Abstract

Space plays an important role in the dynamics of collective adaptive systems (CAS). There are choices between representations to be made when we model these systems with space included explicitly, rather than being abstracted away. Since CAS often involve a large number of agents or components, we focus on scalable modelling and analysis of these models, which may involve approximation techniques. Discrete and continuous space are considered, for both models of individuals and models of populations. The aim of this tutorial is to provide an overview that supports decisions in modelling systems that involve space.

Notes

Acknowledgements

This work is supported by the EU project QUANTICOL, 600708. The author thanks Jane Hillston and Mieke Massink for their useful comments.

References

  1. 1.
    Auger, P., Poggiale, J., Sánchez, E.: A review on spatial aggregation methods involving several time scales. Ecol. Complex. 10, 12–25 (2012)CrossRefGoogle Scholar
  2. 2.
    Baccelli, F., Błaszczyszyn, B.: Stochastic Geometry and Wireless Networks: Volume I and II. NOW Publishers, Hanover (2009)MATHGoogle Scholar
  3. 3.
    Baddeley, A., Bárány, I., Schneider, R.: Spatial point processes and their applications. Stochastic Geometry. Lecture Notes in Mathematics, vol. 1892, pp. 1–75. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Baier, C., Katoen, J.P., Hermanns, H., Wolf, V.: Comparative branching-time semantics for Markov chains. Inf. Comput. 200, 149–214 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Berec, L.: Techniques of spatially explicit individual-based models: construction, simulation, and mean-field analysis. Ecol. Model. 150, 55–81 (2002)CrossRefGoogle Scholar
  6. 6.
    Bittig, A., Haack, F., Maus, C., Uhrmacher, A.: Adapting rule-based model descriptions for simulating in continuous and hybrid space. In: Proceedings of CMSB 2011, pp. 161–170. ACM (2011)Google Scholar
  7. 7.
    Bittig, A., Uhrmacher, A.: Spatial modeling in cell biology at multiple levels. In: Winter Simulation Conference (WSC 2010), pp. 608–619. IEEE (2010)Google Scholar
  8. 8.
    Bolker, B., Pacala, S.: Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor. Popul. Biol. 52, 179–197 (1997)CrossRefMATHGoogle Scholar
  9. 9.
    Bortolussi, L., Gast, N.: Mean-field limits beyond ordinary differential equations. In: Bernardo, M., De Nicola, R., Hillston, J. (eds.) SFM 2016. LNCS, vol. 9700, pp. 61–82. Springer, Switzerland (2016)Google Scholar
  10. 10.
    Bortolussi, L., Hillston, J., Latella, D., Massink, M.: Continuous approximation of collective systems behaviour: a tutorial. Perform. Eval. 70, 317–349 (2013)CrossRefGoogle Scholar
  11. 11.
    Bortolussi, L., Latella, D., Massink, M.: Stochastic process algebra and stability analysis of collective systems. In: De Nicola, R., Julien, C. (eds.) COORDINATION 2013. LNCS, vol. 7890, pp. 1–15. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  12. 12.
    Bortolussi, L., Policriti, A.: Hybrid dynamics of stochastic programs. Theor. Comput. Sci. 411, 2052–2077 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Brauer, F.: Compartmental models in epidemiology. In: Allen, L., Brauer, F., van den Driessche, P., Wu, J. (eds.) Mathematical Epidemiology, pp. 19–80. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Camp, T., Boleng, J., Davies, V.: A survey of mobility models for ad hoc network research. Wirel. Commun. Mob. Comput. 2, 483–502 (2002)CrossRefGoogle Scholar
  15. 15.
    Cerotti, D., Gribaudo, M., Bobbio, A.: Markovian agents models for wireless sensor networks deployed in environmental protection. Reliab. Eng. Syst. Saf. 130, 149–158 (2014)CrossRefGoogle Scholar
  16. 16.
    Cerotti, D., Gribaudo, M., Bobbio, A., Calafate, C.T., Manzoni, P.: A markovian agent model for fire propagation in outdoor environments. In: Aldini, A., Bernardo, M., Bononi, L., Cortellessa, V. (eds.) EPEW 2010. LNCS, vol. 6342, pp. 131–146. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  17. 17.
    Chaintreau, A., Le Boudec, J.Y., Ristanovic, N.: The age of gossip: spatial mean field regime. In: Proceedings of SIGMETRICS/Performance 2009, pp. 109–120. ACM (2009)Google Scholar
  18. 18.
    Chesson, P.: Scale transition theory: its aims, motivations and predictions. Ecol. Complex. 10, 52–68 (2012)CrossRefGoogle Scholar
  19. 19.
    Ciancia, V., Latella, D., Loreti, M., Massink, M.: Spatial logic and spatial model checking for closure spaces. In: Bernardo, M., De Nicola, R., Hillston, J. (eds.) SFM 2016. LNCS, vol. 9700, pp. 156–201. Springer, Switzerland (2016)Google Scholar
  20. 20.
    Codling, E., Plank, M., Benhamou, S.: Random walk models in biology. J. Roy. Soc. Interface 5, 813–834 (2008)CrossRefGoogle Scholar
  21. 21.
    Darling, R., Norris, J.: Differential equation approximations for Markov chains. Probab. Surv. 5, 37–79 (2008)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Davis, M.: Markov Models and Optimization. Chapman & Hall, Boca Raton (1993)CrossRefMATHGoogle Scholar
  23. 23.
    De Masi, A., Presutti, E.: Mathematical Methods for Hydrodynamic Limits. Lecture Notes in Mathematics. Springer, Berlin (1991)CrossRefMATHGoogle Scholar
  24. 24.
    Desharnais, J., Panangaden, P.: Continuous stochastic logic characterizes bisimulation of continuous-time Markov processes. J. Logic Algebraic Program. 56, 99–115 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Diggle, P.: Spatio-temporal point processes: methods and applications. Working paper, Department of Biostatistics, Johns Hopkins University (2005)Google Scholar
  26. 26.
    Diggle, P.: Spatio-temporal point processes, partial likelihood, foot and mouth disease. Stat. Methods Med. Res. 15, 325–336 (2006)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    van den Driessche, P.: Spatial structure: patch models. In: Allen, L., Brauer, F., van den Driessche, P., Wu, J. (eds.) Mathematical Epidemiology, pp. 179–190. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  28. 28.
    Durrett, R., Levin, S.: The importance of being discrete (and spatial). Theor. Popul. Biol. 46, 363–394 (1994)CrossRefMATHGoogle Scholar
  29. 29.
    Durrett, R., Levin, S.: Stochastic spatial models: a user’s guide to ecological applications. Philos. Trans. Roy. Soc. B: Biol. Sci. 343, 329–350 (1994)CrossRefGoogle Scholar
  30. 30.
    Durrett, R., Neuhauser, C.: Particle systems and reaction-diffusion equations. The Ann. Probab. 22, 289–333 (1994)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Ellner, S.: Pair approximation for lattice models with multiple interaction scales. J. Theor. Biol. 210, 435–447 (2001)CrossRefGoogle Scholar
  32. 32.
    Engblom, S.: Computing the moments of high dimensional solutions of the master equation. Appl. Math. Comput. 180, 498–515 (2006)MathSciNetMATHGoogle Scholar
  33. 33.
    Erban, R., Chapman, J., Maini, P.: A practical guide to stochastic simulations of reaction-diffusion processes (2007). arXiv preprint arXiv:0704.1908
  34. 34.
    Fange, D., Berg, O., Sjöberg, P., Elf, J.: Stochastic reaction-diffusion kinetics in the microscopic limit. Proc. Nat. Acad. Sci. 107, 19820–19825 (2010)CrossRefMATHGoogle Scholar
  35. 35.
    Feng, C.: Patch-based hybrid modelling of spatially distributed systems by using stochastic HYPE - ZebraNet as an example. In: Proceedings of QApPL 2014 (2014)Google Scholar
  36. 36.
    Feng, C., Hillston, J., Galpin, V.: Automatic moment-closure approximation of spatially distributed collective adaptive systems. ACM TOMACS 26, 26:1–26:22 (2016)CrossRefGoogle Scholar
  37. 37.
    Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 1972 (1971)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Fishman, G.: Discrete-Event Simulation. Springer, New York (2001)CrossRefMATHGoogle Scholar
  39. 39.
    Fricker, C., Gast, N.: Incentives and regulations in bike-sharing systems with stations of finite capacity (2012). arXiv preprint arXiv:1201.1178
  40. 40.
    Fricker, C., Gast, N., Mohamed, H.: Mean field analysis for inhomogeneous bike sharing systems. DMTCS Proc. 01, 365–376 (2012)MathSciNetMATHGoogle Scholar
  41. 41.
    Galpin, V., Feng, C., Hillston, J., Massink, M., Tribastone, M., Tschaikowski, M.: Review of time-based techniques for modelling space. Technical report TR-QC-05-2014, QUANTICOL (2014)Google Scholar
  42. 42.
    Gear, C.W.: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice Hall, Upper Saddle River (1971)MATHGoogle Scholar
  43. 43.
    Gillespie, D.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340–2361 (1977)CrossRefGoogle Scholar
  44. 44.
    Gillespie, D.: Stochastic simulation of chemical kinetics. Ann. Rev. Phys. Chem. 58, 35–55 (2007)CrossRefGoogle Scholar
  45. 45.
    Gorban, A., Radulescu, O., Zinovyev, A.: Asymptotology of chemical reaction networks. Chem. Eng. Sci. 65, 2310–2324 (2010)CrossRefGoogle Scholar
  46. 46.
    Holmes, E., Lewis, M., Banks, J., Veit, R.: Partial differential equations in ecology: spatial interactions and population dynamics. Ecology 75, 17–29 (1994)CrossRefGoogle Scholar
  47. 47.
    Horton, G., Kulkarni, V., Nicol, D., Trivedi, K.: Fluid stochastic Petri nets: Theory, applications, and solution techniques. Eur. J. Oper. Res. 105, 184–201 (1998)CrossRefMATHGoogle Scholar
  48. 48.
    Hu, H., Myers, S., Colizza, V., Vespignani, A.: WiFi networks and malware epidemiology. Proc. Nat. Acad. Sci. 106, 1318–1323 (2009)CrossRefGoogle Scholar
  49. 49.
    Ilachinski, A.: Cellular Automata: A Discrete Universe. World Scientific, Singapore (2001)CrossRefMATHGoogle Scholar
  50. 50.
    Isham, V.: An introduction to spatial point processes and Markov random fields. Int. Stat. Rev./Rev. Int. Stat. 41, 21–43 (1981)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Kaneko, K.: Diversity, stability, and metadynamics: remarks from coupled map studies. In: Bascompte, J., Solé, R. (eds.) Modeling Spatiotemporal Dynamics in Ecology, pp. 27–45. Springer, New York (1998)Google Scholar
  52. 52.
    Kathiravelu, T., Pears, A.: Reproducing opportunistic connectivity traces using connectivity models. In: 2007 ACM CoNEXT Conference, p. 34. ACM (2007)Google Scholar
  53. 53.
    Kemeny, J., Snell, J.: Finite Markov Chains. Springer, New York (1976)MATHGoogle Scholar
  54. 54.
    Kendall, D.: Mathematical models of the spread of infection. In: Mathematics and Computer Science in Biology and Medicine, pp. 213–225. Medical Research Council London (1965)Google Scholar
  55. 55.
    Kephart, J., White, S.: Directed-graph epidemiological models of computer viruses. In: Proceedings of the IEEE Computer Society Symposium on Research in Security and Privacy, pp. 343–359. IEEE (1991)Google Scholar
  56. 56.
    Klann, M., Koeppl, H.: Spatial simulations in systems biology: from molecules to cells. Int. J. Mol. Sci. 13, 7798–7827 (2012)CrossRefGoogle Scholar
  57. 57.
    Klein, D., Hespanha, J., Madhow, U.: A reaction-diffusion model for epidemic routing in sparsely connected MANETs. In: Proceedings of INFOCOM 2010, pp. 1–9. IEEE (2010)Google Scholar
  58. 58.
    Kurtz, T.: Approximation Popul. Process. SIAM, Philadelphia (1981)CrossRefGoogle Scholar
  59. 59.
    Levin, S., Durrett, R.: From individuals to epidemics. Philos. Trans. Roy. Soc. Lond. Ser. B: Biol. Sci. 351, 1615–1621 (1996)CrossRefGoogle Scholar
  60. 60.
    Lichtenegger, K., Schappacher, W.: A carbon-cycle-based stochastic cellular automata climate model. Int. J. Mod. Phys. C 22, 607–621 (2011)CrossRefGoogle Scholar
  61. 61.
    Marion, G., Mao, X., Renshaw, E., Liu, J.: Spatial heterogeneity and the stability of reaction states in autocatalysis. Phys. Rev. E 66, 051915 (2002)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Marion, G., Swain, D., Hutchings, M.: Understanding foraging behaviour in spatially heterogeneous environments. J. Theor. Biol. 232, 127–142 (2005)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Massink, M., Brambilla, M., Latella, D., Dorigo, M., Birattari, M.: On the use of Bio-PEPA for modelling and analysing collective behaviors in swarm intelligence. Swarm Intell. 7, 201–228 (2013)CrossRefGoogle Scholar
  64. 64.
    Massink, M., Latella, D., Bracciali, A., Harrison, M., Hillston, J.: Scalable context-dependent analysis of emergency egress models. Formal Aspects Comput. 24, 267–302 (2012)CrossRefGoogle Scholar
  65. 65.
    Matis, J., Kiffe, T.: Effects of immigration on some stochastic logistic models: a cumulant truncation analysis. Theor. Popul. Biol. 56, 139–161 (1999)CrossRefMATHGoogle Scholar
  66. 66.
    Matsuda, H., Ogita, N., Sasaki, A., Satō, K.: Statistical mechanics of population: the lattice Lotka-Volterra model. Prog. Theor. Phys. 88, 1035–1049 (1992)CrossRefGoogle Scholar
  67. 67.
    McCaig, C., Norman, R., Shankland, C.: From individuals to populations: a mean field semantics for process algebra. Theor. Comput. Sci. 412, 1557–1580 (2011)MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    Morozov, A., Poggiale, J.C.: From spatially explicit ecological models to mean-field dynamics: the state of the art and perspectives. Ecol. Complex. 10, 1–11 (2012)CrossRefGoogle Scholar
  69. 69.
    Musolesi, M., Mascolo, C.: Mobility models for systems evaluation. In: Garbinato, B., Miranda, H., Rodrigues, L. (eds.) Middleware Netw. Eccentric Mob. Appl., pp. 43–62. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  70. 70.
    Norris, J.: Markov Chains. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
  71. 71.
    Okubo, A., Levin, S.A.: Diffusion and Ecological Problems: Modern Perspectives. Springer, New York (2001)CrossRefMATHGoogle Scholar
  72. 72.
    Othmer, H., Scriven, L.: Instability and dynamic pattern in cellular networks. J. Theor. Biol. 32, 507–537 (1971)CrossRefGoogle Scholar
  73. 73.
    Panangaden, P.: Labelled Markov Processes. Imperial College Press, London (2009)CrossRefMATHGoogle Scholar
  74. 74.
    Pascual, M., Roy, M., Laneri, K.: Simple models for complex systems: exploiting the relationship between local and global densities. Theor. Ecol. 4, 211–222 (2011)CrossRefGoogle Scholar
  75. 75.
    Riley, S.: Large-scale spatial-transmission models of infectious disease. Science 316, 1298–1301 (2007)CrossRefGoogle Scholar
  76. 76.
    Rowe, J.E., Gomez, R.: El Botellón: modeling the movement of crowds in a city. Complex Syst. 14, 363–370 (2003)Google Scholar
  77. 77.
    Ruan, S.: Spatial-temporal dynamics in nonlocal epidemiological models. In: Takeuchi, Y., Iwasa, Y., Sato, K. (eds.) Mathematics for Life Science and Medicine. Biological and Medical Physics, Biomedical Engineering, pp. 97–122. Springer, Heidelberg (2007)Google Scholar
  78. 78.
    Schoenberg, F., Brillinger, D., Guttorp, P.: Point processes, spatial-temporal. In: El-Shaarawi, A., Piegorsch, W. (eds.) Encyclopedia of Environmetrics, pp. 1573–1577. Wiley Online Library, New York (2002)Google Scholar
  79. 79.
    Segel, L., Slemrod, M.: The quasi-steady-state assumption: a case study in perturbation. SIAM Rev. 31, 446–477 (1989)MathSciNetCrossRefMATHGoogle Scholar
  80. 80.
    Simon, H.A., Ando, A.: Aggregation of variables in dynamic systems. Econometrica 29, 111–138 (1961)CrossRefMATHGoogle Scholar
  81. 81.
    Slepchenko, B., Schaff, J., Macara, I., Loew, L.: Quantitative cell biology with the virtual cell. Trends Cell Biol. 13, 570–576 (2003)CrossRefGoogle Scholar
  82. 82.
    Swift, J., Hohenberg, P.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319 (1977)CrossRefGoogle Scholar
  83. 83.
    Takahashi, K., Arjunan, S., Tomita, M.: Space in systems biology of signaling pathways-towards intracellular molecular crowding in silico. FEBS Lett. 579, 1783–1788 (2005)CrossRefGoogle Scholar
  84. 84.
    Thomas, J.W.: Numerical Partial Differential Equations: Finite Difference Methods. Springer, New York (1995)CrossRefMATHGoogle Scholar
  85. 85.
    Tribastone, M., Gilmore, S., Hillston, J.: Scalable differential analysis of process algebra models. IEEE Trans. Softw. Eng. 38, 205–219 (2012)CrossRefGoogle Scholar
  86. 86.
    Tschaikowski, M., Tribastone, M.: A Partial-differential Approximation for Spatial Stochastic Process Algebra. In: Proceedings of VALUETOOLS 2014 (2014)Google Scholar
  87. 87.
    Tschaikowski, M., Tribastone, M.: Exact fluid lumpability in Markovian process algebra. Theor. Comput. Sci. 538, 140–166 (2014)MathSciNetCrossRefMATHGoogle Scholar
  88. 88.
    Tschaikowski, M., Tribastone, M.: Spatial fluid limits for stochastic mobile networks. Performance Evaluation (2015), under minor revisionGoogle Scholar
  89. 89.
    Turing, A.: The chemical basis of morphogenesis. Philos. Trans. Roy. Soc. (Part B) 237, 37–72 (1953)CrossRefGoogle Scholar
  90. 90.
    Vandin, A., Tribastone, M.: Quantitative abstractions for collective adaptive systems. In: Bernardo, M., De Nicola, R., Hillston, J. (eds.) SFM 2016. LNCS, vol. 9700, pp. 202–232. Springer, Switzerland (2016)Google Scholar
  91. 91.
    Webb, S., Keeling, M., Boots, M.: Host-parasite interactions between the local and the mean-field: how and when does spatial population structure matter? J. Theor. Biol. 249, 140–152 (2007)MathSciNetCrossRefGoogle Scholar
  92. 92.
    Wiggins, S.: Normally Hyperbolic Invariant Manifolds in Dynamical Systems. Springer Science & Business Media, New York (1994)CrossRefMATHGoogle Scholar
  93. 93.
    Wu, J., Loucks, O.: From balance of nature to hierarchical patch dynamics: a paradigm shift in ecology. Q. Rev. Biol. 70, 439–466 (1995)CrossRefGoogle Scholar
  94. 94.
    Zhou, X., Ioannidis, S., Massoulié, L.: On the stability and optimality of universal swarms. ACM SIGMETRICS Perform. Eval. Rev. 39, 301–312 (2011)CrossRefGoogle Scholar
  95. 95.
    Zungeru, A., Ang, L.M., Seng, K.P.: Classical and swarm intelligence based routing protocols for wireless sensor networks: a survey and comparison. J. Netw. Comput. Appl. 35, 1508–1536 (2012)CrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Laboratory for Foundations of Computer Science, School of InformaticsUniversity of EdinburghEdinburghUK

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