| Abstract: |
| To describe biological phenomena such as cell migration and cell adhesion many evolutional equations in which a nonlocal interaction of convolution type with a suitable integral kernel is imposed as an advection term are proposed. It is well known that such nonlocal equations can reproduce various behaviors depending on the shape of the integral kernel. For example, the sign of the integral kernel determines whether the cell density aggregates towards the gradient of its density or not. These nonlocal evolutional equations are often difficult to analyze, and the method of analysis is developing. In the light of these background we approximate the nonlocal Fokker-Planck equation by the combination of a Keller-Segel system which is a typical local dynamics. We will show that the solution of the nonlocal Fokker-Planck equation with any even continuous integral kernel can be approximated as a singular limit of the Keller-Segel system by controlling parameters. Furthermore, motivated by pattern formations, we will explain the result and comparison of the linear stability analysis around the equilibrium point of both equations. |
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