# American Institute of Mathematical Sciences

2014, 11(6): 1411-1429. doi: 10.3934/mbe.2014.11.1411

## A stochastic simulation model for Anelosimus studiosus during prey capture: A case study for determination of optimal spacing

 1 Department of Mathematics & Statistics and Institute for Quantitative Biology, East Tennessee State University, Johnson City, TN, 37659 2 Department of Mathematics & Statistics, East Tennessee State University, Johnson City, TN, 37659, United States 3 Department of Biological Sciences, East Tennessee State University, Johnson City, TN, 37659, United States, United States

Received  November 2013 Revised  August 2014 Published  September 2014

In this paper, we develop a stochastic differential equation model to simulate the movement of a social/subsocial spider species, Anelosimus studiosus, during prey capture using experimental data collected in a structured environment. In a subsocial species, females and their maturing offspring share a web and cooperate in web maintenance and prey capture. Furthermore, observations indicate these colonies change their positioning throughout the day, clustered during certain times of the day while spaced out at other times. One key question was whether or not the spiders spaced out optimally'' to cooperate in prey capture. In this paper, we first show the derivation of the model where experimental data is used to determine key parameters within the model. We then use this model to test the success of prey capture under a variety of different spatial configurations for varying colony sizes to determine the best spatial configuration for prey capture.
Citation: Michele L. Joyner, Chelsea R. Ross, Colton Watts, Thomas C. Jones. A stochastic simulation model for Anelosimus studiosus during prey capture: A case study for determination of optimal spacing. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1411-1429. doi: 10.3934/mbe.2014.11.1411
##### References:
 [1] L. Aviles, The Evolution of Social Behavior in Insects and Arachnids, ch. Causes and consequences of cooperation and permanent-sociality in spiders, Cambridge Press, New York, 1997, 477-498. [2] D. R. Billinger, H. K. Preisler, A. A. Ager, J. G. Kie and B. S. Stewart, Modelling Movements of Free-Ranging Animals, Tech. Report 610, Department of Statistics, University of California, Berkeley, 2001. [3] D. R. Brillinger, H. K. Preisler, A. A. Ager and J. G. Kie, An exploratory data analysis (eda) of the paths of moving animals, Journal of Statistical Planning and Inference, 122 (2004), 43-63. doi: 10.1016/j.jspi.2003.06.016. [4] V. Brach, Anelosimus studiosus (araneae: Theridiidae) and the evolution of quasisociality in theridiid spiders, Evolution, 31 (1977), 154-161. doi: 10.2307/2407553. [5] D. R. Brillinger, A particle migrating randomly on a sphere, Journal of Theoretical Probability, 10 (1997), 429-443. doi: 10.1023/A:1022869817770. [6] D. R. Brillinger and B. S. Stewart, Elephant-seal movements: Modelling migrations, The Canadian Journal of Statistics, 26 (1998), 431-443. doi: 10.2307/3315767. [7] R. Furey, Two cooperatively social populations of the theridiid spider Anelosimus studiosus in a temperate region, Animal Behavior, 55 (1998), 727-735. [8] 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. [9] F. Heppner and U. Grenander, Ubiquity of Chaos, ch. A stochastic nonlinear model for coordinated bird flocks, AAAS Publications, Washington, DC, 1990, 233-238. [10] D. Halliday and R. Resnick, Fundamentals of Physics, John Wiley & Sons, Inc., New York, 1988. doi: 10.1063/1.3070817. [11] T. Jones, S. Riechert, S. Dalrymple and P. Parker, Fostering model explains variation in levels of sociality in a spider system, Animal Behavior, 73 (2007), 195-204. doi: 10.1016/j.anbehav.2006.06.006. [12] D. G. Kendall, Pole-seeking brownian motion and bird navigation, Journal of the Royal Statistical Society Series B, 36 (1974), 365-417. [13] P. M. Kareiva and N. Shigesada, Analyzing insect movement as a correlated random walk, Oecologia, 56 (1983), 234-238. doi: 10.1007/BF00379695. [14] MATLAB, Version 8.1.0.604 (r2013a), The MathWorks Inc., Natick, Massachusetts, 2013. [15] Inc Minitab, Minitab, Minitab, 2013. [16] 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. [17] K. B. Newman, State-space modeling of animal movement and mortality with application to salmon, Biometrics, 54 (1998), 1290-1314. doi: 10.2307/2533659. [18] OSP, Tracker Video Analysis and Modeling Too,, 2013., (). [19] H. K. Preisler, A. A. Ager, B. K. Johnson and J. G. Kie, Modeling animal movements using stochastic differential equations, Environmetrics, 15 (2004), 643-657. doi: 10.1002/env.636. [20] H. R. Pulliam and T. Caraco, Behavioural Ecology, an Evolutionary Approach, ch. Living in Groups: Is there Optimal Group Size?, Sinauer, Siunderland, 1984, 122-147. doi: 10.1016/0003-3472(79)90082-4. [21] C. Ross, Ontongeny and Diel Rhythm in Spacing Within a Subsocial web of Anelosimus Studiosus (Araneael Therididdae), Honor's Thesis, East Tennessee State University, May 2013. [22] L. S. Rayor and G. W. Uetz, Trade-offs in foraging success and predation risk with spatial position in colonial spiders, Behavioral Ecology Sociobiology, 27 (1990), 77-85. doi: 10.1007/BF00168449. [23] 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. [24] G. W. Uetz and C. S. Hieber, Evolution of social behaviour in insects and arachnids, ch. Colonial Web-building Spiders: Balancing the Costs and Benefits of Group Living, Cambridge Press, Cambridge, 1997, 458-475.

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##### References:
 [1] L. Aviles, The Evolution of Social Behavior in Insects and Arachnids, ch. Causes and consequences of cooperation and permanent-sociality in spiders, Cambridge Press, New York, 1997, 477-498. [2] D. R. Billinger, H. K. Preisler, A. A. Ager, J. G. Kie and B. S. Stewart, Modelling Movements of Free-Ranging Animals, Tech. Report 610, Department of Statistics, University of California, Berkeley, 2001. [3] D. R. Brillinger, H. K. Preisler, A. A. Ager and J. G. Kie, An exploratory data analysis (eda) of the paths of moving animals, Journal of Statistical Planning and Inference, 122 (2004), 43-63. doi: 10.1016/j.jspi.2003.06.016. [4] V. Brach, Anelosimus studiosus (araneae: Theridiidae) and the evolution of quasisociality in theridiid spiders, Evolution, 31 (1977), 154-161. doi: 10.2307/2407553. [5] D. R. Brillinger, A particle migrating randomly on a sphere, Journal of Theoretical Probability, 10 (1997), 429-443. doi: 10.1023/A:1022869817770. [6] D. R. Brillinger and B. S. Stewart, Elephant-seal movements: Modelling migrations, The Canadian Journal of Statistics, 26 (1998), 431-443. doi: 10.2307/3315767. [7] R. Furey, Two cooperatively social populations of the theridiid spider Anelosimus studiosus in a temperate region, Animal Behavior, 55 (1998), 727-735. [8] 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. [9] F. Heppner and U. Grenander, Ubiquity of Chaos, ch. A stochastic nonlinear model for coordinated bird flocks, AAAS Publications, Washington, DC, 1990, 233-238. [10] D. Halliday and R. Resnick, Fundamentals of Physics, John Wiley & Sons, Inc., New York, 1988. doi: 10.1063/1.3070817. [11] T. Jones, S. Riechert, S. Dalrymple and P. Parker, Fostering model explains variation in levels of sociality in a spider system, Animal Behavior, 73 (2007), 195-204. doi: 10.1016/j.anbehav.2006.06.006. [12] D. G. Kendall, Pole-seeking brownian motion and bird navigation, Journal of the Royal Statistical Society Series B, 36 (1974), 365-417. [13] P. M. Kareiva and N. Shigesada, Analyzing insect movement as a correlated random walk, Oecologia, 56 (1983), 234-238. doi: 10.1007/BF00379695. [14] MATLAB, Version 8.1.0.604 (r2013a), The MathWorks Inc., Natick, Massachusetts, 2013. [15] Inc Minitab, Minitab, Minitab, 2013. [16] 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. [17] K. B. Newman, State-space modeling of animal movement and mortality with application to salmon, Biometrics, 54 (1998), 1290-1314. doi: 10.2307/2533659. [18] OSP, Tracker Video Analysis and Modeling Too,, 2013., (). [19] H. K. Preisler, A. A. Ager, B. K. Johnson and J. G. Kie, Modeling animal movements using stochastic differential equations, Environmetrics, 15 (2004), 643-657. doi: 10.1002/env.636. [20] H. R. Pulliam and T. Caraco, Behavioural Ecology, an Evolutionary Approach, ch. Living in Groups: Is there Optimal Group Size?, Sinauer, Siunderland, 1984, 122-147. doi: 10.1016/0003-3472(79)90082-4. [21] C. Ross, Ontongeny and Diel Rhythm in Spacing Within a Subsocial web of Anelosimus Studiosus (Araneael Therididdae), Honor's Thesis, East Tennessee State University, May 2013. [22] L. S. Rayor and G. W. Uetz, Trade-offs in foraging success and predation risk with spatial position in colonial spiders, Behavioral Ecology Sociobiology, 27 (1990), 77-85. doi: 10.1007/BF00168449. [23] 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. [24] G. W. Uetz and C. S. Hieber, Evolution of social behaviour in insects and arachnids, ch. Colonial Web-building Spiders: Balancing the Costs and Benefits of Group Living, Cambridge Press, Cambridge, 1997, 458-475.
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