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Uniqueness and multiplicity of positive solutions for a diffusive predator–prey model in the heterogeneous environment

Part of: Partial differential equations

Published online by Cambridge University Press:  20 January 2020

Shanbing Li  and
Yaying Dong
Shanbing Li
Affiliation:
School of Mathematics and Statistics, Xidian University, Xi'an 710071, PR China
Yaying Dong
Affiliation:
School of Science, Xi'an Polytechnic University, Xi'an 710048, PR China (lishanbing@xidian.edu.cn)
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Abstract

This is the second part of our study on the spatially heterogeneous predator–prey model where the interaction is governed by a Crowley–Martin type functional response. In part I, we have proved that when the predator competition is strong (i.e. k is large), the model has at most one positive steady-state solution for any $\mu \in \mathbb {R}$, moreover it is globally asymptotically stable for any $\mu >0$. This part is denoted to study the effect of saturation. Our result shows that the large saturation coefficient (i.e. large m) can not only lead to the uniqueness of positive solutions, but also lead to the multiplicity of positive solutions, moreover the stability of the corresponding positive solutions is also completely obtained. This work can be regarded as a supplement of Ref. [10].

Keywords

Predator–prey modelspatial heterogeneitysaturationuniquenessmultiplicity

MSC classification

Primary: 92D40: Ecology
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters 35B32: Bifurcation 92D25: Population dynamics (general)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3321 - 3348
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Copyright
Copyright © The Author(s) 2020. Published by The Royal Society of Edinburgh

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