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

The integral representation of some biological phenomena consists in Volterra equations whose kernels involve a convolution term plus a non convolution one. Some significative applications arise in linearised models of cell migration and collective motion, as described in Di Costanzo et al. (Discrete Contin. Dyn. Syst. Ser. B 25 (2020) 443-472), Etchegaray et al. (Integral Methods in Science and Engineering (2015)), Grec et al. (J. Theor. Biol. 452 (2018) 35-46) where the asymptotic behaviour of the analytical solution has been extensively investigated. Here we consider this type of problems from a numerical point of view and we study the asymptotic dynamics of numerical approximations by linear multistep methods. Through a suitable reformulation of the equation, we collect all the non convolution parts of the kernel into a generalized forcing function, and we transform the problem into a convolution one. This allows us to exploit the theory developed in Lubich (IMA J. Numer. Anal. 3 (1983) 439-465) and based on discrete variants of Paley-Wiener theorem. The main effort consists in the numerical treatment of the generalized forcing term, which will be analysed under suitable assumptions. Furthermore, in cases of interest, we connect the results to the behaviour of the analytical solution.