Dynamics and optimal harvesting of a stochastic predator-prey system with regime switching, S-type distributed time delays and Lévy jumps

Yuanfu Shao; Yuanfu Shao

doi:10.3934/math.2022225

AIMS Mathematics

2022, Volume 7, Issue 3: 4068-4093. doi: 10.3934/math.2022225

Previous Article Next Article

Research article

Dynamics and optimal harvesting of a stochastic predator-prey system with regime switching, S-type distributed time delays and Lévy jumps

Yuanfu Shao ^,

College of Science, Guilin University of Technology, Guilin, Guangxi 541004, China

Received: 07 October 2021 Accepted: 29 November 2021 Published: 14 December 2021
MSC : 60H10, 92B05

This work is concerned with a stochastic predator-prey system with S-type distributed time delays, regime switching and Lévy jumps. By use of the stochastic differential comparison theory and some inequality techniques, we study the extinction and persistence in the mean for each species, asymptotic stability in distribution and the optimal harvesting effort of the model. Then we present some simulation examples to illustrate the theoretical results and explore the effects of regime switching, distributed time delays and Lévy jumps on the dynamical behaviors, respectively.
- regime switching,
- Lévy jumps,
- persistence in the mean,
- asymptotic stability in distribution
Citation: Yuanfu Shao. Dynamics and optimal harvesting of a stochastic predator-prey system with regime switching, S-type distributed time delays and Lévy jumps[J]. AIMS Mathematics, 2022, 7(3): 4068-4093. doi: 10.3934/math.2022225

Related Papers:

Abstract

This work is concerned with a stochastic predator-prey system with S-type distributed time delays, regime switching and Lévy jumps. By use of the stochastic differential comparison theory and some inequality techniques, we study the extinction and persistence in the mean for each species, asymptotic stability in distribution and the optimal harvesting effort of the model. Then we present some simulation examples to illustrate the theoretical results and explore the effects of regime switching, distributed time delays and Lévy jumps on the dynamical behaviors, respectively.

References

[1]	X. Mao, Stochastic differential equations and applications, England: Horwood Publishing Limited, 2007.
[2]	N. Ikeda, S. Watanable, Stochastic differential equations and diffusion processes, New York: North-Holland, 1989.
[3]	X. Mao, G. Marion, E. Renshaw, Environmental Brownian noise suppresses explosions in populations dynamics, Stoch. Proc. Appl., 97 (2002), 95–110. doi: 10.1016/S0304-4149(01)00126-0. doi: 10.1016/S0304-4149(01)00126-0
[4]	M. Liu, K. Wang, Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73 (2011), 1969–2012. doi: 10.1007/s11538-010-9569-5. doi: 10.1007/s11538-010-9569-5
[5]	M. Liu, H. Qiu, K. Wang, A remark on a stochastic predator-prey system with time delays, Appl. Math. Lett., 26 (2013), 318–323. doi: 10.1016/j.aml.2012.08.015. doi: 10.1016/j.aml.2012.08.015
[6]	R. M. May, Stability and complexity in model ecosystems, Princeton: Princeton University Press, 1973.
[7]	R. Z. Khas'minskii, Stochastic stability of differential equations, Netherlands: Sijthoff Noordhoff, Alphen aan den Rijn, 1980.
[8]	M. Liu, Optimal harvesting policy of a stochastic predator-prey model with time delay, Appl. Math. Lett., 48 (2015), 102–108. doi: 10.1016/j.aml.2014.10.007. doi: 10.1016/j.aml.2014.10.007
[9]	X. Mao, C. Yuan, Stochastic differential equations with markovian switching, London: Imperial College Press, 2006.
[10]	Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69–84.
[11]	W. Kong, Y. Shao, Long-time behaviours of a stochastic predator-prey system with Holling-type Ⅱ functional response and regime switching, J. Math., 2021 (2021), 6672030.
[12]	J. Bao, J. Shao, Permanence and extinction of regime-switching predator-prey models, SIAM J. Math. Anal., 48 (2016), 725–739. doi: 10.1137/15M1024512. doi: 10.1137/15M1024512
[13]	H. Chen, P. Shi, C. Lim, Stability analysis for neutral stochastic delay systems with Markovian switching, Syst. Contr. Lett., 110 (2017), 38–48. doi: 10.1016/j.sysconle.2017.10.008. doi: 10.1016/j.sysconle.2017.10.008
[14]	M. Liu, X. He, J. Yu, Dynamics of a stochastic regime-switching predator-prey model with harvesting and distributed delays, Nonlinear Anal.-Hybri., 28 (2018), 87–104. doi: 10.1016/j.nahs.2017.10.004. doi: 10.1016/j.nahs.2017.10.004
[15]	Y. Kuang, Delay differential equations: With applications in population dynamics, Boston: Academic Press, 1993.
[16]	S. Wang, G. Hu, T. Wei, L. Wang, Stability in distribution of a stochastic predator-prey system with S-type distributed time delays, Physica A, 505 (2018), 919–930. doi: 10.1016/j.physa.2018.03.078. doi: 10.1016/j.physa.2018.03.078
[17]	L. Wang, D. Xu, Global asymptotic stability of bidirectional associative memory neural networks with S-type distributed delays, Int. J. Syst. Sci., 33 (2002), 869–877. doi: 10.1080/00207720210161777. doi: 10.1080/00207720210161777
[18]	L. Wang, R. Zhang, Y. Wang, Global exponential stability of reaction-diffusion cellular neural networks with S-type distributed time delays, Nonlinear Anal., 10 (2009), 1101–1113.
[19]	D. Applebaum, Lévy processes and stochastic Calculus, 2nd ed., Lodon: Cambridge University Press, 2009.
[20]	M. Liu, 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. doi: 10.1016/j.jmaa.2013.07.078
[21]	X. Zhang, W. Li, M. Liu, K. Wang, Dynamics of a stochastic Holling Ⅱ one-predator two-prey system with jumps, Physica A, 421 (2015), 571–582. doi: 10.1016/j.physa.2014.11.060. doi: 10.1016/j.physa.2014.11.060
[22]	S. Wang, L. Wang, T. Wei, Permanence and asymptotic behaviors of stochastic predator-prey system with Markovian switching and Lévy noise, Physica A, 495 (2018), 294–311. doi: 10.1016/j.physa.2017.12.088. doi: 10.1016/j.physa.2017.12.088
[23]	Y. Zhao, S. Yuan, Stability in distribution of a stochastic hybrid competitive Lotka-Volterra model with Lévy jumps, Chaos, Solit. Fract., 85 (2016), 98–109. doi: 10.1016/j.chaos.2016.01.015. doi: 10.1016/j.chaos.2016.01.015
[24]	Y. Zhang, Q. Zhang, Dynamic behavior in a delayed stage-structured population model with stochastic fluctuation and harvesting, Nonlinear Dynam., 66 (2011), 231–245. doi: 10.1007/s11071-010-9923-z. doi: 10.1007/s11071-010-9923-z
[25]	J. Yu, M. Liu, Stationary distribution and ergodicity of a stochastic food-chain model with Lévy jumps, Physica A, 482 (2017), 14–28. doi: 10.1016/j.physa.2017.04.067. doi: 10.1016/j.physa.2017.04.067
[26]	G. Hu, K. Wang, Stability in distribution of neutral stochastic functional differential equations with Markovian switching, J. Math. Anal. Appl., 385 (2012), 757–769. doi: 10.1016/j.jmaa.2011.07.002. doi: 10.1016/j.jmaa.2011.07.002
[27]	Y. Shao, Y. Chen, B. X. Dai, Dynamical analysis and optimal harvesting of a stochastic three-species cooperative system with delays and Lévy jumps, Adv. Differ. Equ., 2018 (2018), 423. doi: 10.1186/s13662-018-1874-6. doi: 10.1186/s13662-018-1874-6
[28]	G. D. Liu, X. Z. Meng, Optimal harvesting strategy for a stochastic mutualism system in a polluted environment with regime switching, Physica A, 536 (2019), 120893. doi: 10.1016/j.physa.2019.04.129. doi: 10.1016/j.physa.2019.04.129
[29]	M. Liu, C. Z. Bai, Optimal harvesting of a stochastic mutualism model with regime-switching, Appl. Math. Comput., 373 (2020), 125040. doi: 10.1016/j.amc.2020.125040. doi: 10.1016/j.amc.2020.125040
[30]	I. Barbalat, Systems d''equations differentielles d''oscillations non lineaires, Rev. Roum. Math. Pures Appl., 4 (1959), 267–270.
[31]	D. Prato, J. Zabczyk, Ergodicity for infinite dimensional systems, Cambridge: Cambridge University Press, 1996. doi: 10.1017/CBO9780511662829.
[32]	M. S. Bazaraa, C. M. Shetty, Nonlinear Programming, Wiley, New York: Academic Press, 1979.
[33]	D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525–546. doi: 10.1137/S0036144500378302. doi: 10.1137/S0036144500378302
[34]	D. Amartya, G. P. Samanta, Stochastic prey-predator model with additional food for predator, Physica A, 512 (2018), 121–141. doi: 10.1016/j.physa.2018.08.138. doi: 10.1016/j.physa.2018.08.138
[35]	D. Amartya, G. P. Samanta, Modelling the fear effect on a stochastic prey-predator system with additional food for predator, J. Phys. A-Math. Theor., 51 (2018), 465601. doi: 10.1088/1751-8121/aae4c6. doi: 10.1088/1751-8121/aae4c6

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)