Parameter identification and sensitivity analysis for a phytoplankton competition model
Authors:
Thomas Stojsavljevic, Gabriella Pinter, Istvan Lauko and Nicholas Myers
Journal:
Quart. Appl. Math. 77 (2019), 1-18
MSC (2010):
Primary 35Q92, 35R30, 49K40
DOI:
https://doi.org/10.1090/qam/1514
Published electronically:
August 28, 2018
MathSciNet review:
3897918
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Abstract: Phytoplankton live in a complex environment with two essential resources, light and nutrients, forming various gradients. Light supplied from above is never homogeneously distributed in a body of water due to refraction and absorption from biomass present in the ecosystem and other sources. Nutrients in turn are typically supplied from below mixed-up by diffusion from the benthic region. Here we present a model of two phytoplankton species competing in a deep freshwater lake for light and two nutrients, one of which is assumed to be preferred. The model is comprised of a system of non- linear, non-local partial differential equations with appropriate boundary conditions. The parameter space of the model is analyzed for parameter identifiability - the ability for a parameter’s true value to be recovered through optimization, and for global sensitivity - the influence a parameter has on model response. The results of these analyses are interpreted within their biological context.
References
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- O. M. Phillips, The equilibrium and stability of simple marine biological systems I. Primary nutrient consumers, The American Naturalist (1973), 107-953: 73-93.
- C. S. Reynolds, The ecology of freshwater phytoplankton, Cambridge University Press, 1984.
- G-Y. Rhee and I. J. Gotham, The effect of environment factors on phytoplankton growth: light and the interactions of light with nitrate limitation, Limnology and Oceanography (1981), 26-4: 649-659.
- T. J. Smayda, The suspension and sinking of phytoplankton in the sea, Annual Review of Oceanography and Marine Biology (1970), 8: 353-414.
- Ralph C. Smith, Uncertainty quantification, Computational Science & Engineering, vol. 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2014. Theory, implementation, and applications. MR 3155184
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- Mami T. Wentworth, Ralph C. Smith, and H. T. Banks, Parameter selection and verification techniques based on global sensitivity analysis illustrated for an HIV model, SIAM/ASA J. Uncertain. Quantif. 4 (2016), no. 1, 266–297. MR 3479703, DOI https://doi.org/10.1137/15M1008245
References
- R. Ashino, M. Nagase, and R. Vaillancourt, Behind and beyond the MATLAB ODE suite, Comput. Math. Appl. 40 (2000), no. 4-5, 491–512. MR 1772651, DOI https://doi.org/10.1016/S0898-1221%2800%2900175-9
- H. T. Banks and Kathleen L. Bihari, Modelling and estimating uncertainty in parameter estimation, Inverse Problems 17 (2001), no. 1, 95–111. MR 1818494, DOI https://doi.org/10.1088/0266-5611/17/1/308
- A. S. Brooks and B. G. Torke, Vertical and seasonal distribution of Chlorophyll a in Lake Michigan, Journal of the Fisheries Research Board of Canada (1977), 34:2280-2287.
- A. S. Brooks and D. N. Edginton, Biogeochemical control of phosphorus cycling and primary production in Lake Michigan, Limnology and Oceanography (1994), 39-4: 961-968.
- J. Chattopadhyay, S. Pal, and R. R. Sarkar, Mathematical modeling of harmful algal blooms supported by experimental findings, Ecological Complexity (2004), 1: 225-235.
- Q. Dortch, The interaction between ammonium and nitrate uptake in phytoplankton, Marine ecology press series, Oldendorf (1990), 61-1: 183-201.
- Yihong Du and Sze-Bi Hsu, Concentration phenomena in a nonlocal quasi-linear problem modelling phytoplankton. I. Existence, SIAM J. Math. Anal. 40 (2008), no. 4, 1419–1440. MR 2466162, DOI https://doi.org/10.1137/07070663X
- Yihong Du and Sze-Bi Hsu, Concentration phenomena in a nonlocal quasi-linear problem modelling phytoplankton. II. Limiting profile, SIAM J. Math. Anal. 40 (2008), no. 4, 1441–1470. MR 2466163, DOI https://doi.org/10.1137/070706641
- Yihong Du and Sze-Bi Hsu, On a nonlocal reaction-diffusion problem arising from the modeling of phytoplankton growth, SIAM J. Math. Anal. 42 (2010), no. 3, 1305–1333. MR 2653252, DOI https://doi.org/10.1137/090775105
- Yihong Du and Linfeng Mei, On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity 24 (2011), no. 1, 319–349. MR 2746150, DOI https://doi.org/10.1088/0951-7715/24/1/016
- P. M. Gilbert, F. P. Wilkerson, R. C. Dugdale, J. A. Raven, C. L. Dupont, P. R. Leavitt, and T. M. Kana, Pluses and minuses of ammonium and nitrate uptake and assimilation by phytoplankton and implications for productivity and community composition, with emphasis on nitrogen-enriched conditions, Limnology and Oceanography (2015), 61-1: 165-197.
- G. C. Hays, A. J. Richardson, and C. Robinson, Climate change and marine plankton, Trends in Ecology $\&$ Evolution (2005), 20: 337-334.
- A. Howard, Modeling movement patterns of the cyanobacterium Microcystis, Ecological Applications (2001), 11-1: 304-310.
- J. Huisman and B. Sommeijer, Simulation techniques for the population dynamics of sinking phytoplankton in light-limited environments, Modeling, Analysis and Simulation (2002), 1-17.
- J. Huisman and F. J. Weissing, Light-limited growth and competition for light in well-mixed aquatic environments: an elementary model, Ecology (1994), 75-2: 507-520.
- J. Huisman and F. J. Weissing, Competition for nutrients and light in a mixed water column: a theoretical analysis, The American Naturalist (1995), 146-4: 536-564.
- G. E. Hutchinson, The paradox of the plankton, The American Naturalist (1961), 95-882: 137-145.
- A. D. Jassby and T. Pratt, Mathematical formulation of the relationship between photosynthesis and light for phytoplankton, Limnology and Oceanography (1976), 21-4: 540-547.
- E. F. Keller and L. A. Segel, Travelling bands of chemotactic bacteria: A theeoretical analysis, Journal of Theoretical Biology (1971), 30: 235-248.
- J. T. O. Kirk, Theoretical-analysis of contribution of algal cells to attenuation of light within natural waters 1. General treatment of suspensions of pigmented cells, New Phytologist (1975), 75: 11-20.
- C. A. Klausmeier and E. Lichtman, Algal games: the vertical distribution of phytoplankton in poorly mixed water columns, Limnology and Oceanography (2001), 46-8: 1998-2007.
- D. Krause-Jensen and K. Sand-Jensen, Light attenuation and photosynthesis of aquatic plant communities, Limnology and Oceanography (1998), 43-3: 396-407.
- Jeffrey C. Lagarias, James A. Reeds, Margaret H. Wright, and Paul E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM J. Optim. 9 (1999), no. 1, 112–147. MR 1662563, DOI https://doi.org/10.1137/S1052623496303470
- M. A. Leibold, Resources and predatorss can affect the vertical distributions of zooplankton, Limnology and Oceanography (1990), 35: 33-45.
- R. MacArthur and R. Levins, Competition, habitat selection, and character displacement in a patchy environment, Proceedings of the National Academy of Sciences (USA) (1964), 51: 1207-1210.
- Jarad P. Mellard, Kohei Yoshiyama, Elena Litchman, and Christopher A. Klausmeier, The vertical distribution of phytoplankton in stratified water columns, J. Theoret. Biol. 269 (2011), 16–30. MR 2974462, DOI https://doi.org/10.1016/j.jtbi.2010.09.041
- J. A. Mortonson and A. S. Brooks, Occurrence of a deep nitrate maximum in Lake Michigan, Canadian Journal of Fisheries and Aquatic Science (1980), 37:1025-1027.
- Akira Okubo, Diffusion and ecological problems: mathematical models, Biomathematics, vol. 10, Springer-Verlag, Berlin-New York, 1980. An extended version of the Japanese edition, Ecology and diffusion; Translated by G. N. Parker. MR 572962
- O. M. Phillips, The equilibrium and stability of simple marine biological systems I. Primary nutrient consumers, The American Naturalist (1973), 107-953: 73-93.
- C. S. Reynolds, The ecology of freshwater phytoplankton, Cambridge University Press, 1984.
- G-Y. Rhee and I. J. Gotham, The effect of environment factors on phytoplankton growth: light and the interactions of light with nitrate limitation, Limnology and Oceanography (1981), 26-4: 649-659.
- T. J. Smayda, The suspension and sinking of phytoplankton in the sea, Annual Review of Oceanography and Marine Biology (1970), 8: 353-414.
- Ralph C. Smith, Uncertainty quantification, Computational Science & Engineering, vol. 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2014. Theory, implementation, and applications. MR 3155184
- F. M. Stewart and B. R. Levin, Partitioning of resources and the outcome of interspecific competition: a model and some general considerations, The American Naturalist (1973), 107-954: 171-198.
- Mami T. Wentworth, Ralph C. Smith, and H. T. Banks, Parameter selection and verification techniques based on global sensitivity analysis illustrated for an HIV model, SIAM/ASA J. Uncertain. Quantif. 4 (2016), no. 1, 266–297. MR 3479703, DOI https://doi.org/10.1137/15M1008245
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Additional Information
Thomas Stojsavljevic
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O.Box 413, Milwaukee, WI 53201-0413
Email:
tgs@uwm.edu
Gabriella Pinter
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O.Box 413, Milwaukee, WI 53201-0413
MR Author ID:
635961
Email:
gapinter@uwm.edu
Istvan Lauko
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O.Box 413, Milwaukee, WI 53201-0413
MR Author ID:
639825
Email:
iglauko@uwm.edu
Nicholas Myers
Affiliation:
Department of Mathematics, NC State University, P.O.Box 8205, Raleigh, NC 27695
MR Author ID:
1198833
Email:
nmyers2@ncsu.edu
Received by editor(s):
September 20, 2017
Published electronically:
August 28, 2018
Additional Notes:
This research was supported in part by the National Science Foundation, UBM-Institutional grant: DUE-1129056.
Article copyright:
© Copyright 2018
Brown University
