Previous studies on competition with a finite interaction range have shown that spatial aggregation is possible in the long time limit [ 37- 43].
The analysis is limited to two dimensions and assumes quasi-stationary distributions in the long time limit.
Hence, the importance of investigating fractional equations arises from the necessity to sharpen the concepts of equilibrium, stability states, and time evolution in the long time limit.
The response dynamics of the model have been investigated to identify the possible parameter estimation strategies and in order to predict the long time limit behaviour.
The self-similar solutions of the Fokker Planck equation presented here are based on the assumption of quasi-stationary distributions reached in the long time limit.
In addition, steady-state Laplace PDEs can be formulated to produce the homogenized diffusion tensor that describes the diffusion characteristics of the medium in the long time limit.
In a recent publication an analytical solution of the Fokker Planck continuity equation for the grain size distribution for two-dimensional grain growth in the long time limit (self-similar state) was provided.
The approximate solution of the Fokker Planck equation presented here is limited to two dimensions and is based on the assumption of quasi-stationary distributions reached in the long time limit.
Then, the cumulant generating function in the long time limit is given by F ( χ ; t ) = λ 0 t, where λ0 denotes the minimum eigenvalue of W that develops adiabatically from 0 with χ.
Therefore under this model, the long time limit is exact.
In the long time limit, this should outweigh all time-linear dynamics that have been assumed in PPI network evolution models so far [ 36, 41- 45] (see, however, Discussion).
