Journal of Mathematical Biology

, Volume 66, Issue 4–5, pp 705–717

Group defence and the predator’s functional response



We derive from first principles the functional response of the predator and the reproduction rate of the prey in the case that the prey form groups as a defence against the predator and the latter captures only single prey. We also give some examples of the resulting predator–prey population dynamics.


Predator–prey model Functional response Group defence  Group size distribution Becker–Döring equations 

Mathematics Subject Classification (2000)

92D40 92D50 


  1. Becker R, Döring W (1935) Kinetische Behandlung der Keimbildung in übersättigten Dämpfen. Ann Phys 4:719–752CrossRefGoogle Scholar
  2. Beddington J (1975) Mutual interference between parasites or predators and its effect on searching efficiency. J Animal Ecol 44:331–340CrossRefGoogle Scholar
  3. Cosner C, DeAngelis DL, Ault JS, Olson DB (1999) Effects of spatial grouping on the functional response of predators. Theor Popul Biol 56:65–75MATHCrossRefGoogle Scholar
  4. DeAngelis Goldstein DR, O’Neill R (1975) A model for trophic interaction. Ecology 56:881–892CrossRefGoogle Scholar
  5. Fryxell JM, Mosser A, Sinclair ARE, Packer G (2007) Group formation stabilizes predator–prey dynamics. Nature 449:1041–1044CrossRefGoogle Scholar
  6. Geritz SAH, Gyllenberg M (2012) A mechanistic derivation of the DeAngelis–Beddington functional response. J Theor Biol 314:106–108CrossRefGoogle Scholar
  7. Gueron S, Levin SA (1995) The dynamics of group formation. Math Biosci 128:243–264MATHCrossRefGoogle Scholar
  8. Gueron S (1998) The steady-state distributions of coagulation–fragmentation processes. J Math Biol 37:1–27MathSciNetMATHCrossRefGoogle Scholar
  9. Holling CS (1959) Some characteristics of simple types of predation and parasitism. Can Entomol 91:385–398CrossRefGoogle Scholar
  10. Jabin PE, Niethammer B (2003) On the rate of convergence to equilibrium in the Becker–Döring equations. J Differ Equ 191:518–543MathSciNetMATHCrossRefGoogle Scholar
  11. Jeschke JM, Tollrian R (2005) Effects of predator confusion on functional responses. Oikos 111:547–555CrossRefGoogle Scholar
  12. Ma Q, Johansson A, Sumpter DJT (2011) A first principles derivation of animal group size distributions. J Theor Biol 283:35–43Google Scholar
  13. Metz JAJ, Diekmann O (1986) The dynamics of physiologically structured populations, Lecture Notes in Biomathematics, vol 68. Springer, BerlinGoogle Scholar
  14. Rosenzweig ML, MacArthur RH (1963) Graphical representation and stability conditions for predator–prey interactions. Am Nat 97:209–223CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiFinland

Personalised recommendations