# American Institute of Mathematical Sciences

September  2021, 16(3): 377-411. doi: 10.3934/nhm.2021010

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

 1 DISIM - University of L'Aquila, Via Vetoio 1 (Coppito) 67100 L'Aquila (AQ), Italy 2 Department of Mathematics, An-Najah National University, Al-Junaid street P448 Nablus, Palestine

* Corresponding author: Simone Fagioli

Received  September 2020 Published  September 2021 Early access  May 2021

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.

Citation: Simone Fagioli, Yahya Jaafra. Multiple patterns formation for an aggregation/diffusion predator-prey system. Networks & Heterogeneous Media, 2021, 16 (3) : 377-411. doi: 10.3934/nhm.2021010
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A possible example of a stationary solution to (3) with $N_\rho = N_\eta = 3$ is plotted as described in Definition 1.1
Example of mixed stationary state. Note that by symmetry $L_\rho=-R_\rho$ and $L_\eta=-R_\eta$
An example of a separated stationary state
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 $N = 71$
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 $N = 91$
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
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
Starting from initial data as in (73) with $\alpha = 1$ and $\theta = 0.3$, we get a five bumps steady state
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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