Bulletin of Mathematical Biology

, Volume 41, Issue 4, pp 543–554 | Cite as

Stochastic prey-predator relationships: A random evolution approach

  • Georges A. Bécus


The formalism and results of the theory of random evolutions are used to establish and investigate a model for two randomly interacting populations. The asymptotic stability of the expected solution is studied and contrasted to that of the associated deterministic system.


Limit Theorem Asymptotic Stability Diffusion Approximation Deterministic System Jump Time 
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  1. Abramowitz, M. and I. A. Stegun. 1965. (Eds).Handbook of Mathematical Functions. New York: Dover.Google Scholar
  2. Bartlett, M. S. 1960.Stochastic Models in Ecology and Epidemiology. London: Methuen.Google Scholar
  3. Bécus, G. A. 1979. “Stochastic Prey-Predator Relationships: A Random Differential Equation Approach.”Bull. Math. Biol,41, 91–100.MATHMathSciNetCrossRefGoogle Scholar
  4. Bharucha-Reid, A. T. 1960.Elements of the Theory of Markov Processes and their Applications. New York: McGraw-Hill.Google Scholar
  5. Coppel, W. A. 1965.Stability and Asymptotic Behavior of Differential Equations. Boston: Heath.Google Scholar
  6. Gard, T. C. and D. Kannan. 1976. “On a Stochastic Differential Equation Modeling Prey-Predator Evolution.”J. Appl. Prob.,13, 429–443.MATHMathSciNetCrossRefGoogle Scholar
  7. Goel, N. S., S. C. Maitra and E. W. Montroll. 1971. “On the Volterra and Other Nonlinear Models of Interacting Populations.”Rev. Mod. Phys. 43, 321–376.MathSciNetCrossRefGoogle Scholar
  8. —, and N. R. Dyn. 1974.Stochastic Models in Biology. New York: Academic Press.Google Scholar
  9. Gomatam, J. 1974. “A New Model of Interacting Populations—I: Two Species Systems.”Bull. Math. Biol.,36, 347–353.MATHCrossRefGoogle Scholar
  10. Griego, R. and R. Hersh. 1971. “Theory of Random Evolutions with Applications to Partial Differential Equations.”TAMS,156, 405–418.MATHMathSciNetCrossRefGoogle Scholar
  11. Hersh, R. and M. Pinsky. 1972. “Random Evolutions are Asymptotically Gaussian.”Comm. Pure Appl. Math. 25, 33–44.MATHMathSciNetGoogle Scholar
  12. Kannan, D. 1976. “On Some Markov Models of Certain Interacting Populations.”Bull. Math. Biol.,38, 723–738.MATHMathSciNetCrossRefGoogle Scholar
  13. Kurtz, T. G. 1972. “A Random Trotter Product Formula.”Proc. AMS 35, 147–154.MATHMathSciNetCrossRefGoogle Scholar
  14. — 1973. “A Limit Theorem for Perturbed Operator Semigroups with Applications to Random Evolutions.”J. Funct. Anal. 12, 55–67.MATHMathSciNetCrossRefGoogle Scholar
  15. Lions, J. L. 1976.Sur Quelques Questions d'Analyse, de Mécanique, et de Contrôle Optimal. Montréal, Canada: Presses de l'U. de Montréal.Google Scholar
  16. Prajneshu, 1976a. “A Stochastic Model for Two Interacting Species.”Stoch. Proc. Appl. 4, 271–282.MathSciNetCrossRefGoogle Scholar
  17. Prajneshu, 1976b. “Two-Species Systems in Random Environment.” To appear.Google Scholar
  18. Soong, T. T. 1973.Random Differential Equations in Science and Engineering. New York: Academic Press.Google Scholar

Copyright information

© Society of Mathematical Biology 1979

Authors and Affiliations

  • Georges A. Bécus
    • 1
  1. 1.Department of Engineering ScienceUniversity of CincinnatiCincinnatiUSA

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