Nonlinear dynamics. The projection approach as exampli ed for linearised dynamics in the previous section may seem formal and somewhat indirect, given that bulk degrees of freedom can be eliminated directly. The method comes into its own, however, when we consider the full nonlinear reaction equations (3.1.1), where a direct elimination approach is not feasible. We show in this section that, non-trivially for a nonlinear case, explicit expressions for the rate matrix and memory functions in the projected equations can be found. We will appeal to the small noise limit ϵ → 0 as before, and will need to examine carefully what terms survive in this limit. Note that this was not necessary for the linearised dynamics, where the noise drops out from the equations for the conditionally averaged concentrations, whatever the value of ϵ. Guided by the discussion of the linear case, we will again aim to nd a suitable matrix representation for the operators involved. N N Regarding the choice of observables {aα} for the projection we follow the discussion in Section 3.1.4 and include the concentrations of the subnetwork species and all products of these concentrations, shifted to zero mean as necessary. The list of observables is then aT = (δx1, . . . , δxNs , δx2 − ⟨δx2⟩, δx1δx2 − ⟨δx1δx2⟩, . . . , δx2 s − ⟨δx2 s ⟩), containing in total N s + N s(N s + 1)/2 = N s(N s + 3)/2 distinct quantities. For the steady state distribution we take again a zero mean Gaussian for δx, with a Poissonian covariance matrix. The steady state averages ⟨δxiδxj⟩ are then O(ϵ) and can be neglected against terms of order unity. Applying this simpli cation, the nonlinear projected equations for the subnetwork concentrations then follow from the general result (3.1.7) as N s ∂ δx = Σ δx Ωs,s + Σ δx δx Ωss,s ∂t i j j=1 dtj ∫ t + 0 ji j=1 ,ΣN s 1≤j≤k≤N s j k (jk)i δxj(tj)M s,s(t − tj) + δxj(tj)δxk(tj)M ss,s (t − tj) Σ ji (jk)i 1≤j≤k≤N s + ri(t) (3.2.18) (jk)i Here we have split sums over observables into linear and nonlinear observables to show the structure more clearly. Accordingly there are now linear and nonlinear rate matrix and memory contributions. The bracket on the subscript in the nonlinear terms Ωss,s and M (jk)i indicates the e ect of a concentration product δxjδxk on the time evolution of δxi. As before we have not distinguished in our notation between the original con- centration variables δxi or δxiδxj and their averages conditional on a given initial set of concentrations across the network, b...
