Mathematical analysis of a competition model with mutualism

We perform the mathematical analysis of a model describing interaction two species in chemostat, involving competition and mutualism, simultaneously. The is five-dimensional system differential equations with nonlinear growth functions. give comprehensive description dynamics by determining analytically existence local stability conditions all steady-states, considering large class rates. prove that there exists unique stable coexistence steady-state under which bistability can occur. bifurcation diagrams operating showing rich behavior system.