Abstract
In treating the Volterra-Verhulst prey-predator system with time dependent coefficients, we ask how far this deterministic system represents or approximates the dynamics of the population evolving in a realistic environment which is stochastic in nature. We consider a stochastic system withsmall Gaussian noise type fluctuations. It is shown that the higher moments of the deviation of the deterministic system from the stochastic approach zero as the strength δ of the perturbation decays to zero. For any δ>0 and allT>0, ε>0, the sample population paths that stay within ε distance from the deterministic path during [0,T] form a collection of positive probability. In comparing the stationary distributions of the two systems, we show that the weak limits of those of the stochastic system form a subset of those of the deterministic system. This is in analogy with a result of May connected with the stability of the two systems. Plant and rodent populations possess periodic parameters andexhibit periodic behaivor. We establish theoretically this periodicity under periodicity conditions on the coefficients and perturbing random forces. We also establish a central limit property for the prey-predator system.
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Literature
Bogoljubox, N. N. and Ju. A. Mitropolskii. 1961.Asymptotic Methods in the Theory of Non-Linear Oscillation, New Delhi: Hindustan Publ. Co.
Gard, T. C. and D. Kannan. 1976. “On a Stochastic Differential Equation Modeling of Prey-Predator Evolution.”J. Appl. Prob.,13.
Gikhman, I. I. and A. V. Skorokhod. 1969.Introduction to the Theory of Random Processes. Philadelphia: W. B. Saunders.
— and —. 1972.Stochastic Differential Equations. New York: Springer-Verlag
Girsanov, I. V. 1960. “On Transforming a Certain Class of Stochastic Processes by Absolutely Continuous Substitution of Measures.Theory Prob. Appl.,5, 285–301.
Hasminskii, R. Z. (Khasminskii). 1968. “On the Principle of Averaging for Ito's Stochastic Differential Equations.”Kybernetika (Prague),4, 260–279 (Russian).
Kannan, D. 1976. “On some Markov Models of Certain Interacting Populations.”Bull. Math. Biol.,38, 723–738.
— 1976. “Random Integrodifferential Equations,” inProbabilistic Analysis and Related Topics, (Ed. A. T. Bharucha-Reid), New York: Academic Press.
Kannan, D.Volterra's Prey-Predator Population With Historical Actions, in preparation.
May, R. M. 1973.Stability and Complexity in Model Ecosystems, Princeton, N.J., Princeton Univ. Press.
— 1974. “How Many Species: Some Mathematical Aspects of the Dynamics of Populations” inLectures on Mathematics in the Life Sciences, Vol.6, pp. 65–98 Providence: AMS.
McKean, H. P. 1969.Stochastic Integrals. New York: Academic Press.
Myers, J. H. and C. J. Krebs. 1974. “Population Cycles in Rodents,”Scientific American,230, 38–46.
Papanicolaou, G. C. 1975. “Asymptotic Analysis of Transport Processes.”Bull. Am. Math. Soc.,81, 330–392.
Parthasarathy, K. R. 1967.Probability Measures on Metric Spaces. New York: Academic Press.
Stroock, D. W. 1971. “On the Growth of Stochastic Integrals”.Z. Wahrscheinlickeitstheorie,18, 340–344.
Wiegert, R. G.,et al. 1975. “A Preliminary Ecosystem Model of Coastal Georgia Spartina Marsh.”Estuarine Research,1, 583–601.
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Kannan, D. Volterra-Verhulst prey-predator systems with time dependent coefficients: Diffusion type approximation and periodic solutions. Bltn Mathcal Biology 41, 229–251 (1979). https://doi.org/10.1007/BF02460881
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DOI: https://doi.org/10.1007/BF02460881