Generalized beta models and population growth: so many routes to chaos

M. Fátima Brilhante, M. Ivette Gomes*, Sandra Mendonça, Dinis Pestana, Pedro Pestana

*Corresponding author for this work

Research output: Contribution to journalReview articlepeer-review



Logistic and Gompertz growth equations are the usual choice to model sustainable growth and immoderate growth causing depletion of resources, respectively. Observing that the logistic distribution is geo-max-stable and the Gompertz function is proportional to the Gumbel max-stable distribution, we investigate other models proportional to either geo-max-stable distributions (log-logistic and backward log-logistic) or to other max-stable distributions (Fréchet or max-Weibull). We show that the former arise when in the hyper-logistic Blumberg equation, connected to the Beta (Formula presented.) function, we use fractional exponents (Formula presented.) and (Formula presented.), and the latter when in the hyper-Gompertz-Turner equation, the exponents of the logarithmic factor are real and eventually fractional. The use of a BetaBoop function establishes interesting connections to Probability Theory, Riemann–Liouville’s fractional integrals, higher-order monotonicity and convexity and generalized unimodality, and the logistic map paradigm inspires the investigation of the dynamics of the hyper-logistic and hyper-Gompertz maps.

Original languageEnglish
Article number194
Number of pages40
JournalFractal and Fractional
Issue number2
Publication statusPublished - 15 Feb 2023


  • Beta and BetaBoop
  • Extreme and geo-extreme distributions
  • Fractional calculus
  • Generalized convexity and unimodality
  • Hyper-logistic and hyper-Gompertz growth
  • Nonlinear maps


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