pairs ${(k_{+},k_{-})}$, where $j$ denotes the $j$th branch of the Lambert $W$-function. This formula can be used for studying the stability of the homogeneous state. In particular, if all eigenvalues $\lambda_{j}$ have negative real parts for all possible wavevectors $k_{\pm}$, then the steady state is asymptotically stable.

### Large delay approximation\[sub:ststLong\]

A deeper understanding of the spectrum can be achieved for large delays using the asymptotic methods [@Lichtner2011; @Sieber2013; @Flunkert2010b]. Accordingly to these results, the spectrum splits generically into two parts accordingly to the asymptotics for large delays. The first part is called the *strongly unstable spectrum* and the second part is the *pseudo-continuous spectrum*. The strong spectrum exists for $\alpha>0$ and consists of two complex conjugate, isolated roots which are close to $\lambda_{S,\pm}=\alpha\pm i\beta$. In such a case, the contribution of the term $\frac{1}{\tau}W_{j}\left[\cdot\right]$ in (\[eq:W\]) vanishes. In fact, the strong spectrum always converges to the unstable part of the spectrum of the system with omitted delayed terms [@Yanchuk2005; @Wolfrum2010; @Lichtner2011], i.e. $\frac{\mathrm{d}}{\mathrm{d}t}\delta\tilde{z}(t)=(\alpha+i\beta)\delta\tilde{z}(t)$ in this case. Besides the strong spectrum there are infinitely many more eigenvalues, accumulating on some curves in the complex plane as $\tau\to\infty$. These eigenvalues form the pseudo-continuous spectrum and can be found by substituting the 