Relations between local, nonlocal, discrete and continuous models of bioturbation

A general framework is proposed that classifies existing bioturbation models according to two dividing lines: discrete/semi-discrete/continuous and local/nonlocal. Based on a common stochastic approach, which models biological reworking of particles as a position jump process, the relationships among the different model classes are exposed and the assumptions underlying each model are explicitly derived. We find that discrete/semi-discrete/continuous formulations are principally equivalent, leaving two basic modeling formalisms: (1) the more inclusive nonlocal exchange formalism and (2) the local biodiffusion model, which is obtained as a special case of the former. Three fundamental criteria determine the applicability of these formalisms, termed the frequency (a), symmetry (b), and length criterion (c). These criteria provide a quantitative basis to decide whether a process should be modeled as local (must comply to (a) + (b) + (c)) or nonlocal (must comply to (a) only). An order of magnitude assessment reveals that under natural conditions, most modes of sediment reworking meet the condition for the nonlocal exchange formalism, but violate the additional assumptions of the biodiffusion model, particularly for short-lived tracers. Nevertheless, in practice, the biodiffusion model has proven to be a valuable empirical model for sediment mixing. This apparent contradiction between theory and practice is termed the "biodiffusion paradox." Further exploration of the nonlocal formalism is encouraged to elucidate this paradox.