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Grass carp’s roes should be agglomerated together for maximizing their survival rate against various predators. Any vibration induced by any environmental perturbation should be attenuated immediately. A Toda-like fractal-fractional oscillator is established, which shows a low-frequency property for most cases; however, the grass carp has evolved a very ability to attenuate the perturbated vibration by sticky adhesion. The pull-down stability of the roes’ vibration is discovered through the results of phase diagrams. The mathematical analysis reveals that there is a pull-down plateau for the attenuating process, the plateau’s height and width are discussed graphically, and the main factors affecting the plateau’s properties are elucidated. The paper offers a totally new window for biomechanics, especially for biomimicking design of chatter vibration systems inspired by the agglomerated roes.