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2015, 12(3): 585-607. doi: 10.3934/mbe.2015.12.585

## An aggregate stochastic model incorporating individual dynamics for predation movements of anelosimus studiosus

 1 Department of Mathematics & Statistics, East Tennessee State University, Johnson City, TN, 37614, United States, United States, United States 2 Department of Biological Sciences, East Tennessee State University, Johnson City, TN, 37614, United States, United States, United States

Received  July 2014 Revised  December 2014 Published  February 2015

In this paper, we discuss methods for developing a stochastic model which incorporates behavior differences in the predation movements of Anelosimus studiosus (a subsocial spider). Stochastic models for animal movement and, in particular, spider predation movement have been developed previously; however, this paper focuses on the development and implementation of the necessary mathematical and statistical methods required to expand such a model in order to capture a variety of distinct behaviors. A least squares optimization algorithm is used for parameter estimation to fit a single stochastic model to an individual spider during predation resulting in unique parameter values for each spider. Similarities and variations between parameter values across the spiders are analyzed and used to estimate probability distributions for the variable parameter values. An aggregate stochastic model is then created which incorporates the individual dynamics. The comparison between the optimal individual models to the aggregate model indicate the methodology and algorithm developed in this paper are appropriate for simulating a range of individualistic behaviors.
Citation: Alex John Quijano, Michele L. Joyner, Edith Seier, Nathaniel Hancock, Michael Largent, Thomas C. Jones. An aggregate stochastic model incorporating individual dynamics for predation movements of anelosimus studiosus. Mathematical Biosciences & Engineering, 2015, 12 (3) : 585-607. doi: 10.3934/mbe.2015.12.585
##### References:
 [1] H. Banks, S. Hu and W. Clayton, Modeling and Inverse Problems in the Presence of Uncertainty, CRC Press, Boca Raton, 2014. [2] D. R. Billinger, H. K. Preisler, A. A. Ager, J. G. Kie and B. S. Stewart, Modelling Movements of Free-Ranging Animals, Technical Report 610, Department of Statistics, University of California, Berkeley, 2001. [3] M. Davidian and D. M. Giltinan, Nonlinear models for repeated measurement data: An overview and update, Journal of Agricultural, Biological, and Environmental Statistics, 8 (2003), 387-419. doi: 10.1198/1085711032697. [4] B. Douglas, Tracker: Video Analysis and Modeling Tool, Tracker version 4.80, Copyright(c), 2013, URL http://www.cabrillo.edu/~dbrown/tracker. [5] R. F. Foelix, The Biology of Spiders, 3rd edition, Oxford University Press, 2011. [6] L. Grinstead, J. N. Pruitt, V. Settepani and T. Bilde, Individual personalities shape task differentiation in a social spider, Proceedings of the Royal Society B, 280 (2013). doi: 10.1098/rspb.2013.1407. [7] D. Halliday and R. Resnick, Fundamentals of Physics, John Wiley & Sons, Inc., New York, 1988. [8] P. Hoel and R. Jessen, Basic Statistics for Business and Economics, John Wiley & Sons, Inc., New York, 1971. [9] T. C. Jones and P. G. Parker, Costs and benefits of foraging associated with delayed dispersal in the spider anelosimus studiosus (araneae: Theridiidae), Journal of Arachnology, 28 (2000), 61-69. [10] T. C. Jones and P. G. Parker, Delayed dispersal benefits both mother and offspring in the cooperative spider anelosimus studiosus (araneae: Theridiidae), Behavioral Ecology, 13 (2002), 142-148. doi: 10.1093/beheco/13.1.142. [11] M. Joyner, C. Ross, C. Watts and T. Jones, A stochastic simulation model for anelosimus studiosus during prey capture: A case study for determination of optimal spacing, Mathematical Biosciences and Engineering, 11 (2014), 1411-1429. doi: 10.3934/mbe.2014.11.1411. [12] R. Larson and D. Falvo, Elementary Linear Algebra, 6th edition, Brooks/Cole, Belmont CA, 2010. [13] S. A. Naftilan, Transmission of vibrations in funnel and sheet spider webs, Biological Macromolecules, 24 (1999), 289-293. doi: 10.1016/S0141-8130(98)00092-0. [14] J. N. Pruitt, S. E. Riechert and T. C. Jones, Behavioural syndromes and their fitness consequences in a socially polymorphic spider, anelosimus studiosus, Animal Behaviour, 76 (2008), 871-879. doi: 10.1016/j.anbehav.2008.05.009. [15] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2013, URL http://www.R-project.org/. [16] P. E. Smouse, S. Focardi, P. R. Moorcroft, J. G. Kie, J. D. Forester and J. M. Morales, Stochastic modelling of animal movement, Phi.l Trans.R. Soc. B., 365 (2010), 2201-2211. doi: 10.1098/rstb.2010.0078.

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##### References:
 [1] H. Banks, S. Hu and W. Clayton, Modeling and Inverse Problems in the Presence of Uncertainty, CRC Press, Boca Raton, 2014. [2] D. R. Billinger, H. K. Preisler, A. A. Ager, J. G. Kie and B. S. Stewart, Modelling Movements of Free-Ranging Animals, Technical Report 610, Department of Statistics, University of California, Berkeley, 2001. [3] M. Davidian and D. M. Giltinan, Nonlinear models for repeated measurement data: An overview and update, Journal of Agricultural, Biological, and Environmental Statistics, 8 (2003), 387-419. doi: 10.1198/1085711032697. [4] B. Douglas, Tracker: Video Analysis and Modeling Tool, Tracker version 4.80, Copyright(c), 2013, URL http://www.cabrillo.edu/~dbrown/tracker. [5] R. F. Foelix, The Biology of Spiders, 3rd edition, Oxford University Press, 2011. [6] L. Grinstead, J. N. Pruitt, V. Settepani and T. Bilde, Individual personalities shape task differentiation in a social spider, Proceedings of the Royal Society B, 280 (2013). doi: 10.1098/rspb.2013.1407. [7] D. Halliday and R. Resnick, Fundamentals of Physics, John Wiley & Sons, Inc., New York, 1988. [8] P. Hoel and R. Jessen, Basic Statistics for Business and Economics, John Wiley & Sons, Inc., New York, 1971. [9] T. C. Jones and P. G. Parker, Costs and benefits of foraging associated with delayed dispersal in the spider anelosimus studiosus (araneae: Theridiidae), Journal of Arachnology, 28 (2000), 61-69. [10] T. C. Jones and P. G. Parker, Delayed dispersal benefits both mother and offspring in the cooperative spider anelosimus studiosus (araneae: Theridiidae), Behavioral Ecology, 13 (2002), 142-148. doi: 10.1093/beheco/13.1.142. [11] M. Joyner, C. Ross, C. Watts and T. Jones, A stochastic simulation model for anelosimus studiosus during prey capture: A case study for determination of optimal spacing, Mathematical Biosciences and Engineering, 11 (2014), 1411-1429. doi: 10.3934/mbe.2014.11.1411. [12] R. Larson and D. Falvo, Elementary Linear Algebra, 6th edition, Brooks/Cole, Belmont CA, 2010. [13] S. A. Naftilan, Transmission of vibrations in funnel and sheet spider webs, Biological Macromolecules, 24 (1999), 289-293. doi: 10.1016/S0141-8130(98)00092-0. [14] J. N. Pruitt, S. E. Riechert and T. C. Jones, Behavioural syndromes and their fitness consequences in a socially polymorphic spider, anelosimus studiosus, Animal Behaviour, 76 (2008), 871-879. doi: 10.1016/j.anbehav.2008.05.009. [15] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2013, URL http://www.R-project.org/. [16] P. E. Smouse, S. Focardi, P. R. Moorcroft, J. G. Kie, J. D. Forester and J. M. Morales, Stochastic modelling of animal movement, Phi.l Trans.R. Soc. B., 365 (2010), 2201-2211. doi: 10.1098/rstb.2010.0078.
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