In this paper, based on the new competing nonlocal model proposed by JUNG P S et al, the interaction of the Airy Gaussian beam is numerically investigated by the split-step Fourier scheme. Usually, in nonlocal media with competing nonlinearities, the nonlinear refractive index is induced by two types of independent nonlinearities. However, when the beam propagates in the nematic liquid crystal, the nonlinear refractive index will be induced by molecular orientational effects and thermal effects. In this case, the beam propagation in liquid crystals is guided by the model proposed by JUNG P S. According to the new competitive nonlocal model, the refractive index changes induced by the two nonlinear effects are not independent of each other, so the light-induced refractive index change becomes the refractive index change caused by the two nonlinear effects in the form of multiplication. It is shown that the interaction between Airy Gaussian beams can be controlled by the adjustment of the distribution factor, the beam amplitude, beam separation, phase difference, and the nonlocality. The results show that the distribution factors (the degree of molecular orientation nonlocality, thermal nonlinearity coefficient, and the initial spacing) all affect the interaction between the Airy Gaussian beam. Among the above factors, the degree of thermal nonlocality has the smallest effect on the beam interaction under the condition of in-phase or out-of-phase interaction. For in-phase Airy Gaussian beams interaction, the period of the breather obtained from the fusion of the Airy Gaussian beam decreases with the increase of the distribution factor and gradually fuses into quasi-solitons. For the out-of-phase Airy Gaussian beams interaction, the two Airy Gaussian beams became two independent solitons. The angle between the soliton pairs increases with the decrease of the distribution factor. It is found that an increase in the beam amplitude will increase the interaction force. The increase in the nonlocality of the reorientation nonlinearity will lead to an increase in the width of the Airy Gaussian beam. It is also found that the increase of this nonlocality will lead to the increase of the interaction force between out-of-phase Airy Gaussian beams. Interestingly, it is found that the thermal nonlocality drastically affects the interaction of the out-of-phase Airy Gaussian beams, which leads to the change of the repulsive force into mutual attraction force between two out-of-phase Airy Gaussian beams. So, it will lead to the balance of repulsive force and attraction force under certain conditions. It is also found that the increase of the thermal nonlinear coefficient will lead to the appearance of the repulsive force or attraction force between out-of-phase Airy Gaussian beams. As the initial spacing decreases, the attraction between the in-phase Airy Gaussian beams increases and the vibration period of the formed breathers decreases. In this case, the repulsion between the out-of-phase Airy Gaussian beams increases. When the degree of thermal nonlocality increases, the vibration period of in-phase airy Gaussian beams decreases. When the initial beam amplitude is large, the mutually repulsive Airy Gaussian beams attract each other and fuse into a breather. For the out-of-phase Airy Gaussian beams, when the degree of thermal nonlocality increases to a certain value, the repulsive force and attractive force between the two Airy Gaussian beams can reach a dynamic balance. In this case, it is also found that changing the thermal nonlinear coefficient in the nematic phase liquid crystal can make the two Airy Gaussian beams between the simultaneous existence of attraction and repulsion. When the distribution factor is small, adjusting the thermal nonlinear coefficient makes the change in repulsion between the Airy Gaussian beams more pronounced. These theoretical researches may provide a basis for experiments investigating the interaction between Airy Gaussian beams. In addition, our theory may be useful in achieving all-optical interconnection by the interaction between Airy Gaussian beams.