  This paper studies a two microbial species model in competition for a single
resource in the chemostat including general interspecific density-dependent
growth rates with distinct removal rates for each species. We give the
necessary and sufficient conditions of existence, uniqueness, and local
stability of all steady states. We show that a positive steady state, if it
exists, then it is unique and unstable. In this case, the system exhibits a
bi-stability where the behavior of the process depends on the initial
condition. Our mathematical analysis proves that at most one species can
survive which confirms the competitive exclusion principle. We conclude that
adding only interspecific competition in the classical chemostat model is not
sufficient to show the coexistence of two species even considering mortality in
the dynamics of two species. Otherwise, we focus on the study, theoretically
and numerically, of the operating diagram which depicts the existence and the
stability of each steady state according to the two operating parameters of the
process which are the dilution rate and the input concentration of the
substrate. Using our mathematical analysis, we construct analytically the
operating diagram by plotting the curves that separate their various regions.
Our numerical method using MATCONT software validates these theoretical results
but it reveals new bifurcations that occur by varying two parameters as
Bogdanov-Takens and Zero-Hopf bifurcations. The bifurcation analysis shows that
all steady states can appear or disappear only through transcritical
bifurcations.
