A competition model with dynamically allocated toxin production in the unstirred chemostat

Hua Nie,Jianhua Wu,Sze-Bi Hsu

doi:10.3934/cpaa.2017066

Abstract

This paper deals with a competition model with dynamically allocated toxin production in the unstirred chemostat. First, the existence and uniqueness of positive steady state solutions of the single population model is attained by the general maximum principle, spectral analysis and degree theory. Second, the existence of positive equilibria of the two-species system is investigated by the degree theory, and the structure and stability of nonnegative equilibria of the two-species system are established by the bifurcation theory. The results show that stable coexistence solution can occur with dynamic toxin production, which cannot occur with constant toxin production. Biologically speaking, it implies that dynamically allocated toxin production is sufficiently effective in the occurrence of coexisting. Finally, numerical results illustrate that a wide variety of dynamical behaviors can be achieved for the system with dynamic toxin production, including competition exclusion, bistable attractors, stable positive equilibria and stable limit cycles, which complement the analytic results.

Highlights

IntroductionThe chemostat is a basic resource-based model for competition in an open system and a standard model for the laboratory bio-reactor, which plays an important role in the study of population dynamics and species interactions (see, e.g., [14, 27])
The chemostat is a basic resource-based model for competition in an open system and a standard model for the laboratory bio-reactor, which plays an important role in the study of population dynamics and species interactions.The study on the problem of the influence of toxicants both on the growth of one population and on the competition of two species for a critical nutrient has received considerable attention in the past decades
The focus of this study is to investigate the dynamical behavior of the system (1)(3) in combination with the effects of dynamically allocated toxin production and diffusion, and to explain the coexistence of two species in competition on a single resource in the unstirred chemostat

Summary

Introduction

The chemostat is a basic resource-based model for competition in an open system and a standard model for the laboratory bio-reactor, which plays an important role in the study of population dynamics and species interactions (see, e.g., [14, 27]). The focus of this study is to investigate the dynamical behavior of the system (1)(3) in combination with the effects of dynamically allocated toxin production and diffusion, and to explain the coexistence of two species in competition on a single resource in the unstirred chemostat To this end, we assume that the function K(u, v) satisfies the hypotheses (H1) : K(u, v) is C1 continuous in R+ × R+, where R+ = [0, +∞); (H2) : 0 ≤ K(u, v) < 1 for any u, v ∈ R+; (H3) : K(0, 0) = 0, K(u, v) > 0 for u > 0, v > 0, and Kv(0, v) ≥ 0 for any v ∈ R+.

Note that

Direct computation leads to

In view of the spectral radius r d

Noting that the first eigenvalue of the eigenvalue problem