A stochastic mussel-algae model under regime switching

Yan Xie; Zhijun Liu; Ke Qi; Dongchen Shangguan; Qinglong Wang; Yan Xie; Zhijun Liu; Ke Qi; Dongchen Shangguan; Qinglong Wang

doi:10.3934/mbe.2022224

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 5: 4794-4811. doi: 10.3934/mbe.2022224

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Research article

A stochastic mussel-algae model under regime switching

School of Mathematics and Statistics, Hubei Minzu University, Enshi, Hubei 445000, China

Academic Editor: Tonghua Zhang

Received: 08 December 2021 Revised: 22 February 2022 Accepted: 27 February 2022 Published: 14 March 2022

We investigate a novel model of coupled stochastic differential equations modeling the interaction of mussel and algae in a random environment, in which combined effect of white noises and telegraph noises formulated under regime switching are incorporated. We derive sufficient condition of extinction for mussel species. Then with the help of stochastic Lyapunov functions, a well-grounded understanding of the existence of ergodic stationary distribution is obtained. Meticulous numerical examples are also employed to visualize our theoretical results in detail. Our analytical results indicate that dynamic behaviors of the stochastic mussel-algae model are intimately associated with two kinds of random perturbations.
- mussel-algae model,
- brownian motion,
- regime switching,
- extinction,
- ergodic stationary distribution
Citation: Yan Xie, Zhijun Liu, Ke Qi, Dongchen Shangguan, Qinglong Wang. A stochastic mussel-algae model under regime switching[J]. Mathematical Biosciences and Engineering, 2022, 19(5): 4794-4811. doi: 10.3934/mbe.2022224

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Abstract

We investigate a novel model of coupled stochastic differential equations modeling the interaction of mussel and algae in a random environment, in which combined effect of white noises and telegraph noises formulated under regime switching are incorporated. We derive sufficient condition of extinction for mussel species. Then with the help of stochastic Lyapunov functions, a well-grounded understanding of the existence of ergodic stationary distribution is obtained. Meticulous numerical examples are also employed to visualize our theoretical results in detail. Our analytical results indicate that dynamic behaviors of the stochastic mussel-algae model are intimately associated with two kinds of random perturbations.

References

[1]	H. J. MacIsaac, Potential abiotic and biotic impacts of zebra mussels on the inland waters of North America, Am. Zool., 36 (1996), 287–299. https://doi.org/10.1093/icb/36.3.287 doi: 10.1093/icb/36.3.287
[2]	P. Dolmer, Algal concentration profiles above mussel beds, J. Sea Res., 43 (2000), 113–119. https: //doi.org/10.1016/S1385-1101(00)00005-8 doi: 10.1016/S1385-1101(00)00005-8
[3]	Q. H. Huang, H. Wang, M. A. Lewis, A hybrid continuous/discrete-time model for invasion dynamics of zebra mussels in rivers, SIAM J. Appl. Math., 77 (2017), 854–880. https://doi.org/10.1137/16M1057826 doi: 10.1137/16M1057826
[4]	D. J. Wildish, D. D. Kristmanson, Importance to mussels of the benthic boundary layer, Can. J. Fish. Aquat. Sci., 41 (1984), 1618–1625. https://doi.org/10.1139/f84-200 doi: 10.1139/f84-200
[5]	G. Øie, K. I. Reitan, O. Vadstein, H. Reinertsen, Effect of nutrient supply on growth of blue mussels (Mytilus edulis) in a landlocked bay, Hydrobiologia, 484 (2002), 99–109. https://doi.org/10.1007/978-94-017-3190-4_9 doi: 10.1007/978-94-017-3190-4_9
[6]	J. Widdows, J. S. Lucas, M. D. Brinsley, P. N. Salkeld, F. J. Staff, Investigation of the effects of current velocity on mussel feeding and mussel bed stability using an annular flume, Helgol. Mar. Res., 56 (2002), 3–12. https://doi.org/10.1007/s10152-001-0100-0 doi: 10.1007/s10152-001-0100-0
[7]	H. M. Page, D. M. Hubbard, Temporal and spatial patterns of growth in mussels Mytilus edulis on a offshore platform: relationships to water temperature and food availability, J. Exp. Mar. Biol. Ecol., 111 (1987), 159–179. https://doi.org/10.1016/0022-0981(87)90053-0 doi: 10.1016/0022-0981(87)90053-0
[8]	R. A. Cangelosi, D. J. Wollkind, B. J. Kealy-Dichone, I. Chaiya, Nonlinear stability analyses of Turing patterns for a mussel-algae model, J. Math. Biol., 70 (2015), 1249–1294. https://doi.org/10.1007/s00285-014-0794-7 doi: 10.1007/s00285-014-0794-7
[9]	Z. L. Shen, J. J. Wei, Spatiotemporal patterns in a delayed reaction-diffusion mussel-algae model, Int. J. Bifur. Chaos. Appl. Sci. Engrg., 29 (2019), 1950164. https://doi.org/10.1142/S0218127419501645 doi: 10.1142/S0218127419501645
[10]	S. Djilali, B. Ghanbari, S. Bentout, A. Mezouaghi, Turing-Hopf bifurcation in a diffusive mussel-algae model with time-fractional-order derivative, Chaos Solitons Fractals, 138 (2020), 109954. https://doi.org/10.1016/j.chaos.2020.109954 doi: 10.1016/j.chaos.2020.109954
[11]	M. Holzer, N. Popović, Wavetrain solutions of a reaction-diffusion-advection model of musselalgae interaction, SIAM. J. Appl. Dyn. Syst., 16 (2017), 431–478. https://doi.org/10.1137/15M1040463 doi: 10.1137/15M1040463
[12]	J. V. D. Koppel, M. Rietkerk, N. Dankers, P. M. J. Herman, Scale-dependent feedback and regular spatial patterns in young mussel beds, Am. Nat., 165 (2005), E66–E77. https://doi.org/10.1086/428362 doi: 10.1086/428362
[13]	L. Arnold, W. Horsthemke, J. W. Stucki, The influence of external real and white noise on the Lotka-Volterra model, Biom. J., 21 (1979), 451–471. https://doi.org/10.1002/bimj.4710210507 doi: 10.1002/bimj.4710210507
[14]	M. Liu, Optimal harvesting of stochastic population models with periodic coefficients, J. Nonlinear Sci., 32 (2022), 1–14. https://doi.org/10.1007/s00332-021-09758-6 doi: 10.1007/s00332-021-09758-6
[15]	X. W. Yu, S. L. Yuan, T. H. Zhang, Survival and ergodicity of a stochastic phytoplankton-zooplankton model with toxin-producing phytoplankton in an impulsive polluted environment, Appl. Math. Comput., 347 (2019), 249–264. https://doi.org/10.1016/j.amc.2018.11.005 doi: 10.1016/j.amc.2018.11.005
[16]	X. W. Yu, S. L. Yuan, T. H. 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
[17]	P. P. Mathai, J. H. Bertram, S. K. Padhi, V. Singh, I. E. Tolo, A. Primus, et al., Influence of environmental stressors on the microbiota of zebra mussels (dreissena polymorpha), Microb. Ecol., 81 (2021), 1042–1053. https://doi.org/10.1007/s00248-020-01642-2 doi: 10.1007/s00248-020-01642-2
[18]	R. M. May, Stability and Complexity in Model Ecosystem, Princeton University Press, Princeton, NJ, 2001. https://doi.org/10.1515/9780691206912
[19]	J. Hu, Q. M. Zhang, A. Meyer-Baese, M. Ye, Stationary distribution of a stochastic Alzheimer's disease model, Math. Methods Appl. Sci., 43 (2020), 9706–9718. https://doi.org/10.1002/mma.6642 doi: 10.1002/mma.6642
[20]	M. Y. Song, W. J. Zuo, D. Q. Jiang, T. Hayat, Stationary distribution and ergodicity of a stochastic cholera model with multiple pathways of transmission, J. Franklin Inst., 357 (2020), 10773–10798. https://doi.org/10.1016/j.jfranklin.2020.04.061 doi: 10.1016/j.jfranklin.2020.04.061
[21]	H. K. Qi, X. Z. Meng, Threshold behavior of a stochastic predator-prey system with prey refuge and fear effect, Appl. Math. Lett., 113 (2021), 106846. https://doi.org/10.1016/j.aml.2020.106846 doi: 10.1016/j.aml.2020.106846
[22]	X. R. Mao, S. Sabanis, E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141–156. https://doi.org/10.1016/S0022-247X(03)00539-0 doi: 10.1016/S0022-247X(03)00539-0
[23]	M. Liu, M. L. Deng, Analysis of a stochastic hybrid population model with Allee effect, Appl. Math. Comput., 364 (2020), 124582. https://doi.org/10.1016/j.amc.2019.124582 doi: 10.1016/j.amc.2019.124582
[24]	L. J. Ouyang, X. W. Guo, The effection of temperature on the feeding and growth of fish, Stud. Mar. Sin., 49 (2009), 87–95.
[25]	X. Pang, Research on the Fish Eco-physiological (Thermal Tolerance, Metabolism and Swimming) Response Based on Temperature Changes, Ph.D thesis, College of Resources and Environmental Science of Chongqing University, 2020.
[26]	P. J. Colby, H. Lehtonen, Suggested causes for the collapse of zander, Stizostedion lucioperca (L.), populations in northern and central Finland through comparisons with North American walleye, Stizostedion vitreum (Mitchill), Aqua. Fenn., 24 (1994), 9–20.
[27]	H. Lehtonen, Potential effects of global warming on northern European freshwater fish and fisheries, Fish. Manage. Ecol., 3 (1996), 59–71. https://doi.org/10.1111/j.1365-2400.1996.tb00130.x doi: 10.1111/j.1365-2400.1996.tb00130.x
[28]	J. Lappalainen, Effects of Environmental Factors, Especially Temperature, on the Population Dynamics of Pikeperch (Stizostedion lucioperca (L.)), Ph.D. thesis, University of Helsinki, Finland Academic, 2001.
[29]	E. M. Griebeler, A. Seitz, Effects of increasing temperatures on population dynamics of the zebra mussel dreissena polymorpha: implications from an individual-based model, Oecologia, 151 (2007), 530–543. https://doi.org/10.1007/s00442-006-0591-0 doi: 10.1007/s00442-006-0591-0
[30]	M. Slatkin, The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249–256. https://doi.org/10.2307/1936370 doi: 10.2307/1936370
[31]	A. Settati, A. Lahrouz, Stationary distribution of stochastic population systems under regime switching, Appl. Math. Comput., 244 (2014), 235–243. https://doi.org/10.1016/j.amc.2014.07.012 doi: 10.1016/j.amc.2014.07.012
[32]	R. McVinish, P. K. Pollett, Y. S. Chan, A metapopulation model with Markovian landscape dynamics, Theor. Popul. Biol., 112 (2016), 80–96. https://doi.org/10.1016/j.tpb.2016.08.005 doi: 10.1016/j.tpb.2016.08.005
[33]	D. Li, S. Q. Liu, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, J. Differ. Equ., 263 (2017), 8873–8915. https://doi.org/10.1016/j.jde.2017.08.066 doi: 10.1016/j.jde.2017.08.066
[34]	Q. Liu, D. Q. Jiang, T. Hayat, A. Alsaedi, Dynamical behavior of a multigroup SIRS epidemic model with standard incidence rates and Markovian switching, Discrete Contin. Dyn. Syst. Ser. A., 39 (2019), 5683. https://doi.org/10.3934/dcds.2019249 doi: 10.3934/dcds.2019249
[35]	Y. Zhao, S. L. Yuan, T. H. Zhang, The stationary distribution and ergodicity of a stochastic phytoplankton allelopathy model under regime switching, Commun. Nonlinear Sci. Numer. Simul., 37 (2016), 131–142. https://doi.org/10.1016/j.cnsns.2016.01.013 doi: 10.1016/j.cnsns.2016.01.013
[36]	X. B. Jiang, L. Zu, D. Q. Jiang, D. O'Regan, Analysis of a stochastic Holling type II predator-prey model under regime switching, Bull. Malays. Math. Sci. Soc., 43 (2020), 2171–2197. https://doi.org/10.1007/s40840-019-00798-6 doi: 10.1007/s40840-019-00798-6
[37]	Y. M. Cai, S. Y. Cai, X. R. Mao, Stochastic delay foraging arena predator-prey system with Markov switching, Stoch. Anal. Appl., 38 (2020), 191–212. https://doi.org/10.1080/07362994.2019.1679645 doi: 10.1080/07362994.2019.1679645
[38]	Y. X. Zhou, W. J. Zuo, D. Q. Jiang, M. Y. Song, Stationary distribution and extinction of a stochastic model of syphilis transmission in an MSM population with telegraph noises, J. Appl. Math. Comput., 66 (2021), 645–672. https://doi.org/10.1007/s12190-020-01453-1 doi: 10.1007/s12190-020-01453-1
[39]	C. Q. Xu, S. L. Yuan, T. H. Zhang, Average break-even concentration in a simple chemostat model with telegraph noise, Nonlinear Anal. Hybrid Syst., 29 (2018), 373–382. https://doi.org/10.1016/j.nahs.2018.03.007 doi: 10.1016/j.nahs.2018.03.007
[40]	D. X. Zhou, M. Liu, K. Qi, Z. J. Liu, Long-time behaviors of two stochastic mussel-algae models, Math. Biosci. Eng., 18 (2021), 8392–8414. https://doi.org/10.3934/mbe.2021416 doi: 10.3934/mbe.2021416
[41]	H. Liu, X. X. Li, Q. S. Yang, The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Syst. Control Lett., 62 (2013), 805–810. https://doi.org/10.1016/j.sysconle.2013.06.002 doi: 10.1016/j.sysconle.2013.06.002
[42]	X. R. Mao, C. G. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.
[43]	T. T. Ma, X. Z. Meng, Z. B. Chang, Dynamics and optimal harvesting control for a stochastic one-predator-two-prey time delay system with jumps, Complexity, 2019 (2019), 5342031. https://doi.org/10.1155/2019/5342031 doi: 10.1155/2019/5342031
[44]	M. Yavuz, N. Sene, Stability analysis and numerical computation of the fractional predator-prey model with the harvesting rate, Fractal. Fract., 4 (2021), 35. https://doi.org/10.3390/fractalfract4030035 doi: 10.3390/fractalfract4030035
[45]	P. A. Naik, K. M. Owolabi, M. Yavuz, J. Zu, Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells, Chaos Solitons Fractals, 140 (2020), 110272. https://doi.org/10.1016/j.chaos.2020.110272 doi: 10.1016/j.chaos.2020.110272
[46]	P. A. Naik, M. Yavuz, S. Qureshi, J. Zu, Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan, Eur. Phys. J. Plus., 135 (2020), 1–42. https://doi.org/10.1140/epjp/s13360-020-00819-5 doi: 10.1140/epjp/s13360-020-00819-5
[47]	S. Allegretti, I. M. Bulai, R. Marino, M. A. Menandro, K. Parisi, Vaccination effect conjoint to fraction of avoided contacts for a Sars-Cov-2 mathematical model, Math. Model. Num. Simul., 1 (2021), 56–66. https://doi.org/10.53391/mmnsa.2021.01.006 doi: 10.53391/mmnsa.2021.01.006
[48]	P. Kumar, V. S. Erturk, Dynamics of cholera disease by using two recent fractional numerical methods, Math. Model. Num. Simul., 1 (2021), 102–111. https://doi.org/10.53391/mmnsa.2021.01.010 doi: 10.53391/mmnsa.2021.01.010

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