Structure analysis of the attracting sets for plankton models driven by bounded noises

Zhihao Ke; Chaoqun Xu; Zhihao Ke; Chaoqun Xu

doi:10.3934/mbe.2023277

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 4: 6400-6421. doi: 10.3934/mbe.2023277

Previous Article Next Article

Research article Special Issues

Structure analysis of the attracting sets for plankton models driven by bounded noises

Zhihao Ke ,
Chaoqun Xu ^,

School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China

Received: 01 December 2022 Revised: 30 December 2022 Accepted: 08 January 2023 Published: 01 February 2023

In this paper, we study the attracting sets for two plankton models perturbed by bounded noises which are modeled by the Ornstein-Uhlenbeck process. Specifically, we prove the existence and uniqueness of the solutions for these random models, as well as the existence of the attracting sets for the random dynamical systems generated by the solutions. In order to further reveal the survival of plankton species in a fluctuating environment, we analyze the internal structure of the attracting sets and give sufficient conditions for the persistence and extinction of the plankton species. Some numerical simulations are shown to support our theoretical results.
- plankton model,
- bounded noise,
- Ornstein-Uhlenbeck process,
- attracting set,
- internal structure
Citation: Zhihao Ke, Chaoqun Xu. Structure analysis of the attracting sets for plankton models driven by bounded noises[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 6400-6421. doi: 10.3934/mbe.2023277

Related Papers:

Abstract

In this paper, we study the attracting sets for two plankton models perturbed by bounded noises which are modeled by the Ornstein-Uhlenbeck process. Specifically, we prove the existence and uniqueness of the solutions for these random models, as well as the existence of the attracting sets for the random dynamical systems generated by the solutions. In order to further reveal the survival of plankton species in a fluctuating environment, we analyze the internal structure of the attracting sets and give sufficient conditions for the persistence and extinction of the plankton species. Some numerical simulations are shown to support our theoretical results.

References

[1]	S. Chakraborty, S. Chatterjee, E. Venturino, J. Chattopadhyay, Recurring plankton ploom dynamics modeled via toxin-producing phytoplankton, J. Biol. Phys., 33 (2007), 271–290. https://doi.org/10.1007/s10867-008-9066-3 doi: 10.1007/s10867-008-9066-3
[2]	C. Subhendu, P. K. Tiwari, A. K. Misra, J. Chattopadhyay, Spatial dynamics of a nutrient-phytoplankton system with toxic effect on phytoplankton, Math. Biosci., 264 (2015), 94–100. https://doi.org/10.1016/j.mbs.2015.03.010 doi: 10.1016/j.mbs.2015.03.010
[3]	S. Zhao, S. Yuan, T. Zhang, The impact of environmental fluctuations on a plankton model with toxin-producing phytoplankton and patchy agglomeration, Chaos Soliton. Fract., 162 (2022), 112426. https://doi.org/10.1016/j.chaos.2022.112426 doi: 10.1016/j.chaos.2022.112426
[4]	E. J. Philips, S. Badylak, S. Youn, K. Kelley, The occurrence of potentially toxic dinoflagellates and diatoms in a subtropical lagoon, the Indian river lagoon, Florida, USA, Harmful Algae, 3 (2004), 39–49. https://doi.org/10.1016/j.hal.2003.08.003 doi: 10.1016/j.hal.2003.08.003
[5]	S. P. Colin, H. G. Dam, Effects of the toxic dinoflagellate Alexandrium fundyense on the copepod Acartia hudsonica: A test of the mechanisms that reduce ingestion rates, Mar. Ecol. Prog. Ser., 248 (2003), 55–65. https://doi.org/10.3354/meps248055 doi: 10.3354/meps248055
[6]	Y. Peng, Y. Li, T. Zhang, Global bifurcation in a toxin producing phytoplankton-zooplankton system with prey-taxis, Nonlinear Anal.-Real., 61 (2021), 103326. https://doi.org/10.1016/j.nonrwa.2021.103326 doi: 10.1016/j.nonrwa.2021.103326
[7]	L. E. Schmidt, P. J. Hansen, Allelopathy in the prymnesiophyte Chrysochromulina polylepis: Effect of cell concentration, growth phase and pH, Mar. Ecol.: Prog. Ser., 216 (2001), 67–81. https://doi.org/10.3354/meps216067 doi: 10.3354/meps216067
[8]	F. Rao, Spatiotemporal dynamics in a reaction-diffusion toxic-phytoplankton-zooplankton model, J. Stat. Mech.: Theory Exp., 2013 (2013), 08014. https://doi.org/10.1088/1742-5468/2013/08/P08014 doi: 10.1088/1742-5468/2013/08/P08014
[9]	J. Chattopadhyay, E. Venturino, S. Chatterjee, Aggregation of toxin-producing phytoplankton acts as a defencemechanism-a model-based study, Math. Comput. Model. Dyn., 19 (2013), 159–174. https://doi.org/10.1080/13873954.2012.708876 doi: 10.1080/13873954.2012.708876
[10]	J. Chattopadhayay, R. R. Sarkar, S. Mandal, Toxin-producing plankton may act as a biological control for planktonic blooms-field study and mathematical modelling, J. Theor. Biol., 215 (2002), 333–344. https://doi.org/10.1006/jtbi.2001.2510 doi: 10.1006/jtbi.2001.2510
[11]	T. Scotti, M. Mimura, J. Y. Wakano, Avoiding toxic prey may promote harmful algal blooms, Ecol. Complex, 21 (2015), 157–165. https://doi.org/10.1016/j.ecocom.2014.07.004 doi: 10.1016/j.ecocom.2014.07.004
[12]	S. Jang, J. Baglama, L. Wu, Dynamics of phytoplankton-zooplankton systems with toxin producing phytoplankton, Appl. Math. Comput., 227 (2014), 717–740. https://doi.org/10.1016/j.amc.2013.11.051 doi: 10.1016/j.amc.2013.11.051
[13]	C. Ji, D. Jiang, N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482–498. https://doi.org/10.1016/j.jmaa.2009.05.039 doi: 10.1016/j.jmaa.2009.05.039
[14]	S. Zhang, X. Meng, T. Feng, T. Zhang, Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects, Nonlinear Anal.-Hybrid Syst., 26 (2017), 19–37. https://doi.org/10.1016/j.nahs.2017.04.003 doi: 10.1016/j.nahs.2017.04.003
[15]	Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69–84. https://doi.org/10.1016/j.jmaa.2006.12.032 doi: 10.1016/j.jmaa.2006.12.032
[16]	M. Liu, K. Wang, Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment Ⅱ, J. Theor. Biol., 267 (2010), 283–291. https://doi.org/10.1016/j.jtbi.2010.08.030 doi: 10.1016/j.jtbi.2010.08.030
[17]	M. Liu, K. Wang, Persistence, extinction and global asymptotical stability of a non-autonomous predator-prey model with random perturbation, Appl. Math. Modell., 36 (2012), 5344–5353. https://doi.org/10.1016/j.apm.2011.12.057 doi: 10.1016/j.apm.2011.12.057
[18]	X. Yu, S. Yuan, T. Zhang, The effects of toxin-producing phytoplankton and environmental fluctuations on the planktonic blooms, Nonlinear Dyn., 91 (2018), 1653–1668. https://doi.org/10.1007/s11071-017-3971-6 doi: 10.1007/s11071-017-3971-6
[19]	X. Yu, S. Yuan, T. Zhang, Asymptotic properties of stochastic nutrient-plankton food chain models with nutrient recycling, Nonlinear Anal.–Hybrid Syst., 34 (2019), 209–225. https://doi.org/10.1016/j.nahs.2019.06.005 doi: 10.1016/j.nahs.2019.06.005
[20]	Q. Zhao, S. Liu, X. Niu, Stationary distribution and extinction of a stochastic nutrien-phytoplankton-zooplankton model with cell size, Math. Methods Appl. Sci., 43 (2020), 3886–3902. https://doi.org/10.1002/mma.6114 doi: 10.1002/mma.6114
[21]	T. Caraballo, X. Han, Applied Nonautonomous and Random Dynamical Systems: Applied Dynamical Systems, Springer, Berlin, 2016.
[22]	T. Caraballo, R. Colucci, J. López-De-La-Cruz, A. Rapaport, A way to model stochastic perturbations in population dynamics models with bounded realizations, Commun. Nonlinear Sci., 77 (2019), 239–257. https://doi.org/10.1016/j.cnsns.2019.04.019 doi: 10.1016/j.cnsns.2019.04.019
[23]	X. Zhang, R. Yuan, Pullback attractor for random chemostat model driven by colored noise, Appl. Math. Lett., 112 (2021), 106833. https://doi.org/10.1016/j.aml.2020.106833 doi: 10.1016/j.aml.2020.106833
[24]	L. F. de Jesus, C. M. Silva, H. Vilarinho, Random perturbations of an eco-epidemiological model, Discrete Contin. Dyn.-Ser. B, 27 (2022), 257–275. https://doi.org/10.3934/dcdsb.2021040 doi: 10.3934/dcdsb.2021040
[25]	J. López-de-la-Cruz, Random and stochastic disturbances on the input flow in chemostat models with wall growth, Stoch. Anal. Appl., 37 (2019), 668–698. https://doi.org/10.1080/07362994.2019.1605911 doi: 10.1080/07362994.2019.1605911
[26]	T. Caraballo, R. Colucci, X. Han, Predation with indirect effects in fluctuating environments, Nonlinear Dyn., 84 (2016), 115–126, https://doi.org/10.1007/s11071-015-2238-3 doi: 10.1007/s11071-015-2238-3
[27]	T. Caraballo, M. J. Garrido-Atienza, J. López-de-la-Cruz, Dynamics of some stochastic chemostat models with multiplicative noise, Commun. Pur. Appl. Anal., 16 (2017), 1893–1914. https://doi.org/10.3934/cpaa.2017092 doi: 10.3934/cpaa.2017092
[28]	X. Zhang, R. Yuan, Forward attractor for stochastic chemostat model with multiplicative noise, Chaos, Solitons Fractals, 153 (2021), 111585. https://doi.org/10.1016/j.chaos.2021.111585 doi: 10.1016/j.chaos.2021.111585
[29]	D. Wu, H. Wang, S. Yuan, Stochastic sensitivity analysis of noise-induced transitions in a predatorprey model with environmental toxins, Math. Biosci. Eng., 16 (2019), 2141–2153. https://doi.org/10.3934/mbe.2019104 doi: 10.3934/mbe.2019104
[30]	O. E. Barndorff-Nielsen, N. Shephard, Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics, R. Stat. Soc., 63 (2001), 167–241. https://doi.org/10.1111/1467-9868.00282 doi: 10.1111/1467-9868.00282
[31]	X. Mu, D. Jiang, T. Hayat, A. Alsaedi, Y. Liao, A stochastic turbidostat model with Ornstein-Uhlenbeck process: dynamics analysis and numerical simulations, Nonlinear Dyn., 107 (2022), 2805–2817. https://doi.org/10.1007/s11071-021-07093-9 doi: 10.1007/s11071-021-07093-9
[32]	B. Zhou, D. Jiang, T. Hayat, Analysis of a stochastic population model with mean-reverting Ornstein-Uhlenbeck process and Allee effects, Commun. Nonlinear Sci. Numer. Simul., 111 (2022), 106450. https://doi.org/10.1016/j.cnsns.2022.106450 doi: 10.1016/j.cnsns.2022.106450
[33]	Q. Liu, D. Jiang, Analysis of a stochastic logistic model with diffusion and Ornstein-Uhlenbeck process, J. Math. Phys., 63 (2022), 053505. https://doi.org/10.1063/5.0082036 doi: 10.1063/5.0082036
[34]	T. Caraballo, M. Garrido-Atienza, J. López-De-La-Cruz, A. Rapaport, Modeling and analysis of random and stochastic input flows in the chemostat model, Discrete Contin. Dyn.–Ser. B, 24 (2019), 3591–3614. https://doi.org/10.3934/dcdsb.2018280 doi: 10.3934/dcdsb.2018280
[35]	L. Arnold, Random Dynamical Systems, Springer, Berlin, 1998.
[36]	T. Caraballo, P. E. Kloeden, B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183–207. https://doi.org/10.1007/s00245-004-0802-1 doi: 10.1007/s00245-004-0802-1
[37]	S. Al-Azzawi, J. Liu, X. Liu, Convergence rate of synchronization of systems with additive noise, Discrete Contin. Dyn.-Ser. B, 22 (2017), 227–245. https://doi.org/10.3934/dcdsb.2017012 doi: 10.3934/dcdsb.2017012
[38]	H. Crauel, P. Kloeden, Nonautonomous and random attractors, Jahresber. Deutsch. Math., 117 (2015), 173–206. https://doi.org/10.1365/s13291-015-0115-0 doi: 10.1365/s13291-015-0115-0
[39]	T. Caraballo, G. Lukaszewicz, J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal.-Theory, Methods Appl., 64 (2006), 484–498. https://doi.org/10.1016/j.na.2005.03.111 doi: 10.1016/j.na.2005.03.111
[40]	F. Flandoli, B. Schmalfuss, Random attractors for the 3D stochastic navier-stokes equation with multiplicative white noise, Stoch. Stoch. Rep., 59 (1996), 21–45. https://doi.org/10.1080/17442509608834083 doi: 10.1080/17442509608834083

Reader Comments

Your name:*

Email:*
© 2023 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)