To solve these problems, Thompson et al. (2006) suggested that the nudging be limited to frequency bands centered on climatologically relevant frequencies (e.g., 0 and 1 cycle per year); outside of these frequency bands the model is not nudged and can evolve freely.

This corresponds to replacing (2) by equation(4) dxdt=Φx+f+γ〈c-x〉where 〈·〉〈·〉 denotes a quantity that has been bandpass filtered to pass variations in the vicinity of climatologically relevant frequencies. If γγ is sufficiently small that Eq. (4) remains stable, the Fourier transform of the nudged state is still given by (3) if we replace γγ by γΓ(ω)γΓ(ω) where Γ(ω)Γ(ω) is the transfer function of the bandpass filter. It follows that, away from the climatological frequencies where Γ(ω)=0,X(ω)=Xu(ω)Γ(ω)=0,X(ω)=Xu(ω) as expected. VX-809 clinical trial Biogeochemical models are highly nonlinear and so we now generalize Eq. (1) to equation(5) dxdt=ϕ(x,t)where the dependence of ϕϕ on time t allows for the possibility of time-dependent parameters and external forcing. Based on the above discussion www.selleckchem.com/products/CP-690550.html we propose the following form of frequency dependent nudging: equation(6)

dxdt=ϕ(x,t)+γ〈c-x〉+δ(c-x) Note that this equation differs from Eq. (2) through the addition of a conventional nudging term with nudging coefficient δδ. This term was added to increase the stability of the nudged system. Details on the implementation of the bandpass filter are given below. Biogeochemical models can generate, and couple, variability across a wide range of time scales. Hence, it is not clear a priori that the frequency dependent

nudging defined in Eq. (6) will work nor that it will work better than conventional nudging. In the next section the effectiveness of the scheme is evaluated using one of the simplest models of predator–prey interactions: a modified Lotka–Volterra model. A highly idealized model of the interaction of prey (x1x1) and predators (x2x2) is equation(7) dx1dt=α1×1(1-x1/α3)-α4x1x2dx2dt=α5x1x2(1-x2/α6)-α2x2where α1α1 and α3α3 control the growth of the prey and α4α4 controls the rate of of predation, α5α5 and α6α6 control the growth of the predators and α2α2 is their mortality rate. This pair of equations differs from the well known Lotka–Volterra model in one important respect: the growth terms for prey and predators use the logistic growth parameterization instead of a constant growth rate. The constant growth rates in the standard Lotka–Volterra equations assume infinite carrying capacities. The above modification addresses this issue, implicitly representing resources via an imposed carrying capacity for both prey and predators. The carrying capacities for prey and predators are α3α3 and α6α6, respectively. Modified Lotka–Volterra (LV) equations such as Eq. (7) have been discussed extensively in the ecological literature (e.g. MacArthur, 1970, May, 1973, Chesson, 1990 and Berryman, 1992). To simplify Eq.