The energy spectra of three types of two-dimensional potentials (we will call them ‘Davidson-like potentials’ (DLPs)), characterized by four minima separated by barriers, are investigated. The predictions for spectra and wave functions are obtained by using the nine-point finite-difference method. For these potentials, with the existence of a single configuration, a transition of spectra, as a function of barrier height, is covered from tunneling splitting modes to fluctuations phenomena, with equal peaks wave functions, crossing to the spectra of purely anharmonic oscillator potentials (AOPs). A different type of phase transition occurs when two (or more) configurations coexist. With the change of the parameters, a transition of spectra is covered from coexistence phenomena of two distinct quantum tunneling modes to fluctuations phenomena, with unequal peaks wave functions, crossing to the spectra of purely AOPs. Using DLPs, a particular application of the coherent quadrupole–octupole model to describe energy bands with alternating parity of the nuclei 100Mo, 146,148Nd, 148,150Sm, 220Ra, 220,222Rn, and 220,222Th is presented. The global parametrization of the model for the selected nuclei is achieved.