TABLE 1.

Joining the theoretical and experimental ecologists' perspectives on terms and interpretations of spatial‐temporal dynamics

Term	Interpretation	Mathematical expression
Thermodynamic phase separation	Two distinct phases generate from a single homogeneous mixture, driven by an associated reduction in free energy of the system. A classic example is that when water and oil are mixed together, they spontaneously separate to form a water phase and an oil phase. The mobility of the phase‐separating components is similar.	$\frac{\partial C}{\partial t}=D_1{\mathrm{\nabla }}^2\left(C^3‐C‐\epsilon {\mathrm{\nabla }}^{2}C\right),$ $C$ is the volume fraction or mass fraction of one component in binary mixtures systems.
Motility‐induced phase separation	The mobility of self‐propelled organism/particle is featured with density‐dependent movement. Self‐propelled individuals tend to accumulate where they move more slowly, whereas dissipate from over‐dense areas. This positive feedback makes the system separate into dense and dilute phases.	Equation (2) and Equation (5)
Spinodal instability	In phase‐separation principle, one homogeneous mix phase spontaneously separates into two distinct phases with infinitesimal disturbance.	$\frac{{\partial }^2\mathcal{F}(C)}{\partial C^2}<0,$ $\mathcal{F}(C)$ is the free‐energy function of the systems.
Velocity‐induced drift flux	Drift is the slow movement of an object toward something, and the drift velocity is the average‐velocity of an object during drift. Drift flux is defined as the volumetric flux of either component relative to a surface moving at drift velocity.	$J_H=‐\frac{V\mathrm{\nabla }(VH)}{d\tau }$
Biased random walk	Non‐Brownian diffusion, a random walk that is biased in certain directions, leading to a net drift on average of particles in one specific direction. A random walk can be based on three circumstances: 1) a higher probability of moving to certain directions under uniform moving step lengths; 2) suppose the probabilities of moving to all directions remain equal, but nonuniform moving step lengths; 3) nonuniform moving directions and moving step lengths.	Codling et al., 2010
Coarsening/Ostwald ripening	A phenomenon originally observed in solid solutions or liquid sols that describes the change of an inhomogeneous structure over time: the growth of large clusters at the expense of smaller ones (Movie S1).	$\mathcal{l}\left(t\right)\propto t^{\chi }$, where $\mathcal{l}$ is patch size or spatial wavelength, $\chi >0$ is the exponent of scaling law, and $\chi =\frac{1}{3}$ is the classic Lifshitz–Slyozov law.
Hyperuniform structure	In d‐dimensional Euclidean space, a hyperuniform structure is characterized by an anomalous suppression of large‐scale density fluctuations relative to those in typical disordered systems. Thus, the hyperuniform structure is more uniformly distributed than Poisson spatial distribution in large spatial scales.	${\sigma }^2(L)\sim L^{‐\gamma }$, and $\gamma \in \left(d\right.,\left.d+1\right]$.