[en] 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 πΔ and σN(σΔ) channels where the σ -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 Δ(1232) and the N(1440) resonances compared to the bare quark values. (orig.)