Investigation on the interactions of Airy and nonlinear accelerating beams in a biased photovoltaic-photorefractive crystal was presented theoretically by means of split-step Fourier method. The results shown that, by adjusting the initial beams interval and incident angle, two finite-energy Airy beams in the in-phase or out-of-phase case can attract or repel each other. In the in-phase case, not only single breathing solitons and soliton pairs can generate, but also splitting solitons with oscillation are obtained. While only soliton pairs can be formed in the out-of-phase case. Interaction of two in-phase nonlinear truncated accelerating beams can generate an odd number of breathing solitons, and an even number of soliton pairs can be produced in the out-of-phase case. Moreover, the peak intensity, breathing period and magnitude of the interaction force of the breathing solitons can be effectively regulated by adjusting the external bias and the incident angle. The results can provide a theoretical basis for the interaction regulation of Airy beams, and also have potential application prospects in the fields of all-optical information processing and optical network device fabrication.