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Punctuated Evolution due to Delayed Carrying Capacity


  • V.I. Yukalov
  • E.P. Yukalova
  • D. Sornette



A new delay equation is introduced to describe the punctuated evolution of complex nonlinear systems. A detailed analytical and numerical investigation provides the classification of all possible types of solutions for the dynamics of a population in the four main regimes dominated respectively by: (i) gain and competition, (ii) gain and cooperation, (iii) loss and competition and (iv) loss and cooperation. Our delay equation may exhibit bistability in some parameter range, as well as a rich set of regimes, including monotonic decay to zero, smooth exponential growth, punctuated unlimited growth, punctuated growth or alternation to a stationary level, oscillatory approach to a stationary level, sustainable oscillations, finite-time singularities as well as finite-time death.

Suggested Citation

  • V.I. Yukalov & E.P. Yukalova & D. Sornette, "undated". "Punctuated Evolution due to Delayed Carrying Capacity," Working Papers CCSS-09-004, ETH Zurich, Chair of Systems Design.
  • Handle: RePEc:stz:wpaper:ccss-09-004

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