# American Institute of Mathematical Sciences

2016, 13(4): 653-671. doi: 10.3934/mbe.2016013

## An adaptive feedback methodology for determining information content in stable population studies

 1 Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212 2 Undergraduate Research Opportunities Center (UROC), California State University, Monterey Bay, United States 3 Center for Research in Scienti c Computation, North Carolina State University, Raleigh, NC 27695-8212, United States 4 Ecotoxicology Program, WSU Puyallup Research, Extension Center, Puyallup, WA 98371-4998, United States

Received  November 2015 Revised  February 2016 Published  May 2016

We develop statistical and mathematical based methodologies for determining (as the experiment progresses) the amount of information required to complete the estimation of stable population parameters with pre-specified levels of confidence. We do this in the context of life table models and data for growth/death for three species of Daphniids as investigated by J. Stark and J. Banks [17]. The ideas developed here also have wide application in the health and social sciences where experimental data are often expensive as well as difficult to obtain.
Citation: H. T. Banks, John E. Banks, R. A. Everett, John D. Stark. An adaptive feedback methodology for determining information content in stable population studies. Mathematical Biosciences & Engineering, 2016, 13 (4) : 653-671. doi: 10.3934/mbe.2016013
##### References:
 [1] K. Adoteye, H. T. Banks, K. Cross, S. Eytcheson, K. B. Flores, G. A. LeBlanc, T. Nguyen, C. Ross, E. Smith, M. Stemkovski and S. Stokely, Statistical validation of structured population models for Daphnia magna, Mathematical Biosciences, 266 (2015), 73-84. doi: 10.1016/j.mbs.2015.06.003. [2] K. Adoteye, H. T. Banks, K. B. Flores and G. A. LeBlanc, Estimation of time-varying mortality rates using continuous models for Daphnia magna, Applied Mathematical Letters, 44 (2015), 12-16. doi: 10.1016/j.aml.2014.12.014. [3] H. T. Banks, J. E. Banks, L. K. Dick and J. D. Stark, Estimation of dynamic rate parameters in insect populations undergoing sublethal exposure to pesticides, Bulletin of Mathematical Biology, 69 (2007), 2139-2180. doi: 10.1007/s11538-007-9207-z. [4] H. T. Banks, J. E. Banks, R. Everett and J. Stark, An Adaptive Feedback Methodology for Determining Information Content in Population Studies, CRSC-TR15-12, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, November, 2015. [5] H. T. Banks, J. E. Banks, S. J. Joyner and J. D. Stark, Dynamic models for insect mortality due to exposure to insecticides, Mathematical and Computer Modeling, 48 (2008), 316-332. doi: 10.1016/j.mcm.2007.10.005. [6] J. E. Banks, L. K. Dick, H. T. Banks and J. D. Stark, Time-varying vital rates in ecotoxicology: Selective pesticides and aphid population dynamics, Ecological Modeling, 210 (2008), 155-160. doi: 10.1016/j.ecolmodel.2007.07.022. [7] H. T. Banks, S. Hu and W. C. Thompson, Modeling and Inverse Problems in the Presence of Uncertainty, CRC Press, New York, 2014. [8] H. T. Banks and H. T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes, CRC Press, New York, 2009. [9] J. R. Carey, Applied Demography for Biologists with Special Emphasis on Insects, Oxford University Press, Oxford, 1993. [10] V. E. Forbes and P. Calow, Is the per capita rate of increase a good measure of population-level effects in ecotoxicology? Environmental Toxicology and Chemistry, 18 (1999), 1544-1556. doi: 10.1002/etc.5620180729. [11] V. E. Forbes and P. Calow, Extrapolation in ecological risk assessment: Balancing pragmatism and precaution in chemical controls legislation, Bioscience, 52 (2002), 249-257. [12] V. E. Forbes and P. Calow, Population growth rate as a basis for ecological risk assessment of toxic chemicals, Philosophical Transaction of the Royal Society, London, B: Biological Sciences, 357 (2002), 1299-1306. [13] N. Hanson and J. D. Stark, A comparison of simple and complex population models to reduce uncertainty in ecological risk assessments of chemicals: Example with three species of Daphnia, Ecotoxicology, 20 (2011), 1268-1276. doi: 10.1007/s10646-011-0675-4. [14] N. Hanson and J. D. Stark, Utility of population models to reduce uncertainty and increase value relevance in ecological risk assessments of pesticides: An example based on acute mortality data for Daphnids, Integrated Environmental Assessment and Management, 8 (2012), 262-270. doi: 10.1002/ieam.272. [15] U. Hommen, J. M. Baveco, N. Galic and P. J. van den Brink, Potential application of ecological models in the European environmental risk assessment of chemicals I: review of protection goals in EU directives and regulations, Integrated Environmental Assessment and Management, 6 (2010), 325-337. doi: 10.1002/ieam.69. [16] M. Kot, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511608520. [17] J. D. Stark and J. E. Banks, Developing Demographic Toxicity Data: Optimizing Effort for Predicting Population Outcomes, PeerJ, submitted, 2015.

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##### References:
 [1] K. Adoteye, H. T. Banks, K. Cross, S. Eytcheson, K. B. Flores, G. A. LeBlanc, T. Nguyen, C. Ross, E. Smith, M. Stemkovski and S. Stokely, Statistical validation of structured population models for Daphnia magna, Mathematical Biosciences, 266 (2015), 73-84. doi: 10.1016/j.mbs.2015.06.003. [2] K. Adoteye, H. T. Banks, K. B. Flores and G. A. LeBlanc, Estimation of time-varying mortality rates using continuous models for Daphnia magna, Applied Mathematical Letters, 44 (2015), 12-16. doi: 10.1016/j.aml.2014.12.014. [3] H. T. Banks, J. E. Banks, L. K. Dick and J. D. Stark, Estimation of dynamic rate parameters in insect populations undergoing sublethal exposure to pesticides, Bulletin of Mathematical Biology, 69 (2007), 2139-2180. doi: 10.1007/s11538-007-9207-z. [4] H. T. Banks, J. E. Banks, R. Everett and J. Stark, An Adaptive Feedback Methodology for Determining Information Content in Population Studies, CRSC-TR15-12, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, November, 2015. [5] H. T. Banks, J. E. Banks, S. J. Joyner and J. D. Stark, Dynamic models for insect mortality due to exposure to insecticides, Mathematical and Computer Modeling, 48 (2008), 316-332. doi: 10.1016/j.mcm.2007.10.005. [6] J. E. Banks, L. K. Dick, H. T. Banks and J. D. Stark, Time-varying vital rates in ecotoxicology: Selective pesticides and aphid population dynamics, Ecological Modeling, 210 (2008), 155-160. doi: 10.1016/j.ecolmodel.2007.07.022. [7] H. T. Banks, S. Hu and W. C. Thompson, Modeling and Inverse Problems in the Presence of Uncertainty, CRC Press, New York, 2014. [8] H. T. Banks and H. T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes, CRC Press, New York, 2009. [9] J. R. Carey, Applied Demography for Biologists with Special Emphasis on Insects, Oxford University Press, Oxford, 1993. [10] V. E. Forbes and P. Calow, Is the per capita rate of increase a good measure of population-level effects in ecotoxicology? Environmental Toxicology and Chemistry, 18 (1999), 1544-1556. doi: 10.1002/etc.5620180729. [11] V. E. Forbes and P. Calow, Extrapolation in ecological risk assessment: Balancing pragmatism and precaution in chemical controls legislation, Bioscience, 52 (2002), 249-257. [12] V. E. Forbes and P. Calow, Population growth rate as a basis for ecological risk assessment of toxic chemicals, Philosophical Transaction of the Royal Society, London, B: Biological Sciences, 357 (2002), 1299-1306. [13] N. Hanson and J. D. Stark, A comparison of simple and complex population models to reduce uncertainty in ecological risk assessments of chemicals: Example with three species of Daphnia, Ecotoxicology, 20 (2011), 1268-1276. doi: 10.1007/s10646-011-0675-4. [14] N. Hanson and J. D. Stark, Utility of population models to reduce uncertainty and increase value relevance in ecological risk assessments of pesticides: An example based on acute mortality data for Daphnids, Integrated Environmental Assessment and Management, 8 (2012), 262-270. doi: 10.1002/ieam.272. [15] U. Hommen, J. M. Baveco, N. Galic and P. J. van den Brink, Potential application of ecological models in the European environmental risk assessment of chemicals I: review of protection goals in EU directives and regulations, Integrated Environmental Assessment and Management, 6 (2010), 325-337. doi: 10.1002/ieam.69. [16] M. Kot, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511608520. [17] J. D. Stark and J. E. Banks, Developing Demographic Toxicity Data: Optimizing Effort for Predicting Population Outcomes, PeerJ, submitted, 2015.
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