Sensitive dependence, divergence and unpredictable behaviour in a stratigraphic forward model of a carbonate system

Abstract Although conceptual models of carbonate systems typically assume a dominance of external forcing and linear behaviour to generate metre-scale carbonate parasequences, there is no reason to preclude autocyclic and non-linear behaviour in such systems. Component parts of the carbonate system represented in this numerical forward model are entirely deterministic, but several parts are non-linear and exhibit complex interactions. Onshore sediment transport during relative sea-level rise generates autocyclic quasi-periodic shallowing upward parasequences but model behaviour is sufficiently complex that water depth evolution and parasequence thickness distributions are not predictable in any detail. The model shows sensitive dependence on initial conditions, resulting in divergence of two model cases, despite only a small difference in starting topography. Divergence in water-depth history at one point takes ∼ 10 ka, and for the whole model grid takes ∼ 100 ka. Fischer plots from the two cases show that divergence leads to entirely different parasequence thickness evolution in each case. Chaotic behaviour is a specific type of sensitive dependence, and calculation of trajectory divergence in a 3-D pseudo-phase space indicates that water depth evolution is not truly chaotic. If sensitive dependence, divergence and complex processes generating random products turn out to be common in real carbonate systems, predictions should be limited to elements of the system unaffected by these phenomena, or limited to cases where an element of periodic external forcing over-rides their affects. These results also suggest that increasingly complex and sophisticated stratigraphic forward models are not necessarily going to lead directly to more deterministic predictive power, although they may well be useful sources of statistical data on carbonate strata.