  We derive a method to calculate the multi-channel K matrix applicable to a
broad class of models in which mesons linearly couple to the quark core. The
method is used to calculate pion scattering amplitudes in the energy region of
low-lying P11 and P33 resonances. A good agreement with experimental data is
achieved if in addition to the elastic channel we include the $\pi\Delta$ and
$\sigma N$ ($\sigma\Delta$) channels where the $\sigma$-meson models the
correlated two-pion decay. We solve the integral equation for the K matrix in
the approximation of separable kernels; it yields a sizable increase of the
widths of the $\Delta(1232)$ and the N(1440) resonances compared to the bare
quark values.
