  In this paper, we consider the Cauchy problem of a two-component b-family
system, which includes the two-component Camassa-Holm system and the
two-component Degasperis-Procesi system. It is shown that the solution map of
the two-component b-family system is not uniformly continuous on the initial
data in Besov spaces $B_{p, r}^{s-1}(\mathbb{R})\times B_{p, r}^s(\mathbb{R})$
with $s>\max\{1+\frac{1}{p}, \frac{3}{2}\}$, $1\leq p, r< \infty$. Our result
covers and extends the previous non-uniform continuity in Sobolev spaces
$H^{s-1}(\mathbb{R})\times H^s(\mathbb{R})$ for $s>\frac{5}{2}$ (Nonlinear
Anal., 2014) to Besov spaces.
