Fractional models, non-locality, and complex systems

In this paper, a new approach to the deterministic modelling of dynamics of certain processes in an anomalous environment is proposed. To this end, the standard assumptions that are usually justified by the experiments and led to the classical dynamics models are rewritten in the way that takes into consideration the non-local features of the anomalous environment. The new class of models obtained in this way is characterized by the memory functions that have to be properly determined for a concrete process. In particular, the so-called fractional dynamics models described in terms of the fractional differential equations are among particular cases of the general model. When a concrete process is observed and its characteristics are measured within a certain time interval, the memory functions that characterize the non-locality of the medium can be found by solving an inverse problem for a system of the Volterra integral equations. Special attention is given to the population dynamics examples to highlight the advantages of the new way to focus the model of the dynamics of complex processes compared with the classical ones.