Global dynamics and bifurcation analysis of a chemostat model with obligate

We propose a system of differential equations modeling the competition between two obligate mu-5 tualistic species for a single nutrient in a chemostat. Each species promotes the growth of the other, 6 and growth occurs only in the presence of its partner. The three-dimensional model incorporates 7 interspecific density-dependent growth functions and distinct removal rates. We perform a math-8 ematical analysis by characterizing the multiplicity of equilibria and deriving conditions for their 9 existence and stability. Using MatCont, we construct numerical operating diagrams in the pa-10 rameter space of dilution rate and input substrate concentration, providing a global view of the 11 qualitative dynamics of the system. One-parameter bifurcation diagrams with respect to the input 12 substrate then reveal a variety of dynamical transitions, including saddle–node, Hopf, limit point 13 of cycles (LPC), period-doubling (PD), and homoclinic bifurcations. When mortality is included, 14 the system exhibits a richer dynamical repertoire than in the mortality-free case, with stable and 15 unstable periodic orbits, tri-stability between equilibria and limit cycles, and several codimension-16 two bifurcations, including Bogdanov–Takens (BT), cusp of cycles (CPC), resonance points (R1 and 17 R2), and generalized Hopf (GH) points. These features allow coexistence not only around positive 18 equilibria but also along stable limit cycles, reflecting more realistic ecological dynamics. In contrast, 19 neglecting mortality restricts coexistence to equilibria only. Overall, this study highlights the critical 20 role of mortality in shaping complex dynamics in obligate mutualism, producing multistability and 21 oscillatory coexistence patterns that may better represent natural microbial or ecological systems.