Spatial patterns for a predator-prey system with Beddington-DeAngelis functional response and fractional cross-diffusion

Pan Xue; Cuiping Ren; Pan Xue; Cuiping Ren

doi:10.3934/math.2023990

AIMS Mathematics

2023, Volume 8, Issue 8: 19413-19426. doi: 10.3934/math.2023990

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Spatial patterns for a predator-prey system with Beddington-DeAngelis functional response and fractional cross-diffusion

Pan Xue ^,,
Cuiping Ren

School of General Education, Xi'an Eurasia University, Xi'an, Shaanxi 710065, China

Received: 07 March 2023 Revised: 23 May 2023 Accepted: 28 May 2023 Published: 08 June 2023
MSC : 35B32, 35J65, 92D25

In this paper, we investigate a predator-prey system with fractional type cross-diffusion incorporating the Beddington-DeAngelis functional response subjected to the homogeneous Neumann boundary condition. First, by using the maximum principle and the Harnack inequality, we establish a priori estimate for the positive stationary solution. Second, we study the non-existence of non-constant positive solutions mainly by employing the energy integral method and the Poincaré inequality. Finally, we discuss the existence of non-constant positive steady states for suitable large self-diffusion $ d_2 $ or cross-diffusion $ d_4 $ by using the Leray-Schauder degree theory, and the results reveal that the diffusion $ d_2 $ and the fractional type cross-diffusion $ d_4 $ can create spatial patterns.
- fractional type cross-diffusion,
- predator-prey system,
- Beddington-DeAngelis functional response,
- stationary solution,
- Leray-Schauder degree
Citation: Pan Xue, Cuiping Ren. Spatial patterns for a predator-prey system with Beddington-DeAngelis functional response and fractional cross-diffusion[J]. AIMS Mathematics, 2023, 8(8): 19413-19426. doi: 10.3934/math.2023990

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Abstract

In this paper, we investigate a predator-prey system with fractional type cross-diffusion incorporating the Beddington-DeAngelis functional response subjected to the homogeneous Neumann boundary condition. First, by using the maximum principle and the Harnack inequality, we establish a priori estimate for the positive stationary solution. Second, we study the non-existence of non-constant positive solutions mainly by employing the energy integral method and the Poincaré inequality. Finally, we discuss the existence of non-constant positive steady states for suitable large self-diffusion $ d_2 $ or cross-diffusion $ d_4 $ by using the Leray-Schauder degree theory, and the results reveal that the diffusion $ d_2 $ and the fractional type cross-diffusion $ d_4 $ can create spatial patterns.

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