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

Simone Fagioli; Yahya Jaafra; Simone Fagioli; Yahya Jaafra

doi:10.3934/nhm.2021010

Networks and Heterogeneous Media

2021, Volume 16, Issue 3: 377-411. doi: 10.3934/nhm.2021010

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

Simone Fagioli ^{1
,
,},
Yahya Jaafra ^{2
,}

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

Received: 01 September 2020 Published: 20 May 2021
35B40, 35B36, 35Q92, 45K05, 92D25

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.
- Aggregation-diffusion systems,
- predator-prey model,
- patterns formation
Citation: Simone Fagioli, Yahya Jaafra. Multiple patterns formation for an aggregation/diffusion predator-prey system[J]. Networks and Heterogeneous Media, 2021, 16(3): 377-411. doi: 10.3934/nhm.2021010

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Abstract

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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