Abstract
A mathematical model for spatial patterns and the segregation of a population is presented. Individuals in a population are assumed to move at random under the influence of a given environment potential V(x). The notion of kinetic excitation K(x) and intensity excitation Q(x) of a population is introduced. Then equilibrium states of a population are defined through a macroscopic relation K(x) + Q(x) + V(x) = constant. The problem of finding out equilibrium distributions is reduced to an eigenvalue problem. It is shown that a population is segregated by the nodal surfaces of the eigenfunctions, if it is excited. Some applications of the model to biological and ecological problems are indicated.
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References
Chung, K. L., Walsh, J. B.: To reverse a Markov process. Acta Math. 123, 225–251 (1979)
Dynkin, E. B.: Infinitesimal operators of Markov processes. Theory of Probability and its Appl. (English) 1, 34–54 (1956)
Feller, W.: Diffusion processes in one dimension. Trans. AMS. 77, 1–31 (1954)
Hasminsky, R. Z.: Diffusion processes and elliptic differential equations degenerating at the boundary of the domain. Theory of Probability and its Appl. (English) 3, 400–419 (1958)
Kolmogoroff, A.: Ueber die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104, 415–458 (1931)
Kolmogoroff, A.: Zur Umkehrbarkeit der statistischen Naturgesetze. Math. Ann. 113, 766–772 (1937)
Meyer, P. A.: Le retournement du temps, d'après Chung et Walsh. Lecture notes in Math. 191, pp. 213–236, Berlin, Heidelberg, New York: Springer 1971
Morisita, M.: Measuring of habitat value by the ‘Environmental density’ method. Statistical Ecology, vol. 1, pp. 379–401, (G. P. Patil, E. C. Pielou, N. E. Waters, eds.) University Park, London: The Pennsylvania State Univ. Press. 1971
Nagasawa, M.: The adjoint process of a diffusion with reflecting barrier. Kōdai Math. Sem. Rep. 13, 235–248 (1961)
Nagasawa, M., Sato, K.: Remarks to ‘ The adjoint process of a diffusion with reflecting barrier ’. Kōdai Math. Sem. Rep. 14, 119–122 (1962)
Nagasawa, M.: Time reversions of Markov processes. Nagoya Math. Journal, 24, 177–204 (1964)
Nagasawa, M., Maruyama, T.: An application of time reversal of Markov processes to a problem of population genetics. Advances in Applied Probability 11, 457–478 (1979)
Nagasawa, M.: An application of segregation model for septation of Escherichia Coli, submitted to J. Th. Biology
Nelson, E.: Derivation of Schrödinger equation from Newtonian Mechanics. Phys. Rev. 150, 1076–1085 (1966)
Pauling, L., Wilson, E. B.: Introduction to Quantum Mechanics (with application to Chemistry) New York, London: McGraw-Hill Book Co. Inc. 1935
Shigesada, N., Teramoto, E.: A consideration on the theory of environment density. (Japanese with English summary) Japanese J. Ecol. 28, 1–8 (1978)
Titchmarsh, E. C.: Eigenfunction Expansions. Associated with second-order differential equations, Part 1, second edition, Oxford: Clarendon Press 1962
Yamada, M., Maruyama, T., Hirota, Y.: A model for distribution of septation sites in Escherichia coli. U.S.-Japan Intersociety Microbiology Congress, Los Angeles-Honolulu, May 1979
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Nagasawa, M. Segregation of a population in an environment. J. Math. Biology 9, 213–235 (1980). https://doi.org/10.1007/BF00276026
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DOI: https://doi.org/10.1007/BF00276026