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Multiple patterns formation for an aggregation/diffusion predator-prey system

  • * Corresponding author: Simone Fagioli

    * Corresponding author: Simone Fagioli 
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  • We investigate existence of stationary solutions to an aggregation/diffusion system of PDEs, modelling a two species predator-prey interaction. In the model this interaction is described by non-local potentials that are mutually proportional by a negative constant $ -\alpha $, with $ \alpha>0 $. Each species is also subject to non-local self-attraction forces together with quadratic diffusion effects. The competition between the aforementioned mechanisms produce a rich asymptotic behavior, namely the formation of steady states that are composed of multiple bumps, i.e. sums of Barenblatt-type profiles. The existence of such stationary states, under some conditions on the positions of the bumps and the proportionality constant $ \alpha $, is showed for small diffusion, by using the functional version of the Implicit Function Theorem. We complement our results with some numerical simulations, that suggest a large variety in the possible strategies the two species use in order to interact each other.

    Mathematics Subject Classification: 35B40, 35B36, 35Q92, 45K05, 92D25.


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  • Figure 1.  A possible example of a stationary solution to (3) with $ N_\rho = N_\eta = 3 $ is plotted as described in Definition 1.1

    Figure 2.  Example of mixed stationary state. Note that by symmetry $ L_\rho=-R_\rho $ and $ L_\eta=-R_\eta $

    Figure 3.  An example of a separated stationary state

    Figure 4.  In this figure, a mixed steady state is plotted by using initial data given by (70), $ \alpha = 0.1 $, $ \theta = 0.4 $. Number of particles are chosen equal to number of cells in the finite volume method, which is $ N = 71 $

    Figure 5.  A separated steady state is presented in this figure. Initial data are given by (71). The parameters are $ \alpha = 0.2 $ and $ \theta = 0.4 $ with $ N = 91 $

    Figure 6.  This figure shows how from the initial densities $ \rho_0,\eta_0 $ and $ \theta $, as in Figure 4, a transition between mixed and a sort of separated steady state appears by choosing the value of $ \alpha = 6 $. This large value of $ \alpha $ suggests an unstable behavior in the profile, see Remark 3.1

    Figure 7.  A steady state of four bumps is showed in this figure starting from initial data as in (72) with $ \alpha = 0.05 $ and $ \theta = 0.3 $. The number of particles $ N = 181 $, which is the same as number of cells

    Figure 8.  Starting from initial data as in (73) with $ \alpha = 1 $ and $ \theta = 0.3 $, we get a five bumps steady state

    Figure 9.  This last figure shows a possible existence of traveling waves by choosing initial data as in (74), $ \alpha = 1 $ and $ \theta = 0.2 $. The first two plots are performed by particles method, while the third one is done by finite volume method. Here we fix $ N = 101 $

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