Advertisement

Journal of Mathematical Biology

, Volume 26, Issue 3, pp 263–298 | Cite as

Models of dispersal in biological systems

  • H. G. Othmer
  • S. R. Dunbar
  • W. Alt
Article

Abstract

In order to provide a general framework within which the dispersal of cells or organisms can be studied, we introduce two stochastic processes that model the major modes of dispersal that are observed in nature. In the first type of movement, which we call the position jump or kangaroo process, the process comprises a sequence of alternating pauses and jumps. The duration of a pause is governed by a waiting time distribution, and the direction and distance traveled during a jump is fixed by the kernel of an integral operator that governs the spatial redistribution. Under certain assumptions concerning the existence of limits as the mean step size goes to zero and the frequency of stepping goes to infinity the process is governed by a diffusion equation, but other partial differential equations may result under different assumptions. The second major type of movement leads to what we call a velocity jump process. In this case the motion consists of a sequence of “runs” separated by reorientations, during which a new velocity is chosen. We show that under certain assumptions this process leads to a damped wave equation called the telegrapher's equation. We derive explicit expressions for the mean squared displacement and other experimentally observable quantities. Several generalizations, including the incorporation of a resting time between movements, are also studied. The available data on the motion of cells and other organisms is reviewed, and it is shown how the analysis of such data within the framework provided here can be carried out.

Key words

Dispersal Cell movement Random walks Stochastic processes 

References

  1. Abramowitz, M., Stegun, I.: Handbook of mathematical functions. New York: Dover, 1965Google Scholar
  2. Alt, W.: Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9, 147–177 (1980)Google Scholar
  3. Alt, W., Eisele, T., Schaaf, R.: Chemotaxis of gametes: A diffusion approximation IMA. J. Math. Appl. Med. Biol. 2, 109–129 (1985)Google Scholar
  4. Aronson, D. G.: The role of diffusion in mathematical population biology: Skellam revisited. In: Capasso, V., Grosso, E., Paveri-Fontana, S. L. (eds.) Mathematics in biology and medicine (Lect. Notes Biomath., vol. 57, pp. 2–6). Berlin Heidelberg New York Tokyo: Springer 1985Google Scholar
  5. Berg, H.: How bacteria swim. Sci. Am. 233, 36–44 (1975)Google Scholar
  6. Berg, H.: Random walks in biology. Princeton: Princeton University Press 1983Google Scholar
  7. Berg, H. C., Brown, D. A.: Chemotaxis in Escherichia coli analysed by three dimensional tracking. Nature, 239, 500–504 (1972)Google Scholar
  8. Boyarsky, A.: A Markov chain model for human granulocyte movement. J. Math. Biol. 2, 69–78 (1975)Google Scholar
  9. Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 2–89 (1943)Google Scholar
  10. Dunbar, S., Othmer, H. G.: On a nonlinear hyperbolic equation describing transmission lines, cell movement and branching random walks. In: Othmer, H. G. (ed.) Nonlinear oscillations in biology and chemistry (Lect. Notes Biomath., vol. 66, pp. 274–289). Berlin Heidelberg New York Tokyo: Springer 1986Google Scholar
  11. Dunbar, S.: A branching random evolution and a nonlinear hyperbolic equation. To appear in SIAM J. Appld. Math. (1988)Google Scholar
  12. Dunn, G. A.: Characterizing a kinesis response: time averaged measures of cell speed and directional persistence. In: Keller, H. O., Till, G. O. (eds.) Leukocyte locomotion and chemotaxis, pp. 14–33. Basel: Birkhäuser 1983Google Scholar
  13. Einstein, A.: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Physik 17, 549–560 (1905)Google Scholar
  14. Feller, W.: An introduction to probability theory. Wiley: New York 1968Google Scholar
  15. Fürth, R.: Die Brownsche Bewegung bei Berücksichtigung einer Persistenz der Bewegungsrichtung. Z. Physik 2, 244–256 (1920)Google Scholar
  16. Gail, M. H., Boone, C. W.: The locomotion of mouse fibroblasts in tissue culture. Biophys. J. 10, 980–993 (1970)Google Scholar
  17. Goldstein, S.: On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Applied Math. VI, 129–156 (1951)Google Scholar
  18. Greenberg, E., Canale-Parola, E.: Chemotaxis in Spirocheta aurantia. J. Bacteriol. 130, 485–494 (1977)Google Scholar
  19. Gruler, H., Bültman, B. D.: Analysis of cell movement. Blood Cells 10, 61–77 (1984)Google Scholar
  20. Hall, R. L.: Amoeboid movement as a correlated walk. J. Math. Biol. 4, 327–335 (1977)Google Scholar
  21. Hall, R. L., Peterson, S. C.: Trajectories of human granulocytes. Biophys. J. 25, 365–372 (1979)Google Scholar
  22. Henderson, R., Renshaw, E.: Spatial stochastic models and computer simulation applied to the study of tree root systems. In: Barritt, M., Wishart, D. (eds.) Proceedings in Computational Statistics 4th Symposium, Edinburgh 1980, pp. 389–395. Wien: Physica 1980Google Scholar
  23. Jones, R.: Movement patterns and egg distribution in cabbage butterflies. J. An. Ecol. 46, 195–212 (1977)Google Scholar
  24. Johnson, N. L., Kotz, S.: Distributions in statistics — continuous univariate distributions, vol. 2. New York: Wiley 1970Google Scholar
  25. Kac, M.: A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math. 3, 497–509 (1974)Google Scholar
  26. Kareiva, P.: Local movement in herbivorous insects: applying a passive diffusion model to markrecapture field experiments. Oecologica 57, 322–327 (1983)Google Scholar
  27. Kareiva, P., Shigesada, N.: Analyzing insect movement as a correlated random walk. Oecologica 56, 234–238 (1983)Google Scholar
  28. Karlin, S., Taylor, H.: A first course in stochastic processes. New York: Academic Press 1975Google Scholar
  29. Keller, H. U., Zimmerman, A.: Orthokinetic and klinokinetic responses of human polymorphonuclear leukocytes. Cell Motility 5, 447–461 (1985)Google Scholar
  30. Koshland, D.: Bacterial chemotaxis as a model behavioral system. New York: Raven Press 1980Google Scholar
  31. Lackie, J. H.: Cell movement and cell behaviour. London: Allen and Unwin 1986Google Scholar
  32. Levin, S. A.; Random walk models of movement and their implications. In: Hallam, T. G., Levin, S. A. (eds.) Mathematical ecology. An introduction (Lect. Notes Biomath., vol. 17, pp. 149–155). Berlin Heidelberg New York Tokyo: Springer 1986Google Scholar
  33. Lovely, P. S., Dahlquist, F. W.: Statistical measures of bacterial motility and chemotaxis. J. Theor. Biol. 50, 477–496 (1975)Google Scholar
  34. McKean, H.: Chapman-Enskog-Hilbert expansions for a class of solutions of the telegraph equation. J. Math. Phys. 75, 1–10 (1967)Google Scholar
  35. Morse, P. M., Feshbach, H.: Methods of theoretical physics. New York: McGraw-Hill 1953Google Scholar
  36. Noble, P. B., Levine, M.: Computer-assisted analyses of cell locomotion and chemotaxis. Boca Raton: CRC Press 1986Google Scholar
  37. Nossal, R.: Stochastic aspects of biological locomotion. J. Stat. Phys. 30, 391–399 (1983)Google Scholar
  38. Nossal, R., Weiss, G. H.: A descriptive theory of cell migration on surfaces. J. Theor. Biol. 47, 103–113 (1974)Google Scholar
  39. Okubo, A.: Diffusion and ecological problems: mathematical models. New York Heidelberg Berlin: Springer 1980Google Scholar
  40. Othmer, H. G.: Interactions of reaction and diffusion in open systems. Ph.D. Thesis, Minneapolis: Univ. of Minnesota (1969)Google Scholar
  41. Othmer, H. G.: On the significance of finite propagation speeds in multicomponent reacting systems. J. Chem. Phys. 64, 460–470 (1976)Google Scholar
  42. Patlak, C. S.: Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311–338 (1953)Google Scholar
  43. Resibois, P., DeLeener, M.: Classical kinetic theory of fluids. New York: Wiley 1977Google Scholar
  44. Segel, L. A.: Mathematical models for cellular behavior. In: Levin, S. A. (ed.) Studies in mathematical biology, vol. 15, pp. 156–190. Washington: MAA 1978Google Scholar
  45. Shigesada, N.: Spatial distribution of dispersing animals. J. Math. Biol. 9, 85–96 (1980)Google Scholar
  46. Siniff, D. P., Jessen, C. R.: A simulation model of animal movement patterns. Adv. Ecol. Res. 6 185–219 (1969)Google Scholar
  47. Skellam, J. G.: The formulation and interpretation of diffusionary processes in population biology. In: Bartlett, M. S., Hiorns, R. W. (eds.) The mathematical theory of the dynamics of biological populations. New York: Academic Press 1973Google Scholar
  48. Smith, J. N. M.: The food searching behaviour of two European thrushes. I.: Behavior 48, 276–302 (1974); II.: Behavior 49, 1–61 (1974)Google Scholar
  49. Tranquillo, R., Lauffenburger, D.: Stochastic models of leukocyte chemosensory movement. J. Math. Biol. 25, 229–262 (1987)Google Scholar
  50. Widder, D.: The Laplace transform. Princeton: Princeton University Press 1946Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • H. G. Othmer
    • 1
  • S. R. Dunbar
    • 2
  • W. Alt
    • 3
  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  2. 2.Department of Mathematics and StatisticsUniversity of Nebraska-LincolnLincolnUSA
  3. 3.Abteilung Theoretische BiologieUniversität BonnBonn 1Federal Republic of Germany

Personalised recommendations