that the delay $\tau$ is large comparing to the timescale of the system (we will mention it each time explicitly), which simplifies analytical calculations.

Stability and bifurcations of homogeneous stationary state
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### Characteristic equation and its solutions

System (\[eq: SL\]) has a homogeneous steady state $z_{m,n}\equiv0$. Let us investigate stability of this stationary state. Linearizing the equation of motion (\[eq: SL\]) around $z_{m,n}=0$ yields the following equation for the evolution of small perturbations $\delta z_{m,n}(t)$: $$\frac{\mathrm{d}}{\mathrm{d}t}\delta z_{m,n}(t)=(\alpha+i\beta)\delta z_{m,n}(t)+\frac{C}{2}\left(\delta z_{m-1,n}(t-\tau)+\delta z_{m,n-1}(t-\tau)\right).$$ This equation can be diagonalized by a spatial discrete Fourier-transformation $$\delta\tilde{z}_{k_{1},k_{2}}=\sum_{m=1}^{M}\sum_{n=1}^{N}e^{ik_{1}m+ik_{2}n}\delta z_{m,n},$$ where the wavevector $\mathbf{k=}(k_{1},k_{2})^{T}$ admits the following discrete values: $$(k_{1},k_{2})^{T}=2\pi\left(\frac{l}{M},\frac{j}{N}\right)^{T},\quad l=1,\ldots,M,\quad j=1,\ldots,N.\label{eq:k}$$ We obtain $$\frac{\mathrm{d}}{\mathrm{d}t}\delta\tilde{z}_{k_{1},k_{2}}(t)=(\alpha+i\beta)\delta\tilde{z}_{k_{1},k_{2}}(t)+\frac{C}{2}\left(e^{ik_{1}}+e^{ik_{2}}\right)\delta\tilde{z}_{k_{1},k_{2}}(t-\tau).\label{eq:SLld}$$ Since the equation for the Fourier modes is uncoupled, one can drop the indices $k_{1}$ and $k_{2}$ for simplicity and introduce the notations $$k_{\pm}=\frac{1}{2}\left(k_{1}\pm k_{2}\right),\label{eq:Q}$$ which is just a rotation of coordinates in the Fourier space. Note that $k_{\pm}$ admits discrete values accordingly to (\[eq:k\]). Then the system (\[eq:SLld\]) can be rewritten as $$\frac{\mathrm{d}}{\mathrm{d}t}\delta\tilde{z}(t)=(\alpha+i\beta)\delta\tilde{z}(t)+Ce^{ik_{+}}\cos\left(k_{-}\right)\delta\tilde{z}(t-\tau).$$ Since the modes are decoupled in the Fourier space, the corresponding characteristic equation factorizes and reads

$$0=\prod_{\{(k_{_{+}},k_{-})\}} \left( -\lambda+\alpha+i\beta+C\cos\left(k_{-}\right)e^{ik_{+}-\lambda\tau} \right).\label{eq: SLststChEq-1}$$

This is a transcendental equation in $\lambda$, which can be solved as $$\lambda_{j}=\alpha+i\beta+\frac{1}{\tau}W_{j}\left(\tau C\cos\left(k_{-}\right)e^{ik_{+}-(\alpha+i\beta)\tau}\right),\label{eq:W}$$ for all admitted 