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Researchers developed a generalized Allee-logistic (GAL) map, a mathematical model that combines population growth dynamics with the Allee effect, where populations below a certain threshold face extinction. The model reveals a tricritical point—a special parameter value where the nature of the extinction transition changes from continuous to discontinuous—and demonstrates that the Allee effect suppresses the emergence of chaotic behavior. The study provides exact mathematical solutions and connects chaotic dynamics with phase transition theory from statistical physics.
Why it matters
This work advances our understanding of population collapse and extinction dynamics in ecology, providing analytical tools to predict when populations might face abrupt versus gradual decline. The mathematical framework could inform conservation strategies and help identify warning signs before populations reach critical thresholds.
arXiv:2606.05351v1 Announce Type: cross
Abstract: We present a novel nonlinear dynamical model, the generalized Allee-logistic (GAL) map given by $x_{t+1} = r x_t (1 – x_t) G(x_t)$ where $G(x_t) = m (x_t – h) + 1 – m$ incorporates the Allee effect with magnitude $m$ and threshold $h$. The case $m = 0$ yields the logistic map with a continuous transition to extinction. Conversely, $m = 1$ recovers a previously studied model that undergoes only a discontinuous extinction-to-active transition. Between these extremes, the GAL map exhibits nontrivial phenomena, including tricriticality with a closed-form expression for the tricritical point and a universal crossover function. Under a small external input, we verify Widom-like relations. We also note that the Allee effect disfavors the onset of chaos. Our work establishes additional bridges between analytically tractable chaotic maps, nonequilibrium tricriticality, and Allee effects.
Source: Tricriticality and chaos in a generalized Allee-logistic map