Nonlinear analysis in a nutrient-algae-zooplankton system with sinking of algae. (English) Zbl 1470.92236

Summary: A reaction-diffusion-advection model is proposed for the Zeya Reservoir to study interactions between algae and zooplankton, including the diffusive spread of algae and zooplankton and the sinking of algae. The model is investigated both with and without sinking. Conditions of Hopf and Turing bifurcation in the spatial domain are obtained, and conditions for differential-flow instability that gives rise to the formation of spatial patterns are derived. Using numerical simulation, the authors examine the impacts on algae of different nutrient concentrations, different sinking rates, and various diffusive spreading patterns. Finally, the models with and without sinking are compared, revealing that the sinking of algae plays an important role in the oscillations of algae and zooplankton. All these results may help to achieve a better understanding of the impact of algae in the Zeya Reservoir.


92D25 Population dynamics (general)
35K40 Second-order parabolic systems
35K57 Reaction-diffusion equations
92D40 Ecology
Full Text: DOI


[1] Malchow, H., Spatio-temporal pattern formation in nonlinear non-equilibrium plankton dynamics, Proceedings of the Royal Society B: Biological Sciences, 251, 1331, 103-109 (1993) · doi:10.1098/rspb.1993.0015
[2] Malchow, H.; Hilker, F. M.; Petrovskii, S. V.; Brauer, K., Oscillations and waves in a virally infected plankton system—part I: the lysogenic stage, Ecological Complexity, 1, 3, 211-223 (2004) · doi:10.1016/j.ecocom.2004.03.002
[3] Dai, C.; Zhao, M.; Chen, L., Bifurcations and periodic solutions for an algae-fish semicontinuous system, Abstract and Applied Analysis, 2013 (2013) · Zbl 1470.34115 · doi:10.1155/2013/619721
[4] Wang, R.; Liu, Q.; Sun, G.; Jin, Z.; van de Koppel, J., Nonlinear dynamic and pattern bifurcations in a model for spatial patterns in young mussel beds, Journal of the Royal Society Interface, 6, 37, 705-718 (2009) · doi:10.1098/rsif.2008.0439
[5] van de Koppel, J.; Rietkerk, M.; Dankers, N.; Herman, P. M. J., Scale-dependent feedback and regular spatial patterns in young mussel beds, The American Naturalist, 165, 3, E66-E77 (2005) · doi:10.1086/428362
[6] Wang, Y.; Zhao, M.; Dai, C.; Pan, X., Nonlinear dynamics of a nutrient-plankton model, Abstract and Applied Analysis, 2014 (2014) · Zbl 1470.37116 · doi:10.1155/2014/451757
[7] Pearson, J. E., Complex patterns in a simple system, Science, 261, 5118, 189-192 (1993) · doi:10.1126/science.261.5118.189
[8] Sherratt, J. A., Periodic travelling waves in cyclic predator-prey systems, Ecology Letters, 4, 1, 30-37 (2001) · doi:10.1046/j.1461-0248.2001.00193.x
[9] Sherratt, J. A.; Smith, M. J., Periodic travelling waves in cyclic populations: field studies and reaction-diffusion models, Journal of the Royal Society Interface, 5, 22, 483-505 (2008) · doi:10.1098/rsif.2007.1327
[10] Medvinsky, A. B.; Petrovskii, S. V.; Tikhonova, I. A.; Malchow, H.; Li, B. L., Spatiotemporal complexity of plankton and fish dynamics, SIAM Review, 44, 3, 311-370 (2002) · Zbl 1001.92050 · doi:10.1137/S0036144502404442
[11] Levin, S. A., The problem of pattern and scale in ecology, Ecology, 73, 6, 1943-1967 (1992) · doi:10.2307/1941447
[12] Wang, W. M.; Liu, Q. X.; Zhen, J., Spatiotemporal complexity of a ratio-dependent predator-prey system, Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 75, 5, 051913-051921 (2007) · doi:10.1103/PhysRevE.75.051913
[13] Turing, A., The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London B, 231, 37-72 (1952) · Zbl 1403.92034
[14] Klausmeier, C. A., Regular and irregular patterns in semiarid vegetation, Science, 284, 5421, 1826-1828 (1999) · doi:10.1126/science.284.5421.1826
[15] Murray, J. D., Mathematical Biology, Biomathematics, 19 (1993), Berlin, Germany: Springer, Berlin, Germany · Zbl 0779.92001
[16] Hassell, M. P.; Comins, H. N.; May, R. M., Spatial structure and chaos in insect population dynamics, Nature, 353, 6341, 255-258 (1991) · doi:10.1038/353255a0
[17] Huisman, J.; Pham Thi, N. N.; Karl, D. M.; Sommeijer, B., Reduced mixing generates oscillations and chaos in the oceanic deep chlorophyll maximum, Nature, 439, 7074, 322-325 (2006) · doi:10.1038/nature04245
[18] Liu, J. K., Advanced Hydrobiology (1999), Beijing, China: Science Press, Beijing, China
[19] Rinke, K.; Huber, A. M. R.; Kempke, S.; Eder, M.; Wolf, T.; Probst, W. N.; Rothhaupt, K. O., Lake-wide distributions of temperature, phytoplankton, zooplankton, and fish in the pelagic zone of a large lake, Limnology and Oceanography, 54, 4, 1306-1322 (2009) · doi:10.4319/lo.2009.54.4.1306
[20] Malchow, H.; Radtke, B.; Kallache, M.; Medvinsky, A. B.; Tikhonov, D. A.; Petrovskii, S. V., Spatio-temporal pattern formation in coupled models of plankton dynamics and fish school motion, Nonlinear Analysis: Real World Applications, 1, 1, 53-67 (2000) · Zbl 0986.92041 · doi:10.1016/S0362-546X(99)00393-4
[21] Lazzaro, X.; Drenner, R. W.; Stein, R. A.; Smith, J. D., Planktivores and plankton dynamics: effects of fish biomass and planktivore type, Canadian Journal of Fisheries and Aquatic Sciences, 49, 7, 1466-1473 (1992) · doi:10.1139/f92-161
[22] Sieber, M.; Malchow, H.; Schimansky-Geier, L., Constructive effects of environmental noise in an excitable prey-predator plankton system with infected prey, Ecological Complexity, 4, 4, 223-233 (2007) · doi:10.1016/j.ecocom.2007.06.005
[23] Ptacnik, R.; Diehl, S.; Berger, S., Performance of sinking and nonsinking phytoplankton taxa in a gradient of mixing depths, Limnology and Oceanography, 48, 5, 1903-1912 (2003) · doi:10.4319/lo.2003.48.5.1903
[24] Pitcher, G. C.; Walker, D. R.; Mitchell-Innes, B. A., Phytoplankton sinking rate dynamics in the southern Benguela upwelling system, Marine Ecology Progress Series, 55, 261-269 (1989)
[25] Bienfang, P. K., Phytoplankton sinking rates in oligotrophic waters off Hawaii, USA, Marine Biology, 61, 1, 69-77 (1980) · doi:10.1007/BF00410342
[26] Condie, S. A.; Bormans, M., The influence of density stratification on particle settling, dispersion and population growth, Journal of Theoretical Biology, 187, 1, 65-75 (1997) · doi:10.1006/jtbi.1997.0417
[27] Lucas, L. V.; Cloern, J. E.; Koseff, J. R.; Monismith, S. G.; Thompson, J. K., Does the Sverdrup critical depth model explain bloom dynamics in estuaries?, Journal of Marine Research, 56, 2, 375-415 (1998) · doi:10.1357/002224098321822357
[28] Diehl, S., Phytoplankton, light, and nutrients in a gradient of mixing depths: theory, Ecology, 83, 2, 386-398 (2002) · doi:10.1890/0012-9658(2002)083[0386:PLANIA]2.0.CO;2
[29] Wang, H. L.; Feng, J. F., Ecological Dynamics and Prediction of Red Tide (2006), Tianjin, China: Tianjin University Press, Tianjin, China
[30] Rosenzweig, M. L., Paradox of enrichment: destabilization of exploitation ecosystems in ecological time, Science, 171, 3969, 385-387 (1971) · doi:10.1126/science.171.3969.385
[31] Hanski, I., The functional response of predators: worries about scale, Trends in Ecology and Evolution, 6, 5, 141-142 (1991) · doi:10.1016/0169-5347(91)90052-Y
[32] Hsu, S. B.; Hwang, T. W.; Kuang, Y., Rich dynamics of a ratio-dependent predator-prey models, Bulletin of Mathematical Biology, 61, 489-506 (2011)
[33] Jost, C.; Arino, O.; Arditi, R., About deterministic extinction in ratio-dependent predator-prey models, Bulletin of Mathematical Biology, 61, 1, 19-32 (1999) · Zbl 1323.92173 · doi:10.1006/bulm.1998.0072
[34] Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 36, 4, 389-406 (1998) · Zbl 0895.92032 · doi:10.1007/s002850050105
[35] Haque, M., Ratio-dependent predator-prey models of interacting populations, Bulletin of Mathematical Biology, 71, 2, 430-452 (2009) · Zbl 1170.92027 · doi:10.1007/s11538-008-9368-4
[36] Flores, J. D.; Olivares, E. G., Dynamics of a predator-prey model with Allee effect on prey and ratio-dependent functional response, Ecological Complexity, 18, 59-65 (2014) · doi:10.1016/j.ecocom.2014.02.005
[37] Sen, M.; Banerjee, M.; Morozov, A., Stage-structured ratio-dependent predator-prey models revisited: when should the maturation lag result in system destabilization?, Ecological Complexity, 19, 23-34 (2014) · doi:10.1016/j.ecocom.2014.02.001
[38] Song, Y.; Zou, X., Spatiotemporal dynamics in a diffusive ratio-dependent predator-prey model near a Hopf-Turing bifurcation point, Computers & Mathematics with Applications, 67, 10, 1978-1997 (2014) · Zbl 1366.92111 · doi:10.1016/j.camwa.2014.04.015
[39] Kuznetsov, S. P.; Mosekilde, E.; Dewel, G.; Borckmans, P., Absolute and convective instabilities in a one-dimensional Brusselator flow model, The Journal of Chemical Physics, 106, 18, 7609-7616 (1997) · doi:10.1063/1.473763
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.