The bulletin of mathematical biophysics

, Volume 15, Issue 4, pp 431–476 | Cite as

A mathematical contribution to the study of orientation of organisms

  • Clifford S. Patlak


Organisms orient themselves to a stimulus by two general methods. One method is by directed orientation (taxis); the other is by undirected locomotory reaction (kinesis). An equation, and the methods for finding the necessary parameters of this equation, is derived for the distribution of organisms within a container, with the following limitations: (1) the organisms have no accommodation, (2) they are always active, and (3) the stimulus changes slowly with position. Necessary modifications of the equation are then derived, so that the last two limitations may be eliminated. The equation cannot be solved excatly because of its complexity; hence an approximation method must be used. This method is discussed, an approximate solution is found, and a time constant for equilibrium to be established is derived. Applications tovarious experiments in the literature are then made with fairly satisfactory results. A new interpretation of the theory of klino-kinesis with accommodation is found upon application of the equations developed to experimental work. Further limitations and uses of these equations are then discussed.


Free Path Steep Gradient Avoidance Reaction Active Organism Taxic Component 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bentley, E. W. 1944. “The Biology and Behaviour of Ptinus Tectus Boie. (Coleoptera, Ptinidae), a Pest of Stored Products. V. Humidity Reactions.”Jour. Exp. Biol.,20, 152–58.Google Scholar
  2. —, D. L. Gunn, and D. W. Ewer. 1941. “The Biology and Behaviour of Ptinus Tectus Boie. (Coleoptera, Ptinidae), a Pest of Stored Products. I. The Daily Rhythm of Locomotory Activity, Especially in Relation to Light and Temperature.”Jour. Exp. Biol.,18, 182–95.Google Scholar
  3. Bursell, E. and D. W. Ewer. 1950. “On the Reactions to Humidity of Peripatopsis Moseleyi (Wood-Mason).”Jour. Exp. Biol.,26, 335–53.Google Scholar
  4. Ewer, D. W. and E. Bursell. 1950. “A Note on the Classification of Elementary Behaviour Patterns.”Behaviour,3, 40–47.Google Scholar
  5. Fraenkel, G. 1931. “Die Mechanik der Orientierung der Tiere im Raum.”Biol. Rev.,6, 36–87.Google Scholar
  6. — and D. L. Gunn. 1940.The Orientation of Animals. Oxford: Oxford University Press.Google Scholar
  7. Gunn, D. L. 1937. “The Humidity Reactions of the Wood-Louse, Porcellio Scaber (Latreille).”Jour. Exp. Biol.,14, 178–86.Google Scholar
  8. — and H. S. Hopf. 1942. “The Biology and Behaviour of Ptinus Tectus Boie. (Coleoptera, Ptinidae), a Pest of Stored Products. II. The Amount of Locomotory Activity in Relation to Experimental and to Previous Temperatures.”Jour. Exp. Biol.,18, 278–89.Google Scholar
  9. — and J. S. Kennedy. 1936. “Apparatus for Investigating the Reactions of Land Arthropods to Humidity.”Jour. Exp. Biol.,13, 450–59.Google Scholar
  10. —, J. S. Kennedy, and D. P. Pielou. 1937. “Classification of Taxes and Kineses.”Nature,140, 1064.Google Scholar
  11. — and D. P. Pielou. 1940. “The Humidity Behaviour of the Mealworm Beetle, Tenebrio Molitor L. III. The Mechanism of the Reaction.”Jour. Exp. Biol.,17, 307–16.Google Scholar
  12. — and B. M. Walshe. 1942. “The Biology and Behaviour of Ptinus Tectus Boie. (Coleoptera, Ptinidae), a Pest of Stored Products. IV. Temperature Preference.”Jour. Exp. Biol.,19, 133–40.Google Scholar
  13. Jennings, H. S. 1904.Contributions to the Study of the Behavior of Lower Organisms. Washington, D.C.: Carnegie Institute.Google Scholar
  14. — 1906.Behavior of the Lower Organisms. New York: Macmillan Co.Google Scholar
  15. Kennedy, J. S. 1937. “The Humidity Reactions of the African Migratory Locust, Locusta Migratoria Migratorioides R. and F., Gregarious Phase.”Jour. Exp. Biol.,14, 187–97.Google Scholar
  16. Lees, A. D. 1943. “On the Behavior of Wireworms of the Genus Agriotes Esch. (Coleoptera, Elasteridae). I. Reactions to Humidity.”Jour. Exp. Biol.,20, 43–53.Google Scholar
  17. — 1948. “The Sensory Physiology of the Sheep Tick Ixodes Ricinus L.”Jour. Exp. Biol.,25, 145–207.Google Scholar
  18. Mast, S. O. 1911.Light and the Behavior of Organisms. New York: John Wiley and Sons.Google Scholar
  19. — 1938. “Factors Involved in the Process of Orientation of Lower Organisms in Light.”Biol. Rev.,13, 186–224.Google Scholar
  20. Patlak, C. S. 1952. “Random Walk with Persistence and External Bias.”Bull. Math. Biophysics,15, 311–38.MathSciNetGoogle Scholar
  21. Pielou, D. P. and D. L. Gunn. 1940. “The Humidity Behaviour of the Mealworm Beetle, Tenebrio Molitor L. I. The Reaction to Differences of Humidity.”Jour. Exp. Biol.,17, 286–94.Google Scholar
  22. Rashevsky, N. 1948.Mathematical Biophysics. Rev. Ed. Chicago: University of Chicago Press.zbMATHGoogle Scholar
  23. Thorpe, W. H., A. C. Crombie, R. Hill, and J. H. Darrah. 1947. “The Behavior of Wireworms in Response to Chemical Stimulation.”Jour. Exp. Biol.,23, 234–66.Google Scholar
  24. Ullyott, P. 1936. “The Behavior of Dendrocoelum Lacteum. II. Responses in Non-Directional Gradients.”Jour. Exp. Biol.,13, 265–78.Google Scholar
  25. Waloff, N. 1941. “The Mechanisms of Humidity Reactions of Terrestrial Isopods.”Jour. Exp. Biol.,18, 115–35.Google Scholar
  26. Wigglesworth, V. B. 1941. “The Sensory Physiology of the Human Louse Pediculus Humanus Corporis de Geer (Anoplura).”Parasitology,33, 67–109.CrossRefGoogle Scholar

Copyright information

© University of Chicago 1953

Authors and Affiliations

  • Clifford S. Patlak
    • 1
  1. 1.Committee on Mathematical BiologyThe University of ChicagoChicagoUSA

Personalised recommendations