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Spatiotemporal Dynamics in a Spatial PlanktonSystem

Published online by Cambridge University Press:  27 July 2010

R. K. Upadhyay ,
W. Wang  and
N. K. Thakur
R. K. Upadhyay*
Affiliation:
Department of Applied Mathematics, Indian School of Mines, Dhanbad, 826004, India
W. Wang
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai, 200433 P.R. China School of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang, 325035 P.R.China
N. K. Thakur
Affiliation:
Department of Applied Mathematics, Indian School of Mines, Dhanbad, 826004, India
*
* Corresponding author. E-mail:[email protected]. Present Address is: Institute of Biology,Department of Plant Taxonomy and Ecology, Research group of Theoretical Biology andEcology, Eötvös Lorand University, H-1117, Pazmany P.S. 1/A, Budapest, Hungary.
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Abstract

In this paper, we investigate the complex dynamics of a spatial plankton-fish system withHolling type III functional responses. We have carried out the analytical study for bothone and two dimensional system in details and found out a condition for diffusiveinstability of a locally stable equilibrium. Furthermore, we present a theoreticalanalysis of processes of pattern formation that involves organism distribution and theirinteraction of spatially distributed population with local diffusion. The results ofnumerical simulations reveal that, on increasing the value of the fish predation rates,the sequences spots → spot-stripe mixtures → stripes → hole-stripe mixtures holes → wavepattern is observed. Our study shows that the spatially extended model system has not onlymore complex dynamic patterns in the space, but also has spiral waves.

Keywords

spatial plankton systempredator-prey interactionglobally asymptotically stablepattern formation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 5: Reaction-diffusion waves , 2010 , pp. 102 - 122
Copyright
© EDP Sciences, 2010

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References

M. Abbott. Phytoplankton patchiness: ecological implications and observation methods. In: Patch dynamics (Levin, S. A., Powell, T. M. and Steele, J. H., eds.), Lecture Notes in Biomath., 96 (1993), 37-49.
A. D. Bazykin, A.I. Khibnik, B. Krauskopf, B. Nonlinear dynamics of interacting populations. World Scientific, Singapore, 1998.
Chen, B., Wang, M.. Qualitative analysis for a diffusive predator-prey model. Comp. Math. with Appl., 55 (2008), 339-355.CrossRefGoogle Scholar
Dubey, B., Hussain, J.. Modelling the interaction of two biological species in polluted environment. J. Math. Anal. Appl., 246 (2000), 58-79.CrossRefGoogle Scholar
Fasham, M. J. R.. The statistical and mathematical analysis of plankton patchiness. Oceanogr. Mar. Biol. Annu. Rev., 16 (1978), 43-79.Google Scholar
Fu, C., Mohn, R., Fanning, L.P.. Why the Atlantic cod stock off eastern Nova Scotia has not recovered. Can. J. Fish. Aquat. Sci., 58 (2001), 1613-1623.CrossRefGoogle Scholar
Gao, H., Wei, H., Sun, W., Zhai, X.. Functions used in biological models and their influence on simulations. Indian J. Marine Sci., 29 (2000), 230-237.Google Scholar
Greene, C.H., Widder, E. A., Youngbluth, M. J., Tamse, A., Johnson, G. E.. The migration behavior, fine structure and bioluminescent activity of krill sound-scattering layers. Limnology and Oceanography, 37 (1992), 650-658.CrossRefGoogle Scholar
Grimm, V., Revilla, E., Berger, U., Jeltsch, F., Mooij, W., Railsback, S., Thulke, H., Weiner, J., Wiegand, T., DeAngelis, D. Pattern-oriented modeling of agent-based complex systems: lessons from ecology, Science, 310 (2005), 987–991.CrossRefGoogle ScholarPubMed
Hammer, A.C., Pitchford, J.W.. The role of mixotrophy in plankton bloom dynamics and the consequences for productivity. ICES J. Marine Sci., 62 (2005), 833-840.CrossRefGoogle Scholar
Kar, T.K., Matsuda, H.. Global dynamics and controllability of a harvested prey-predator system with Holling type III functional response. Nonlinear Anal.: Hybrid Systems, 1 (2007), 59-67.Google Scholar
Liermann, M., Hilborn, R.. Depensation: Evidence, models and implications. Fish and Fisheries, 2 (2001), 33-58.CrossRefGoogle Scholar
Loehle, C.. Challenges of ecological complexity. Ecological Complexity, 1 (2004), 3-6.Google Scholar
Ludwig, D., Jones, D., Holling, C.. Qualitative analysis of an insect outbreak system: the spruce budworm and forest. J. Animal Eco., 47 (1978), 315-332.CrossRefGoogle Scholar
Mackas, F., Boyd, C. M.. Spectral analysis of zooplankton spatial heterogeneity. Science, 204 (1979), 62-64. CrossRefGoogle ScholarPubMed
Magnusson, K. G., Palsson, O.K.. Predator-prey interactions of cod and capelin in Icelandic waters. ICES Marine Science Symposium, 193 (1991), 153-170.Google Scholar
Malchow, H.. Spatio-temporal pattern formation in nonlinear nonequilibrium plankton dynamics. Proc. Roy. Soc. Lond. Series B, 251 (1993), 103-109.CrossRefGoogle Scholar
Malchow, H.. Nonlinear plankton dynamics and pattern formation in an ecohydrodynamic model system. J. Marine Systems, 7 (1996), 193-202.CrossRefGoogle Scholar
Malchow, H.. Non-equilibrium spatio-temporal patterns in models of non-linear plankton dynamics. Freshwater Biol., 45 (2000), 239-251.CrossRefGoogle Scholar
Malchow, H., Petrovskii, S. V., Medvinsky, A. B.. Numerical study of plankton-fish dynamics in a spatially structured and noisy environment. Ecol. Model., 149 (2002), 247-255.CrossRefGoogle Scholar
H. Malchow, S. V. Petrovskii, E. Venturino. Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models and Simulation, CRC Press, UK, 2008.
R. M. May. Stability and Complexity in model ecosystems. Princeton University press, Princeton, NJ. 1973.
Medvinsky, A. B., Petrovskii, S. V., Tikhonova, I. A., Malchow, H., Li, B.-L.. Spatiotemporal complexity of plankton and fish dynamics. SIAM Review, 44 (2002), 311-370.CrossRefGoogle Scholar
Medvinsky, A. B., Petrovskii, S. V., Tikhonova, I. A., Venturino, E., Malchow, H.. Chaos and regular dynamics in a model multi-habitat plankton-fish community. J. Biosciences, 26 (2001), 109-120.CrossRefGoogle Scholar
Medvinsky, A. B., Tikhonova, I. A., Aliev, R. R., Li, B. -L., Lin, Z. S., Malchow, H.. Patchy environment as a factor of complex plankton dynamics. Phys. Rev. E, 64 (2001), 021915-021917.CrossRefGoogle ScholarPubMed
Michaelis, L., Menten, M. L.. Die Kinetik der Invertinwirkung. Biochem. Z., 49 (1913), 333-369.Google Scholar
Morozov, A.. Emergence of Holling type III zooplankton functional response: Bringing together field evidence and mathematical modelling. J. Theor. Biol., 265 (2010), 45-54.CrossRefGoogle Scholar
Morozov, A., Arashkevich, E., Reigstad, M., Falk-Petersen, S.. Influence of spatial heterogeneity on the type of zooplankton functional response: A study based on field observations. Deep-Sea Research II, 55 (2008), 2285-2291.CrossRefGoogle Scholar
J. D. Murray. Mathematical biology. Springer-Verlag, New York, 1989.
Nilssen, K. T., Pedersen, O.-P., Folkow, L., Haug, T.. Food consumption estimates of Barents Sea harp seals. NAMMCO Scientific Publications, 2 (2000), 9-27.CrossRefGoogle Scholar
A. Okubo. Diffusion and ecological problems: mathematical models. Springer-Verlag, Berlin. 1980.
Pascual, M.. 1993. Diffusion-induced chaos in a spatial predator-prey system. Proc. Royal Soc. B, 251 (1993), 1-7.CrossRefGoogle Scholar
Petrovskii, S. V., Malchow, H.. Critical phenomena in plankton communities: KISS model revisited. Nonlinear Anal.: RWA, 1 (2000), 37-51.CrossRefGoogle Scholar
Petrovskii, S. V., Malchow, H.. Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics. Theor. Popul. Biol., 59 (2001), 157-174.CrossRefGoogle ScholarPubMed
Poggiale, J. -C., Gauduchon, M., Auger, P.. Enrichment paradox induced by spatial heterogeneity in a phytoplankton- zooplankton system. Math. Model. Natural Phenom., 3 (2008), 87-102.CrossRefGoogle Scholar
Real, L. A.. The kinetic of functional response. Am. Nat., 111 (1977), 289-300.CrossRefGoogle Scholar
M. Scheffer. Ecology of shallow lakes. Chapman and Hall, London. 1998.
Scheffer, M., De Boer, R. J.. Implications of spatial heterogeneity for the paradox of enrichment. Ecology, 76 (1996), 2270-2277.CrossRefGoogle Scholar
Schweder, T., Hagen, G. S., Hatlebakk, E.. Direct and indirect effects of minke whale abundance on cod and herring fisheries: A scenario experiment for the Greater Barents Sea. NAMMCO Scientific Publications, 1 (2000), 120-133.CrossRefGoogle Scholar
Segel, L. A., Jackson, J. L.. Dissipative structure: An explanation and an ecological example. J. Theo. Biol., 37 (1972), 545-559.CrossRefGoogle ScholarPubMed
Sherratt, J. A., Eagan, B. T., Lewis, M. A.. Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality? Phil. Trans. Roy. Soc. Lond. B, 352 (1997), 21-38. CrossRefGoogle Scholar
Sherratt, J. A., Lewis, M. A., Fowler, A. C.. Ecological chaos in the wake of invasion. PNAS, 92 (1995), 2524-2528.CrossRefGoogle ScholarPubMed
J. H. Steele. Spatial pattern in plankton communities. Plenum Press, New York, 1978.
Steele, J. H., Henderson, E. W.. A simple plankton model. Am. Nat., 117 (1981), 676-691.CrossRefGoogle Scholar
Steele, J. H., Henderson, E. W.. A simple model for plankton patchiness. J. Plankton Research, 14 (1992), 1397-1403.CrossRefGoogle Scholar
Steele, J. H., Henderson, E. W.. The role of predation in plankton models. J. Plankton Research, 14 (1992), 157-172.CrossRefGoogle Scholar
Truscott, J. E., Brindley, J.. Equilibria, stability and excitability in a general class of plankton population models. Phil. Trans. Roy. Soc. Lond. A, 347 (1994), 703-718.CrossRefGoogle Scholar
Truscott, J. E., Brindley, J.. Ocean plankton populations as excitable media. Bull. Math. Biol., 56 (1994), 981-998.CrossRefGoogle Scholar
P. Turchin. Complex population dynamics: a theoretical/empirical Synthesis. Princeton University Press, Princeton, NJ, 2003.
Upadhyay, R. K., Kumari, N., Rai, V.. Wave of chaos and pattern formation in a spatial predator-prey system with Holling type IV functional response. Math. Model. Natural Phenom., 3 (2008), 71-95.CrossRefGoogle Scholar
Upadhyay, R. K., Kumari, N., Rai, V.. Wave of chaos in a diffusive system: Generating realistic patterns of patchiness in plankton-fish dynamics. Chaos Solit. Fract., 40 (2009), 262-276.CrossRefGoogle Scholar
Upadhyay, R. K., Thakur, N. K., Dubey, B.. Nonlinear non-equilibrium pattern formation in a spatial aquatic system: Effect of fish predation. J. Biol. Sys., 18 (2010), 129-159.CrossRefGoogle Scholar
Xiao, J., Li, H., Yang, J., Hu, G.. Chaotic Turing pattern formation in spatiotemporal systems. Frontier of Physics in China, 1 (2006), 204-208.CrossRefGoogle Scholar