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.
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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
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A possible example of a stationary solution to (3) with
Example of mixed stationary state. Note that by symmetry
An example of a separated stationary state
In this figure, a mixed steady state is plotted by using initial data given by (70),
A separated steady state is presented in this figure. Initial data are given by (71). The parameters are
This figure shows how from the initial densities
A steady state of four bumps is showed in this figure starting from initial data as in (72) with
Starting from initial data as in (73) with
This last figure shows a possible existence of traveling waves by choosing initial data as in (74),