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Pattern formation in a two-species aggregation model Evers, Joep


We consider a system of two aggregation equations. This system consists of two continuity equations for the densities \(\rho_1\) and \(\rho_2\), coupled via the the velocities \(v_1\) and \(v_2\). Each \(v_i\) is given by \(v_i = -\nabla K_{ii} * \rho_i -\nabla K_{ij} * \rho_j\), where the former convolution term accounts for self-interactions and the latter one for cross-interactions. Each kernel is chosen such that the interactions exhibit Newtonian repulsion and linear attraction (i.e., the kernel is quadratic). The free parameters in the model are the coefficients multiplying the repulsion and attraction parts. We assume that the interactions are symmetric, in the sense that \(K_{11}=K_{22}\) and \(K_{12}=K_{21}\). Our aim is to characterise the steady states of the model and their stability for varying model parameters. For our specific interaction kernels, it is known that the density can take only two different values in a steady state, depending on whether the two densities coexist at a certain position or not. However, the geometry of the supports of \(\rho_1\) and \(\rho_2\) is far from trivial.

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