Some mathematical considerations on two-mode searching I

  • Hiromi Seno


Purpose of this paper is to consider mathematically the relation between the efficiency of two-mode searching behavior and the target’s patchy distribution. Two-mode searching includes patch-searching and target-catching. Two intuitive models are presented: Model 1 constructed by a Wiener process onR 1; Model 2 by a time-discrete Markov process onS 1, that is, on a circle. These two different models give different results depending on the characteristics of each model. We apply our results to a coevolutionary game between the searcher’s searching behavior and the target’s distribution. Compared with a simple mode searching, the superiority of two-mode searching is shown to depend seriously on the target’s distribution.

Key words

searching mathematical model Wiener process Markov process 


  1. [1]
    M.L. Cain, Random search by herbivorous insects: A simulation model. Ecology,66 (1985), 876–888.CrossRefGoogle Scholar
  2. [2]
    R.J. Cowie and J.R. Krebs, Optimal foraging in patchy environments. Population Dynamics (eds. R.M. Anderson, B.D. Tumer and L.R. Taylor), Blackwell Scientific Publications, New York, 1979, 183–205.Google Scholar
  3. [3]
    D.R. Cox, H.D. Miller, The Theory of Stochastic Processes. Chapman and Hall, London, 1972.Google Scholar
  4. [4]
    W.D. Hamilton, Geometry for the selfish herd. J. Theoret. Biol.,31 (1971), 295–311.CrossRefGoogle Scholar
  5. [5]
    Y. Iwasa, M. Higashi and N. Yamamura, Prey distribution as a factor determining the choice of optimal foraging strategy. Am. Nat.,117 (1981), 710–723.CrossRefMathSciNetGoogle Scholar
  6. [6]
    M. Kato, Mining pattern of the honeysuckle leaf-miner Phytomyza lonicerae. Res. Popul. Ecol.,26 (1984), 84–96.CrossRefGoogle Scholar
  7. [7]
    M. Kato, The adaptive significance of leaf-mining pattern as an anti-parasitoid strategy: A theoretical study. Res. Popul. Ecol.,27 (1985), 265–275.CrossRefGoogle Scholar
  8. [8]
    P. Knoppien and J. Reddingius, Predators with two modes of searching: A mathematical model. J. Theoret. Biol.,114 (1985), 273–301.CrossRefMathSciNetGoogle Scholar
  9. [9]
    B.O. Koopman, The theory of search: I. Kinematic bases. Oper. Res.,4 (1956), 324–346.CrossRefMathSciNetGoogle Scholar
  10. [10]
    G. Murdie, Simulation of the effects of predator/parasite models on prey/host spatial distribution. Statistical Ecology (eds. G.P. Patil, E.C. Pielou and W.E. Waters), Penn. State Statistical Series, Pennsilvania, 1971, 215–233.Google Scholar
  11. [11]
    G. Murdie and M.P. Hassell, Food distribution, searchign success and predator-prey models. The Mathematical Theory of The Dynamics of Biological Populations (eds. M.S. Bartlett and R.W. Hiorns), Academic Press, London, 1973, 87–101.Google Scholar
  12. [12]
    G. Nackman, A mathematical model of the functional relationship between density and spatial distribution of a population. J. Anim. Ecol.,50 (1981), 453–460.CrossRefGoogle Scholar
  13. [13]
    J.E. Paloheimo, On a theory of search. Biometrika,58 (1971), 61–75.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    J.E. Paloheimo, A stochastic theory of search: Implications for predator-prey situations. Math. Biosci.,12 (1971), 105–132.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    T. Royama, A comparative study of models for predation and parasitism. Researches on Population Ecology, Kyoto University, Supplement1, Kyoto, 1971, 1–91.CrossRefGoogle Scholar
  16. [16]
    A. Stewart-Oaten, Minimax strategies for a predator-prey game. Theoret. Population Biol.,22 (1982), 410–424.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    H. Ubukata, A model of mate searching and territorial behaviour for “Flier” type dragonflies. J. Ethol.,4 (1986), 105–112.CrossRefGoogle Scholar
  18. [18]
    J.A. Wiens, Population responses to patchy environments. Ann. Rev. Ecol. Syst.,7 (1976), 81–120.CrossRefGoogle Scholar

Copyright information

© the JJIAM Publishing Committee 1991

Authors and Affiliations

  • Hiromi Seno
    • 1
    • 2
  1. 1.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di NapoliNapoliItaly
  2. 2.Information Processing Center of Medical SciencesNippon Medical SchoolTokyoJapan

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