A spatially heterogeneous predator-prey model

Julián López-Gómez; Eduardo Muñoz-Hernández

doi:10.3934/dcdsb.2020081

Article Contents

2021, Volume 26, Issue 4: 2085-2113. Doi: 10.3934/dcdsb.2020081

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A spatially heterogeneous predator-prey model

Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Madrid, 28040, Spain

^* Corresponding author
^* Corresponding author

Received: April 2019

Early access: January 2020

Published: April 2021

This paper has been supported by the IMI of the Complutense University of Madrid and the Ministry of Science and Universities of Spain under Research Grant PGC2018-097104-B-100. The second author, ORCID: 0000-0003-1184-6231, has been also supported by contract CT42/18-CT43/18 of Complutense University of Madrid

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Abstract

This paper introduces a spatially heterogeneous diffusive predator–prey model unifying the classical Lotka–Volterra and Holling–Tanner ones through a prey saturation coefficient, $ m(x) $, which is spatially heterogenous and it is allowed to 'degenerate'. Thus, in some patches of the territory the species can interact according to a Lotka–Volterra kinetics, while in others the prey saturation effects play a significant role on the dynamics of the species. As we are working under general mixed boundary conditions of non-classical type, we must invoke to some very recent technical devices to get some of the main results of this paper.

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Mathematics Subject Classification: Primary: 35J57, 35Q92, 35A16.

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Figure 1. Stability of the semitrivial solutions

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Figure 2. Stability of $ (0, 0) $

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Figure 3. An admissible component $ \mathfrak{C} $

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Figure 4. By Theorem 5.1, (5) admits a coexistence state for each $ (\lambda, \mu) $ in the green region, while it cannot admit a coexistence state in the white one. Within the yellow region, (5) might admit, or not, a coexistence state as it will become apparent later

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Figure 5. The coexistence wedges of Figure 4 when $ \mathrm{Int\, }m^{-1}(0)\neq \emptyset $

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