of stochastic delay differential equations and applications to logistic equation and neural networks, J. Korean Math. Soc., 50 (2013), 1165-1181.
doi: 10.4134/JKMS.2013.50.6.1165. |
| [22] |
R. S. Liptser,
A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228.
doi: 10.1080/17442508008833146. |
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M. Liu and C. Z. Bai,
On a stochastic delayed predator-prey model with Lévy jumps, Appl. Math. Comput., 228 (2014), 563-570.
doi: 10.1016/j.amc.2013.12.026. |
| [24] |
M. Liu and K. Wang,
Stochastic Lotka-Volterra systems with Lévy noise, J. Math. Anal. Appl., 410 (2014), 750-763.
doi: 10.1016/j.jmaa.2013.07.078. |
| [25] |
M. Liu and Y. Zhu,
Stationary distribution and ergodicity of a stochastic hybrid competition model with Lévy jumps, Nonlinear Anal. Hybrid Syst., 30 (2018), 225-239.
doi: 10.1016/j.nahs.2018.05.002. |
| [26] |
Q. Liu and Q. Chen,
Asymptotic behavior of a stochastic non-autonomous predator-prey system with jumps, Appl. Math. Comput., 271 (2015), 418-428.
doi: 10.1016/j.amc.2015.08.040. |
| [27] |
Q. Liu and D. Jiang,
Periodic solution and stationary distribution of stochastic predator-prey models with higher-order perturbation, J. Nonlinear Sci., 28 (2018), 423-442.
doi: 10.1007/s00332-017-9413-2. |
| [28] |
Q. Liu, D. Jiang, N. Shi, T. Hayat and A. Alsaedi,
Stochastic mutualism model with Lévy jumps, Commun. Nonlinear Sci. Numer. Simul., 43 (2017), 78-90.
doi: 10.1016/j.cnsns.2016.05.003. |
| [29] |
R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 1973.
|
| [30] |
D. Nguyen and G. G. Yin,
Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differ. Equations, 262 (2017), 1192-1225.
doi: 10.1016/j.jde.2016.10.005. |
| [31] |
S. J. Schreiber, M. Benaim and K. A. S. Atchadé,
Persistence in fluctuating environments, J. Math. Biol., 62 (2011), 655-683.
doi: 10.1007/s00285-010-0349-5. |
| [32] |
H. Wang, C. Du and M. Liu,
Dynamics of a stochastic service resource mutualism model with Lévy noises and harvesting, J. Nonlinear Sci. Appl., 10 (2017), 6205-6218.
doi: 10.22436/jnsa.010.12.07. |
| [33] |
Y. Wang and Z. Liu,
Almost periodic solutions for stochastic differential equations with Lévy noise, Nonlinearity, 25 (2012), 2803-2821.
doi: 10.1088/0951-7715/25/10/2803. |
| [34] |
J. Yang, Threshold behavior in a stochastic predator-prey model with general functional response, Physica A, 551 (2020), 124610 12 pp.
doi: 10.1016/j.physa.2020.124610. |
| [35] |
B. G. Zhang and K. Gopalsamy,
On the periodic solution of $n$-dimensional stochastic population models, Stoc. Anal. Appl., 18 (2000), 323-331.
doi: 10.1080/07362990008809671. |
| [36] |
Q. Zhang, D. Jiang, Y. Zhao and D. O'Regan,
Asymptotic behavior of a stochastic population model with Allee effect by Lévy jumps, Nonlinear Anal. Hybrid Syst., 24 (2017), 1-12.
doi: 10.1016/j.nahs.2016.10.005. |
| [37] |
X. Zhang, K. Wang and D. Li,
Stochastic periodic solutions of stochastic differential equations driven by Lévy process, J. Math. Anal. Appl., 430 (2015), 231-242.
doi: 10.1016/j.jmaa.2015.04.090. |
| [38] |
Y. Zhao and S. Yuan,
Stability in distribution of a stochastic hybrid competitive lotka-volterra model with lévy jumps, Chaos Solitons Fract., 85 (2016), 98-109.
doi: 10.1016/j.chaos.2016.01.015. |
| [39] |
X. Zou and K. Wang,
Numerical simulations and modeling for stochastic biological systems with jumps, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1557-1568.
doi: 10.1016/j.cnsns.2013.09.010. |
| [40] |
L. Zu, D. Jiang, D. O'Regan and B. Ge,
Periodic solution for a non-autonomous Lotka-Volterra predator-prey model with random perturbation, J. Math. Anal. Appl., 430 (2015), 428-437.
doi: 10.1016/j.jmaa.2015.04.058. |
show all references
References:
| [1] |
D. Applebaum, Lévy Processes and Stochastics Calculus, 2$^nd$ edition, Cambridge University Press, 2009.
doi: 10.1017/CBO9780511809781.
|
| [2] |
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. |
| [3] |
J. Bao and C. Yuan,
Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl, 391 (2012), 363-375.
doi: 10.1016/j.jmaa.2012.02.043. |
| [4] |
B.-E. Berrhazi, M. E. Fatini, T. Caraballo and R. Pettersson,
A stochastic SIRI epidemic model with Lévy noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2415-2431.
doi: 10.3934/dcdsb.2018057. |
| [5] |
R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman Hall/CRC, 2004. |
| [6] |
J. Cyr, P. Nguyen and R. Temam,
Stochastic one layer shallow water equations with Lévy noise, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3765-3818.
doi: 10.3934/dcdsb.2018331. |
| [7] |
J. M. Cushing,
Periodic time-dependent predator-prey system, SIAM J. Appl. Math., 32 (1977), 82-95.
doi: 10.1137/0132006. |
| [8] |
G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996.
doi: 10.1017/CBO9780511662829.
|
| [9] |
M. Deng, Dynamics of a stochastic population model with Allee effect and Lévy jumps, Physica A, 531 (2019), 121745, 11 pp.
doi: 10.1016/j.physa.2019.121745. |
| [10] |
J. Duan, An Introduction to Stochastic Dynamics, Cambridge University Press, New York, 2015.
|
| [11] |
S. N. Evans, P. L. Ralph, S. J. Schreiber and A. Sen,
Stochastic population growth in spatially heterogeneous environments, J. Math. Biol., 66 (2013), 423-476.
doi: 10.1007/s00285-012-0514-0. |
| [12] |
C. Feng and H. Zhao,
Random periodic processes, periodic measures and ergodicity, J. Differ. Equations, 269 (2020), 7382-7428.
doi: 10.1016/j.jde.2020.05.034. |
| [13] |
T. G. Hallam and Z. E. Ma,
Persistence in population models with demographic fluctuations, J. Math. Biol., 24 (1986), 327-339.
doi: 10.1007/BF00275641. |
| [14] |
P. R. Halmos, Measure Theory, Springer-Verlag, New York, 1970.
doi: 10.1007/978-1-4684-9440-2. |
| [15] |
A. Hening and D. H. Nguyen,
Stochastic Lotka-Volterra food chains, J. Math. Biol., 77 (2018), 135-163.
doi: 10.1007/s00285-017-1192-8. |
| [16] |
G. Hu and Y. Li,
Asymptotic behaviors of stochastic periodic differential equations with Markovian switching, Appl. Math. Compt., 264 (2015), 403-416.
doi: 10.1016/j.amc.2015.04.033. |
| [17] |
D. Jiang and N. Shi,
A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164-172.
doi: 10.1016/j.jmaa.2004.08.027. |
| [18] |
D. Jiang, N. Shi and X. Li,
Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008), 588-597.
doi: 10.1016/j.jmaa.2007.08.014. |
| [19] |
R. Khasminskii, Stochastic Stability of Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2012.
doi: 10.1007/978-3-642-23280-0. |
| [20] |
D. Li, J. Cui and G. Song,
Permanence and extinction for a single-species systems with jump-diffusion, J. Math. Anal. Appl., 430 (2015), 438-464.
doi: 10.1016/j.jmaa.2015.04.050. |
| [21] |
D. Li and D. Xu,
Periodic solutions of stochastic delay differential equations and applications to logistic equation and neural networks, J. Korean Math. Soc., 50 (2013), 1165-1181.
doi: 10.4134/JKMS.2013.50.6.1165. |
| [22] |
R. S. Liptser,
A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228.
doi: 10.1080/17442508008833146. |
| [23] |
M. Liu and C. Z. Bai,
On a stochastic delayed predator-prey model with Lévy jumps, Appl. Math. Comput., 228 (2014), 563-570.
doi: 10.1016/j.amc.2013.12.026. |
| [24] |
M. Liu and K. Wang,
Stochastic Lotka-Volterra systems with Lévy noise, J. Math. Anal. Appl., 410 (2014), 750-763.
doi: 10.1016/j.jmaa.2013.07.078. |
| [25] |
M. Liu and Y. Zhu,
Stationary distribution and ergodicity of a stochastic hybrid competition model with Lévy jumps, Nonlinear Anal. Hybrid Syst., 30 (2018), 225-239.
doi: 10.1016/j.nahs.2018.05.002. |
| [26] |
Q. Liu and Q. Chen,
Asymptotic behavior of a stochastic non-autonomous predator-prey system with jumps, Appl. Math. Comput., 271 (2015), 418-428.
doi: 10.1016/j.amc.2015.08.040. |
| [27] |
Q. Liu and D. Jiang,
Periodic solution and stationary distribution of stochastic predator-prey models with higher-order perturbation, J. Nonlinear Sci., 28 (2018), 423-442.
doi: 10.1007/s00332-017-9413-2. |
| [28] |
Q. Liu, D. Jiang, N. Shi, T. Hayat and A. Alsaedi,
Stochastic mutualism model with Lévy jumps, Commun. Nonlinear Sci. Numer. Simul., 43 (2017), 78-90.
doi: 10.1016/j.cnsns.2016.05.003. |
| [29] |
R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 1973.
|
| [30] |
D. Nguyen and G. G. Yin,
Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differ. Equations, 262 (2017), 1192-1225.
doi: 10.1016/j.jde.2016.10.005. |
| [31] |
S. J. Schreiber, M. Benaim and K. A. S. Atchadé,
Persistence in fluctuating environments, J. Math. Biol., 62 (2011), 655-683.
doi: 10.1007/s00285-010-0349-5. |
| [32] |
H. Wang, C. Du and M. Liu,
Dynamics of a stochastic service resource mutualism model with Lévy noises and harvesting, J. Nonlinear Sci. Appl., 10 (2017), 6205-6218.
doi: 10.22436/jnsa.010.12.07. |
| [33] |
Y. Wang and Z. Liu,
Almost periodic solutions for stochastic differential equations with Lévy noise, Nonlinearity, 25 (2012), 2803-2821.
doi: 10.1088/0951-7715/25/10/2803. |
| [34] |
J. Yang, Threshold behavior in a stochastic predator-prey model with general functional response, Physica A, 551 (2020), 124610 12 pp.
doi: 10.1016/j.physa.2020.124610. |
| [35] |
B. G. Zhang and K. Gopalsamy,
On the periodic solution of $n$-dimensional stochastic population models, Stoc. Anal. Appl., 18 (2000), 323-331.
doi: 10.1080/07362990008809671. |
| [36] |
Q. Zhang, D. Jiang, Y. Zhao and D. O'Regan,
Asymptotic behavior of a stochastic population model with Allee effect by Lévy jumps, Nonlinear Anal. Hybrid Syst., 24 (2017), 1-12.
doi: 10.1016/j.nahs.2016.10.005. |
| [37] |
X. Zhang, K. Wang and D. Li,
Stochastic periodic solutions of stochastic differential equations driven by Lévy process, J. Math. Anal. Appl., 430 (2015), 231-242.
doi: 10.1016/j.jmaa.2015.04.090. |
| [38] |
Y. Zhao and S. Yuan,
Stability in distribution of a stochastic hybrid competitive lotka-volterra model with lévy jumps, Chaos Solitons Fract., 85 (2016), 98-109.
doi: 10.1016/j.chaos.2016.01.015. |
| [39] |
X. Zou and K. Wang,
Numerical simulations and modeling for stochastic biological systems with jumps, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1557-1568.
doi: 10.1016/j.cnsns.2013.09.010. |
| [40] |
L. Zu, D. Jiang, D. O'Regan and B. Ge,
Periodic solution for a non-autonomous Lotka-Volterra predator-prey model with random perturbation, J. Math. Anal. Appl., 430 (2015), 428-437.
doi: 10.1016/j.jmaa.2015.04.058. |
Figure 2.
Dynamical behaviors of model (46) with $ c(t) = 0.85+0.75\sin t $ : (a) $ h = 0.4 $ ; (b) $ h = -0.6 $ . Other parameters are given by (47)
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