Spatial patterns of the Brusselator model with asymmetric Lévy diffusion

The formation of spatial patterns is an important issue in reaction–diffusion systems.Previous studies have mainly focused on the spatial patterns in reaction–diffusion models equipped with symmetric diffusion(such as normal or fractional Laplace diffusion),namely,assuming that spatial environments of the systems are homogeneous.However,the complexity and heterogeneity of spatial environments of biochemical reactions in vivo can lead to asymmetric diffusion of reactants.Naturally,there arises an open question of how the asymmetric diffusion affects dynamical behaviors of biochemical reaction systems.To answer this,we build a general asymmetric L´evy diffusion model based on the theory of a continuous time random walk.In addition,we investigate the two-species Brusselator model with asymmetric L´evy diffusion,and obtain a general condition for the formation of Turing and wave patterns.More interestingly,we find that even though the Brusselator model with symmetric diffusion cannot produce steady spatial patterns for some parameters,the asymmetry of L´evy diffusion for this model can produce wave patterns.This is different from the previous result that wave instability requires at least a three-species model.In addition,the asymmetry of L´evy diffusion can significantly affect the amplitude and frequency of the spatial patterns.Our results enrich our knowledge of the mechanisms of pattern formation.

Influence of asymmetric tidal mixing on sediment dynamics in a partially mixed estuary

To investigate the influence of asymmetric tidal mixing(ATM) on sediment dynamics in tidal estuaries, we developed a vertically one-dimensional idealized analytical model, in which the M_2 tidal flow, residual flow and suspended sediment concentration(SSC) are described. Model solutions are obtained in terms of tidallyaveraged, and tidally-varying components(M_2 and M_4) of both hydrodynamics and sediment dynamics. The effect of ATM was considered with a time-varying eddy viscosity and time-varying eddy diffusivity of SSC. For the first time, an analytical solution for SSC variation driven by varying diffusivity could be derived. The model was applied to York River Estuary, where higher(or lower) eddy diffusivity was observed during flood(or ebb) in a previous study. The model results agreed well with the observation in both hydrodynamics and sediment dynamics. The vertical sediment distribution under the influence of ATM was analyzed in terms of the phase lag of the M_2 component of SSC relative to tidal flow. The phase lag increases significantly in estuaries with typical ATM(higher diffusivity during flood and lower diffusivity during ebb) for the case of seaward-directed net bottom shear stress(e.g., strong river discharge). In contrary, the phase lag is reduced by ATM, if the tidally-averaged bottom shear stress is landward(e.g., strong horizontal density gradient). The dynamics of sediment transport was analyzed as a function of ATM phase lag to identify the time of highest sediment diffusivity, as well as a function of the residual flow, to evaluate the relative importance of seaward and landward residual flows. In estuaries with relative strong fresh water discharge or weak tidal forcing(in case of flood season or neap tide), the near bottom SSC could be higher during ebb than during flood, since the bottom shear stress is higher during ebb due to seaward residual flow. However, landward net sediment transport can be expected in these estuaries in case of a typical ATM, because higher diffusivity causes higher SSC and landward transport during the flood period, while both SSC and seaward transport could be lower during ebb. On the contrary, seaward sediment transport can be expected in estuaries with landward tidally mean bottom shear stress in case of a reverse ATM,where sediment diffusivity is higher during the ebb.