American Institute of Mathematical Sciences

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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

Jiangtao Yang ,

School of Mathematics and Statistics, Yangtze Normal University, Chongqing, 408100, China

* Corresponding author: Jiangtao Yang

Received  July 2020 Published  October 2021 Early access  December 2020

Fund Project: The author is supported by Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJQN202001401)

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.

Keywords: Lévy noise, persistence, periodic solution, periodic measure, extinction.
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 & Continuous Dynamical Systems - B, 2021, 26 (10) : 5641-5660. doi: 10.3934/dcdsb.2020371
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman Hall/CRC, 2004.  Google Scholar

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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.  doi: 10.3934/dcdsb.2018331.  Google Scholar

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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. doi: 10.1016/j.physa.2019.121745.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[14]

P. R. Halmos, Measure Theory, Springer-Verlag, New York, 1970. doi: 10.1007/978-1-4684-9440-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. Khasminskii, Stochastic Stability of Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

R. S. Liptser, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228.  doi: 10.1080/17442508008833146.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 1973.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1] D. Applebaum, Lévy Processes and Stochastics Calculus, 2$^nd$ edition, Cambridge University Press, 2009.  doi: 10.1017/CBO9780511809781.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman Hall/CRC, 2004.  Google Scholar

[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.  Google Scholar

[7]

J. M. Cushing, Periodic time-dependent predator-prey system, SIAM J. Appl. Math., 32 (1977), 82-95.  doi: 10.1137/0132006.  Google Scholar

[8] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511662829.  Google Scholar
[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.  Google Scholar

[10] J. Duan, An Introduction to Stochastic Dynamics, Cambridge University Press, New York, 2015.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

P. R. Halmos, Measure Theory, Springer-Verlag, New York, 1970. doi: 10.1007/978-1-4684-9440-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. Khasminskii, Stochastic Stability of Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

R. S. Liptser, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228.  doi: 10.1080/17442508008833146.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 1973.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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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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