The effect of parameters on positive solutions and asymptotic behavior of an unstirred chemostat model with B\u2013D functional response

Xiaozhou Feng,Suping Sun,Tongqian Zhang,Xiaomin An

doi:10.1186/s13662-018-1587-x

Abstract

This paper deals with the effect of parameters on properties of positive solutions and asymptotic behavior of an unstirred chemostat model with the Beddington–DeAngelis (denote by B–D) functional response under the Robin boundary condition. Firstly, we establish some a priori estimates and a sufficient condition for the existence of positive solutions (see (Feng et al. in J. Inequal. Appl. 2016(1):294, 2016)). Secondly, we study the effect of the small parameter k_{1} and sufficiently large k_{2} in B–D functional response, which shows that the model has at least two positive solutions. Thirdly, we investigate the case of sufficiently large k_{1}. The results show that if k_{1} is sufficiently large, then the positive solution of this model is determined by a limiting equation. Finally, we present an asymptotic behavior of solutions depending on time. The main methods used in this paper include the fixed point index theory, bifurcation theory, perturbation technique, comparison principle, and persistence theorem.

Highlights

In [1], the coexistence of an unstirred chemostat model (1) with B–D functional response is established by fixed point index theory, but in the present paper, we investigate the effect of parameters on the multiplicity and stability of positive solutions of equilibrium state model of (1); the asymptotic behavior of solutions of (1) is established, which further enrich the results for system (1)
The results show that if k1 is sufficiently large, the positive solution of this model is determined by a limiting equation
6 Conclusion This paper deals with plasmid-bearing and plasmid-free models in the unstirred chemostat with the Beddington–DeAngelis functional response

Summary

Introduction

The chemostat is a very important resource-based model for the continuous culture of competition microorganisms and a standard model for the laboratory apparatus on bioreactor, which have been studied from various views such as population dynamics and species interactions [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. We are concerned with the following unstirred chemostat model with the B–D functional response under homogeneous Robin boundary condition in a bounded domain :. This paper deals with plasmid-bearing and plasmid-free models in the unstirred chemostat with the B–D functional response under a homogeneous Robin boundary condition. In [1], the coexistence of an unstirred chemostat model (1) with B–D functional response is established by fixed point index theory, but in the present paper, we investigate the effect of parameters on the multiplicity and stability of positive solutions of equilibrium state model of (1); the asymptotic behavior of solutions of (1) is established, which further enrich the results for system (1). 3, we study the effect of the small parameter k1 and sufficiently large k2 in B–D functional response, which proves that the model has at least two positive solutions.

Preliminaries and lemmas

The effect of mutual interference between predators

Conclusion

Methods