Abstract
Rashevsky's equations for describing the joint variation of cell shape and concentration of a metabolite are discussed. Conditions for the existence of non-spherical equilibria and the location of these are obtained and involve only two parametersa andA. Sufficient (but not necessary) conditions for the stability of these equilibria can also be expressed in terms of these parameters alone. Necessary conditions involve in some cases a third parameterB. Quasi-periodic fluctuations about a stable nonspherical equilibrium may occur, but only in caseB lies on a certain finite range which can be defined in terms ofa andA.
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Literature
Bendixson, Ivar. 1901. “Sur les Courbes Définies par des Équations Differentielles.”Acta Mathematica,24, 1–88.
Rashevsky, N. 1939. “Mathematical Biophysics of the Cell with Reference to the Contractility of Tissues and Amoeboid Movements.”Bull. Math. Biophysics,1, 47–62.
Rashevsky, N. 1940.Advances and Applications of Mathematical Biology. Chicago: The University of Chicago Press.
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Householder, A.S. Mathematical biophysics of cellular forms and movements. Bulletin of Mathematical Biophysics 3, 27–38 (1941). https://doi.org/10.1007/BF02478104
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DOI: https://doi.org/10.1007/BF02478104