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Bifurcation structure of coexistence states for a prey–predator model with large population flux by attractive transition

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  23 August 2021

Kousuke Kuto Open the ORCID record for Kousuke Kuto [Opens in a new window]  and
Kazuhiro Oeda
Kousuke Kuto
Affiliation:
Department of Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan (kuto@waseda.jp)
Kazuhiro Oeda
Affiliation:
Center for Fundamental Education, Kyushu Sangyo University, 2-3-1 Matsukadai, Higashi-ku, Fukuoka 813-8503, Japan (kazuoeda@ip.kyusan-u.ac.jp)
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Abstract

This paper is concerned with a prey–predator model with population flux by attractive transition. Our previous paper (Oeda and Kuto, 2018, Nonlinear Anal. RWA, 44, 589–615) obtained a bifurcation branch (connected set) of coexistence steady states which connects two semitrivial solutions. In Oeda and Kuto (2018, Nonlinear Anal. RWA, 44, 589–615), we also showed that any positive steady-state approaches a positive solution of either of two limiting systems, and moreover, one of the limiting systems is an equal diffusive competition model. This paper obtains the bifurcation structure of positive solutions to the other limiting system. Moreover, this paper implies that the global bifurcation branch of coexistence states consists of two parts, one of which is a simple curve running in a tubular domain near the set of positive solutions to the equal diffusive competition model, the other of which is a connected set characterized by positive solutions to the other limiting system.

Keywords

Asymptotic behaviourattractive transitional fluxbifurcation analysiscoexistence steady statesLyapunov–Schmidt reductionprey–predator model

MSC classification

Primary: 35J57: Boundary value problems for second-order elliptic systems
Secondary: 35J61: Semilinear elliptic equations 35B32: Bifurcation 92D25: Population dynamics (general) 35B20: Perturbations 35B09: Positive solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 4 , August 2022 , pp. 965 - 988
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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