Long time behaviour of the approximate solution to quasi-convolution Volterra equations

In some important biological phenomena Volterra integral and integrodifferential equations represent an appropriate mathematical model for the description of the dynamics involved (see e.g. [1], and the bibliography therein). In most cases, the kernels of these equations are of convolution type, however, some recent applications, as cell migration and collective motion [4-5], are characterized by kernels with a quasi-convolution form, namely involving a convolution contribution plus a non-convolution term. We focus on problems of this type and exploit some known results about convolution equations [2, 3], in order to describe the asymptotic dynamics of numerical approximations and connect the results to the behaviour of the analytical solution