In this paper, we use numerical simulations to demonstrate a half-soliton interaction of waves in a mathematical model of a "prey-predator" system with taxis when of two colliding waves, one annihilates and the other continues to propagate. We show that this effect depends on the "ages" or, equivalently, "widths" of the colliding waves. In two spatial dimensions we demonstrate that the type of interaction, i.e., annihilation, quasisoliton, or half-soliton, depends not only on curvature and width of the colliding waves, but also on the angle of the collision. When conditions of collision are varying in such a way that only a part of a wave survives the collision, then "taxitons," compact pieces of solitary waves, may form, which can exist for a significant time.