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

Brilhante, Maria de Fátima; Gomes, Maria Ivette; Mendonça, Sandra; Pestana, Dinis; Pestana, Pedro Duarte

http://hdl.handle.net/10400.2/20707

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Authors

Brilhante, Maria de Fátima

Gomes, Maria Ivette

Mendonça, Sandra

Pestana, Dinis

Pestana, Pedro Duarte

Abstract(s)

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 (p, q) function, we use fractional exponents p − 1 = 1 ∓ 1/α and q − 1 = 1 ± 1/α, 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.

Keywords

Beta and BetaBoop Fractional Calculus Nonlinear Maps

URI

http://hdl.handle.net/10400.2/20707

Citation

Brilhante, M.F., Gomes, M.I., Mendonça, S., Pestana, D. & Pestana, P.D. (2023). Generalized Beta Models and Population Growth: So Many Routes to Chaos. Fractal and Fractional. 7, 2, 40 p., 194.