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Steady-state solutions of one-dimensional competition models in an unstirred chemostat via the fixed point index theory

Part of: Nonlinear integral equations Mathematical biology in general Nonlinear operators and their properties Boundary value problems

Published online by Cambridge University Press:  11 March 2020

Kunquan Lan  and
Wei Lin
Kunquan Lan
Affiliation:
Department of Mathematics, Ryerson University, Toronto, Ontario, CanadaM5B 2K3 (klan@ryerson.ca)
Wei Lin
Affiliation:
School of Mathematical Sciences, SCMS, and Centre for Computational Systems Biology, Fudan University, Shanghai200433, P. R. China (wlin@fudan.edu.cn)
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Abstract

The existence and nonexistence of semi-trivial or coexistence steady-state solutions of one-dimensional competition models in an unstirred chemostat are studied by establishing new results on systems of Hammerstein integral equations via the classical fixed point index theory. We provide three ranges for the two parameters involved in the competition models under which the models have no semi-trivial and coexistence steady-state solutions or have semi-trivial steady-state solutions but no coexistence steady-state solutions or have semi-trivial or coexistence steady-state solutions. It remains open to find the largest range for the two parameters under which the models have only coexistence steady-state solutions. We apply the new results on systems of Hammerstein integral equations to obtain results on steady-state solutions of systems of reaction-diffusion equations with general separated boundary conditions. Such type of results have not been studied in the literature. However, these results are very useful for studying the competition models in an unstirred chemostat. Our results on Hammerstein integral equations and differential equations generalize and improve some previous results.

Keywords

System of Hammerstein integral equationstwo point boundary value problemsfixed point indexone-dimensional competition modelchemostat

MSC classification

Primary: 45G15: Systems of nonlinear integral equations
Secondary: 34B18: Positive solutions of nonlinear boundary value problems 47H10: Fixed-point theorems 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 92B05: General biology and biomathematics
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 240 - 264
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Copyright
Copyright © Royal Society of Edinburgh 2020

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