Crowding effects on coexistence solutions in the unstirred chemostat

Abstract

This paper deals with the unstirred chemostat model with crowding effects. The introduction of crowding effects makes the conservation law invalid, and the equations cannot be combined to eliminate one of the variables. Consequently, the usual reduction of the system to a competitive system of one order lower is lost. Thus the system with predation and competition is non-monotone, and the single population model cannot be reduced to a scalar system. First, the uniqueness and asymptotic behaviors of the semi-trivial solutions are established. Second, the existence and structure of coexistence solutions are given by the degree theory and bifurcation theory. It turns out that the positive bifurcation branch connects one semi-trivial solution branch with another. Finally, the stability and asymptotic behaviors of coexistence solutions are discussed in some cases. It is shown that crowding effects are sufficiently effective in the occurrence of coexisting, and overcrowding of a species has an inhibiting effect on itself.

Keywords:

• Chemostat
• crowding effects
• the degree theory
• the generalized maximum principle
• the bifurcation theory

AMS Subject Classifications:

• 35K57
• 35B32

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work is supported by the Natural Science Foundation of China [grant number 11271236], [grant number 11401356]; the Program of New Century Excellent Talents in University of Ministry of Education of China [grant number NCET-12-0894]; the Shaanxi New-star Plan of Science and Technology [grant number 2015KJXX-21]; Fundamental Research Funds for the Central Universities [grant number GK201303008].