# American Institute of Mathematical Sciences

August  2018, 15(4): 993-1010. doi: 10.3934/mbe.2018044

## Optimal design for dynamical modeling of pest populations

 1 Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212, USA 2 Undergraduate Research Opportunities Center (UROC), California State University, Monterey Bay, Seaside, CA 93955, USA 3 Department of Entomology and Nematology, Center for Population Biology, University of California, Davis, CA 95616, USA

Received  July 31, 2017 Accepted  November 30, 2017 Published  March 2018

We apply SE-optimal design methodology to investigate optimal data collection procedures as a first step in investigating information content in ecoinformatics data sets. To illustrate ideas we use a simple phenomenological citrus red mite population model for pest dynamics. First the optimal sampling distributions for a varying number of data points are determined. We then analyze these optimal distributions by comparing the standard errors of parameter estimates corresponding to each distribution. This allows us to investigate how many data are required to have confidence in model parameter estimates in order to employ dynamical modeling to infer population dynamics. Our results suggest that a field researcher should collect at least 12 data points at the optimal times. Data collected according to this procedure along with dynamical modeling will allow us to estimate population dynamics from presence/absence-based data sets through the development of a scaling relationship. These Likert-type data sets are commonly collected by agricultural pest management consultants and are increasingly being used in ecoinformatics studies. By applying mathematical modeling with the relationship scale from the new data, we can then explore important integrated pest management questions using past and future presence/absence data sets.

Citation: H. T. Banks, R. A. Everett, Neha Murad, R. D. White, J. E. Banks, Bodil N. Cass, Jay A. Rosenheim. Optimal design for dynamical modeling of pest populations. Mathematical Biosciences & Engineering, 2018, 15 (4) : 993-1010. doi: 10.3934/mbe.2018044
##### References:
 [1] H. T. Banks, J. E. Banks, R. A. Everett and J. D. Stark, An adaptive feedback methodology for determining information content in stable population studies, Mathematical Biosciences and Engineering, 13 (2016), 653-671.  doi: 10.3934/mbe.2016013. [2] H. T. Banks, J. E. Banks, J. Rosenheim and K. Tillman, Modelling populations of Lygus hesperus cotton fields in the San Joaquin Valley of California: The importance of statistical and mathematical model choice, Journal of Biological Dynamics, 11 (2017), 25-39.  doi: 10.1080/17513758.2016.1143533. [3] H. T. Banks, J. E. Banks, N. Murad, J. A Rosenheim and K. Tillman, Modelling pesticide treatment effects on Lygus hesperus in cotton fields, CRSC-TR15-09, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, September, 2015; Proceedings, 27 th IFIP TC7 Conference 2015 on System Modelling and Optimization, L. Bociu et al (Eds.) CSMO 2015 IFIP AICT, 494 (2017), 1-12, Springer. [4] H. T. Banks, K. Bekele-Maxwell, L. Bociu, M. Noorman and K. Tillman, The complex-step method for sensitivity analysis of non-smooth problems arising in biology, Eurasian Journal of Mathematical and Computer Applications, 3 (2015), 15-68. [5] H. T. Banks, A. Cintron-Arias and F. Kappel, Parameter selection methods in inverse problem formulation, CRSC-TR10-03, N. C. State University, February, 2010, Revised, November, 2010; in Mathematical Modeling and Validation in Physiology: Application to the Cardiovascular and Respiratory Systems, (J. J. Batzel, M. Bachar, and F. Kappel, eds.), 43-73, Lecture Notes in Mathematics, 2064, Springer-Verlag, Berlin 2013. [6] H. T. Banks, S. Dediu, S. L. Ernstberger and F. Kappel, Generalized sensitivities and optimal experimental design, Journal of Inverse and Ill-posed Problems, 18 (2010), 25-83. [7] H. T. Banks and M. L. Joyner, Information Content in Data Sets: A Review of Methods for Interrogation and Model Comparison, CRSC-TR17-14, N. C. State University, Raleigh, NC, June, 2017. doi: 10.1515/jiip-2017-0096. [8] H. T. Banks, K. Holm and F. Kappel, Comparison of optimal design methods in inverse problems, Inverse Problems, 27 (2011), 075002, 31 pp. [9] H. T. Banks, S. Hu and W. C. Thompson, Modeling and Inverse Problems in the Presence of Uncertainty, CRC Press, New York, 2014. [10] H. T. Banks and H. T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes, CRC Press, New York, 2009. [11] C. C. Childers and T. R. Fasulo, Citrus red mite, Gainesville: University of Florida Institute of Food and Agricultural Sciences, ENY817,1992. http://ufdc.ufl.edu/IR00004619/00001 [12] S. H. Dreistadt, Review of Integrated Pest Management for Citrus, 3rd ed, Journal of Agricultural & Food Information, University of California Division of Agriculture and Natural Resources, Publication 3303,2012. [13] L. Ferguson and E. E. Grafton-Cardwell, Citrus Production Manual, University of California Agriculture and Natural Resources, Publication 3539,2014. [14] V. P. Jones and M. P. Parrella, Intratree regression sampling plans for the citrus red mite (Acari: Tetranychidae) on lemons in southern California, Journal of Economic Entomology, 77 (1984), 810-813.  doi: 10.1093/jee/77.3.810. [15] M. Kogan, Integrated pest management: historical perspectives and contemporary developments, Annual Review of Entomology, 43 (1998), 243-270.  doi: 10.1146/annurev.ento.43.1.243. [16] R. Likert, A technique for the measurement of attitudes, Archives of Psychology, 22 (1932), p55. [17] G. Livingston, L. Hack, K. Steinmann, E. E. Grafton-Cardwell and J. A. Rosenheim, An ecoinformatics approach to field scale evaluation of pesticide efficacy and hazards in California citrus, in prep. [18] J. A. Rosenheim and C. Gratton, Ecoinformatics (big data) for agricultural entomology: Pitfalls, progress, and promise, Annual Review of Entomology, 62 (2017), 399-417.  doi: 10.1146/annurev-ento-031616-035444. [19] J. A. Rosenheim, S. Parsa, A. A. Forbes, W. A. Krimmel, Y. H. Law, M. Segoli, M. Segoli, F. S. Sivakoff, T. Zaviezo and K. Gross, Ecoinformatics for integrated pest management: Expanding the applied insect ecologist's tool-kit, Journal of Economic Entomology, 104 (2011), 331-342.  doi: 10.1603/EC10380.

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##### References:
 [1] H. T. Banks, J. E. Banks, R. A. Everett and J. D. Stark, An adaptive feedback methodology for determining information content in stable population studies, Mathematical Biosciences and Engineering, 13 (2016), 653-671.  doi: 10.3934/mbe.2016013. [2] H. T. Banks, J. E. Banks, J. Rosenheim and K. Tillman, Modelling populations of Lygus hesperus cotton fields in the San Joaquin Valley of California: The importance of statistical and mathematical model choice, Journal of Biological Dynamics, 11 (2017), 25-39.  doi: 10.1080/17513758.2016.1143533. [3] H. T. Banks, J. E. Banks, N. Murad, J. A Rosenheim and K. Tillman, Modelling pesticide treatment effects on Lygus hesperus in cotton fields, CRSC-TR15-09, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, September, 2015; Proceedings, 27 th IFIP TC7 Conference 2015 on System Modelling and Optimization, L. Bociu et al (Eds.) CSMO 2015 IFIP AICT, 494 (2017), 1-12, Springer. [4] H. T. Banks, K. Bekele-Maxwell, L. Bociu, M. Noorman and K. Tillman, The complex-step method for sensitivity analysis of non-smooth problems arising in biology, Eurasian Journal of Mathematical and Computer Applications, 3 (2015), 15-68. [5] H. T. Banks, A. Cintron-Arias and F. Kappel, Parameter selection methods in inverse problem formulation, CRSC-TR10-03, N. C. State University, February, 2010, Revised, November, 2010; in Mathematical Modeling and Validation in Physiology: Application to the Cardiovascular and Respiratory Systems, (J. J. Batzel, M. Bachar, and F. Kappel, eds.), 43-73, Lecture Notes in Mathematics, 2064, Springer-Verlag, Berlin 2013. [6] H. T. Banks, S. Dediu, S. L. Ernstberger and F. Kappel, Generalized sensitivities and optimal experimental design, Journal of Inverse and Ill-posed Problems, 18 (2010), 25-83. [7] H. T. Banks and M. L. Joyner, Information Content in Data Sets: A Review of Methods for Interrogation and Model Comparison, CRSC-TR17-14, N. C. State University, Raleigh, NC, June, 2017. doi: 10.1515/jiip-2017-0096. [8] H. T. Banks, K. Holm and F. Kappel, Comparison of optimal design methods in inverse problems, Inverse Problems, 27 (2011), 075002, 31 pp. [9] H. T. Banks, S. Hu and W. C. Thompson, Modeling and Inverse Problems in the Presence of Uncertainty, CRC Press, New York, 2014. [10] H. T. Banks and H. T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes, CRC Press, New York, 2009. [11] C. C. Childers and T. R. Fasulo, Citrus red mite, Gainesville: University of Florida Institute of Food and Agricultural Sciences, ENY817,1992. http://ufdc.ufl.edu/IR00004619/00001 [12] S. H. Dreistadt, Review of Integrated Pest Management for Citrus, 3rd ed, Journal of Agricultural & Food Information, University of California Division of Agriculture and Natural Resources, Publication 3303,2012. [13] L. Ferguson and E. E. Grafton-Cardwell, Citrus Production Manual, University of California Agriculture and Natural Resources, Publication 3539,2014. [14] V. P. Jones and M. P. Parrella, Intratree regression sampling plans for the citrus red mite (Acari: Tetranychidae) on lemons in southern California, Journal of Economic Entomology, 77 (1984), 810-813.  doi: 10.1093/jee/77.3.810. [15] M. Kogan, Integrated pest management: historical perspectives and contemporary developments, Annual Review of Entomology, 43 (1998), 243-270.  doi: 10.1146/annurev.ento.43.1.243. [16] R. Likert, A technique for the measurement of attitudes, Archives of Psychology, 22 (1932), p55. [17] G. Livingston, L. Hack, K. Steinmann, E. E. Grafton-Cardwell and J. A. Rosenheim, An ecoinformatics approach to field scale evaluation of pesticide efficacy and hazards in California citrus, in prep. [18] J. A. Rosenheim and C. Gratton, Ecoinformatics (big data) for agricultural entomology: Pitfalls, progress, and promise, Annual Review of Entomology, 62 (2017), 399-417.  doi: 10.1146/annurev-ento-031616-035444. [19] J. A. Rosenheim, S. Parsa, A. A. Forbes, W. A. Krimmel, Y. H. Law, M. Segoli, M. Segoli, F. S. Sivakoff, T. Zaviezo and K. Gross, Ecoinformatics for integrated pest management: Expanding the applied insect ecologist's tool-kit, Journal of Economic Entomology, 104 (2011), 331-342.  doi: 10.1603/EC10380.
Optimized meshes resulting from SE-optimal implementation
Relationship between sampling distribution and corresponding performance (cost)
Average standard errors (over 1000 MC trials) for each parameter, comparing optimized versus uniform grids for N = 6, 12, 18, 24, and 30
Confidence intervals for each parameter for N = 6, 12, 18, 24, and 30 on the optimized grids
Heaviside functions and Dirac delta "functions"
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