Skip to main content
SpringerLink
Log in

Existence of a limiting pattern for a system of nonlinear equations describing interpopulation competition

  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

A system of arbitrarily many nonlinear ordinary differential equations which can be interpreted as describing competition between populations is studied here. It is found that limits exist for the system given specified constraints on the “signal function” which describes input-output relations at the level of single populations. Existence of limits is shown by means of a function which records which variable in the system is growing fastest at a given time i.e. who is winning the competition. Generalizations are discussed to sigmoid signal functions which appear in models of pattern discrimination by neuron populations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature

  • Cronin, J. 1975. “Periodic Solutions inn Dimensions and Volterra Equations”J. Diff. Eqns.,19, 21–35.

    Article  MATH  MathSciNet  Google Scholar 

  • Ellias, S. A. and S. Grossberg. 1975. “Pattern Formation, Contrast Control, and Oscillations in the Short Term Memory of Shunting On-Center Off-surround Networks.”Biol. Cyb.,20, 69–98.

    Article  MATH  MathSciNet  Google Scholar 

  • Gilpin, M. E. and F. J. Ayala. 1973. “Global Models of Growth and Competition.”Proc. Natn. Acad. Sci., U.S.A. 70, 3590–3593.

    Article  MATH  Google Scholar 

  • Grossberg, S. 1973. “Contour Enhancement, Short Term Memory, and Constancies in Reverberating Neural Networks.”Stud. Appl. Math.,52, 213–257.

    MATH  MathSciNet  Google Scholar 

  • — 1976. “On the Development of Feature Detectors in the Visual Cortex with Applications to Learning and Reaction-diffusion Ssytems.”Biol. Cyb.,21, 145–159.

    Article  MATH  MathSciNet  Google Scholar 

  • — 1977. “Pattern Formation by the Global Limits of a Nonlinear Competitive Interaction inn Dimensions.”J. Math. Biol.,4, 237–257.

    Article  MATH  MathSciNet  Google Scholar 

  • — 1978. “Communication, Memory and Development.” In:Progress in Theoretical Biology, Eds. R. Rosen and F. Snell. New York: Academic Press, p. 183.

    Google Scholar 

  • — and D. S. Levine. 1975. “Some Development and Attentional Biases in the Contrast Enhancement and Short Term Memory of Recurrent Neural Networks.”J. Theor. Biol.,53, 341–380.

    Article  MathSciNet  Google Scholar 

  • Levine, D. S. and S. Grossberg. 1976. “Visual Illusions in Neural Networks: Line Neutralization, Tilt Aftereffect, and Angle Expansion.”J. Theor. Biol.,61, 477–504.

    Article  MathSciNet  Google Scholar 

  • May, R. M. 1973.Stability and Complexity in Model Ecosystems. Princeton, N. J.: Princeton University Press.

    Google Scholar 

  • — and W. J. Leonard. 1975. “Nonlinear Aspects of Competition Between Three Species.”SIAM J. Appl. Math.,29, 243–253.

    Article  MATH  MathSciNet  Google Scholar 

  • Pierpont, J. 1912.Lectures on the Theory of Functions of Real Variables. Vol. II. Boston: Ginn.

    Google Scholar 

  • Ratliff, F. 1965.Mach Bands: Quantitative Studies of Neural Networks in the Retina. San Francisco: Holden-Day.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Pittsburgh, 15260, Pittsburgh, PA, USA

    Daniel S. Levine

Authors
  1. Daniel S. Levine
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Levine, D.S. Existence of a limiting pattern for a system of nonlinear equations describing interpopulation competition. Bltn Mathcal Biology 41, 617–628 (1979). https://doi.org/10.1007/BF02462418

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02462418

Keywords

Search

Navigation