### Mathematical Biosciences and Engineering

2019, Issue 1: 1-24. doi: 10.3934/mbe.2019001
Research article Special Issues

# Stochastic modeling of algal bloom dynamics with delayed nutrient recycling

• Received: 25 March 2018 Accepted: 22 June 2018 Published: 05 December 2018
• Using the discrete Markov chain, in this paper we develop a stochastic model for algal bloom, in which white noise terms are introduced to describe the e ects of environmental random fluctuations and time delay to account for the time needed in the conversion of detritus into nutrient. For the proposed model, we firstly discuss the well-posedness, namely the existence and uniqueness of the global positive solution. Then, it is followed by seeking the sufficient conditions for the stochastic stability of its washout equilibrium. Then by using Fourier transform method, the spectral densities of the nutrient and the algae population are estimated. Finally, we show that larger noise can make the algae population extinct exponentially with probability one. Our theoretical and numerical results suggest that the environmental random fluctuations may have more significant influences on the dynamics of the model than the delay. These findings are helpful for a better understanding of the formation mechanism of algal blooms.

Citation: Xuehui Ji, Sanling Yuan, Tonghua Zhang, Huaiping Zhu. Stochastic modeling of algal bloom dynamics with delayed nutrient recycling[J]. Mathematical Biosciences and Engineering, 2019, 16(1): 1-24. doi: 10.3934/mbe.2019001

### Related Papers:

• Using the discrete Markov chain, in this paper we develop a stochastic model for algal bloom, in which white noise terms are introduced to describe the e ects of environmental random fluctuations and time delay to account for the time needed in the conversion of detritus into nutrient. For the proposed model, we firstly discuss the well-posedness, namely the existence and uniqueness of the global positive solution. Then, it is followed by seeking the sufficient conditions for the stochastic stability of its washout equilibrium. Then by using Fourier transform method, the spectral densities of the nutrient and the algae population are estimated. Finally, we show that larger noise can make the algae population extinct exponentially with probability one. Our theoretical and numerical results suggest that the environmental random fluctuations may have more significant influences on the dynamics of the model than the delay. These findings are helpful for a better understanding of the formation mechanism of algal blooms.

 [1] B. Qin, Z. Liu and K. Havens, Eutrophication of shallow lakes with special reference to Lake Taihu, China, Hydrobiologia, 2007. [2] B. Qin, Lake Taihu, China, Springer, New York, 2008. [3] B. Qin, G. Zhu and G. Gao, A drinking water crisis in Lake Taihu, China: linkage to climatic variability and lake management. Environ. Manage., 45 (2010), 105–112. [4] L. Xu, J. Shen and D. Marinova, Changes of public environmental awareness in response to the Taihu blue- green algae bloom incident in China, Environ. Dev. Sustain., 15 (2013), 1281–1302. [5] C. Qiuwen and E. Mynett, Modelling algal blooms in the Dutch coastal waters by integrated numerical and fuzzy cellular automata approaches, Ecol. Model., 199 (2006), 73–81. [6] B. Wang, Cultural eutrophication in the Changjiang (Yangtze River) plume: history and perspective, Estuar. Coast. Shelf. S., 69 (2006), 474–477. [7] W. He, J. Shang, X. Lu and C. Fan, Effects of sludge dredging on the prevention and control of algae-caused black bloom in Taihu Lake, China, J. Environ. Sci., 25 (2013), 430–440. [8] E. Beretta, G. Bischi and F. Solimano, Stability in chemostat equations with delayed nutrient recycling, J. Math. Biol., 28 (1990), 99–111. [9] X. He, S. Ruan and H. Xia, Global stability in chemostat-type plankton models with delayed nutrient recycling, J. Math. Biol., 37 (1998), 253–271. [10] J. Caperon, Time lag in population growth response of isochrysis galbana to a variable nitrate environment, Ecology, 50 (1969), 188–192. [11] A. Misra, P. Chandra and V. Raghavendra, Modeling the depletion of dissolved oxygen in a lake due to algal bloom: effect of time delay, Adv. Water Resour., 34 (2011), 1232–1238. [12] A. Huppert, B. Blasius and R. Olinky, A model for seasonal phytoplankton blooms, J. Theor. Biol., 236 (2005), 276–290. [13] S. Chen, X. Chen and Y. Peng, A mathematical model of the effect of nitrogen and phosphorus on the growth of blue-green algae population, Appl. Math. Model., 33 (2009), 1097–1106. [14] S. Kunikane and M. Kaneko, Growth and nutrient uptake of green alga, Scenedesmus dimorphus, under a wide range of nitrogen/phosphorus ratio-II, Kinetic model, Water Res., 18 (1984), 1313–1326. [15] D. Trolle, J. Elliott and W. Mooij, Advancing projections of phytoplankton responses to climate change through ensemble modelling, Environ. Modell. Softw., 61 (2014), 371–379. [16] D. Anderson, J. Burkholder andW. Cochlan, Harmful algal blooms and eutrophication: examining of linkages from selected coastal regions of the United States, Harmful Algae, 8 (2008), 39–53. [17] R. Sarkar, A stochastic model for autotroph-herbivore system with nutrient reclycing, Ecol. Model., 178 (2004), 429–440. [18] M. Zhu, H. Paerl and G. Zhu, The role of tropical cyclones in stimulating cyanobacterial (Microcystis spp.) blooms in hypertrophic Lake Taihu, China, Harmful Algae, 39 (2014), 310–321. [19] D Huang, H Wang and J. Feng, Modelling algal densities in harmful algal blooms (HAB) with stochastic dynamics, Appl. Math. Model., 32 (2008), 1318–1326. [20] K. Das and N. Gazi, Random excitations in modelling of algal blooms in estuarine systems, Ecol. Model., 222 (2011), 2495–2501. [21] P. Mandal, L. Allen and M. Banerjee, Stochastic modeling of phytoplankton allelopathy, Appl. Math. Model., 38 (2014), 1583–1596. [22] L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equations, 217 (2005), 26–53. [23] R. Durrett, Stochastic calculus: a practical introduction, CRC press, Boston, 1996. [24] M. Frtihcan, Complex time-delay systems: theory and applications, Springer, 2010. [25] X. Mao, Stochastic differential equations and applications, Elsevier, 2007. [26] L. Arnold, Stochastic differential equations: theory and applications, New York, 1974. [27] S. Mohammed, Stochastic functional differential equations, Boston Pitman, 1984. [28] F. Wei and K. Wang, The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, J. Math. Anal. Appl., 331 (2007), 516–531. [29] I. Gradshteyn and I. Ryzhik, Table of integrals, series and products, Academic press, New York, 1980. [30] D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525–546.
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