Two-size fragmentation cascade ($\dot{F}=\alpha SF-\beta F^2$)

A simplified collisional cascade tracks large objects $S$ and small fragments $F$. A fragment can disrupt a large object and multiply, while fragment–fragment collisions remove material. With $\dot{S}=-\kappa SF$ and $\dot{F}=\alpha SF-\beta F^2$, elimination of time gives a linear differential equation, $dF/dS-(\beta /\kappa )F/S=-\alpha /\kappa$. It has solution $F=C{S}^k-(\alpha /\kappa )S/(1-k)$ for $k=\beta /\kappa \ne 1$. Its maximum occurs at $F=(\alpha /\beta )S$; early exponential growth does not imply permanent exponential growth after the large-object reservoir is depleted.