Coexistence of an unstirred chemostat model with B-D functional response by fixed point index theory

Xiao-Zhou Feng,Jian-Hui Tian,Xiao-Li Ma

doi:10.1186/s13660-016-1241-7

Abstract

This paper deals with an unstirred chemostat model with the Beddington-DeAngelis functional response. First, some prior estimates for positive solutions are proved by the maximum principle and the method of upper and lower solutions. Second, the calculation on the fixed point index of chemostat model is obtained by degree theory and the homotopy invariance theorem. Finally, some sufficient condition on the existence of positive steady-state solutions is established by fixed point index theory and bifurcation theory.

Highlights

The chemostat is a laboratory apparatus used for the continuous culture of microorganisms
In Section, some prior estimates for positive solutions are proved by the maximum principle and the upper and lower solution method
5 Conclusion The coexistence of an unstirred chemostat model with B-D functional response is studied by fixed point index theory in our paper

Summary

Introduction

The chemostat is a laboratory apparatus used for the continuous culture of microorganisms. We say that T has property α on Wy if there exists t ∈ ( , ) and ω ∈ Wy\Sy, such that ω – tTω ∈ Sy. Suppose that F : W → W is a compact operator, and y ∈ W is an isolated fixed point of F, such that Fy = y , let L = F (y ) is Fréchet differentiable at y , it follows that L : W → W. ([ ]) Assume that q(x) ∈ C( ), q(x) + p > on , p is a positive real constant, λ is the principal eigenvalue of the following problem:. Let λ , μ be, respectively, the principal eigenvalue of the following problem:. Has a unique positive solution, denoted by , satisfying the following properties:. Suppose b > dμ , we denote the unique positive solution by θ for the following problem:. (ii) u + υ < z, x ∈ ̄ ; Proof The proof is in [ , ], we omit it

Calculations of fixed point index

Conclusion