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Periodicity and boundedness of solutions of generalized differential equations of growth

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Abstract

Sufficient conditions are given for the existence of periodic solutions of differential equations, having as special cases the equations used to describe the competition between two species. The Poincaré bifurcation theory is used to secure one set of conditions, and another set of conditions is secured through a generalization of a method of V. Volterra. The question of boundedness is considered and conditions implying boundedness and conditions implying that populations are bounded away from zero are given. Several integrable classes of systems are discovered and a particular example having periodic solutions is examined in detail.

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Author notes

  1. P. E. Waltman

    Present address: Sandia Corporation, Albuquerque, New Mexico

Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Missouri, USA

    W. R. Utz & P. E. Waltman

Authors
  1. W. R. Utz
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  2. P. E. Waltman
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Additional information

This research was supported by the Air Force Office of Scientific Research under Grant 62-207.

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Utz, W.R., Waltman, P.E. Periodicity and boundedness of solutions of generalized differential equations of growth. Bulletin of Mathematical Biophysics 25, 75–93 (1963). https://doi.org/10.1007/BF02477772

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  • DOI: https://doi.org/10.1007/BF02477772

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