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4. Applied Models

    1. Clustering in biology


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      Problem 4.1.

      [Michael Levy (UC Berkeley)] Let us assume there are some dividing cells communicating with each other and moving around in $\mathbb{R}^3$ or $\mathbb{R}^2$ with both cell growth and not necessarily totally complete cell division. Is there a good model that describes asymptotic cluster size distribution of those cells? For further information see Larson et al. (in preparation), Fairclough et al. 2010, and Dayel et al. 2011 or Racliff et al. 2015.

        • Gang Dynamics


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          Problem 4.2.

          [Alethea Barbaro (Case Western Reserve University)] Let us consider two opponent gangs of graffiti (A and B). We can derive a model describing their behavior from the lattice model. Indeed, if $g_A$, $g_B$ are the densities corresponding to the graffiti of the two gangs, and $\rho_A$, $\rho_B$ the densities of agents in A and B, we have the following system of PDEs: \begin{equation*} \begin{cases} \partial_t g_A=c\rho_A-g_A\\ \partial_t g_B=c\rho_B-g_B\\ \partial_t\rho_A=\frac{1}{4}\nabla\cdot(\nabla\rho_A+2\beta\rho_A\nabla g_B)\\ \partial_t\rho_B=\frac{1}{4}\nabla\cdot(\nabla\rho_B+2\beta\rho_B\nabla g_A), \end{cases} \end{equation*} for some constants $c,\beta$. Mathematical theory for such system is missing. What can we say about the energy, existence of solutions, phase transitions, numerics?

              Cite this as: AimPL: Nonlocal differential equations in collective behavior, available at http://aimpl.org/nonlocalde.