This paper is focused on the behavior near the extinction time of solutions of systems of ordinary differential equations with a sublinear dissipation term. Suppose the dissipation term is a product of a linear mapping $A$ and a positively homogeneous scalar function $H$ of a negative degree $-\alpha$. Then any solution with an extinction time $T_*$ behaves like $(T_*-t)^{ 1/\alpha}\xi_*$ as time $t\to T_*^-$, where $\xi_*$ is an eigenvector of $A$. The result allows the higher order terms to be general and the nonlinear function $H$ to take very complicated forms. As a demonstration, our theoretical study is applied to an inhomogeneous population model.
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