Tricriticality and chaos in a generalized Allee-logistic map

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.

8 pages, 7 figures and 1 table