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Coexistence in a model of competition in the chemostat incorporating discrete delays. (English) Zbl 0676.92013

Consider the following model of two microbial populations competing for a single nutrient in a chemostat with delays in uptake conversion:

(1)ds(t)/dt=1-s(t)-x 1 (t)p 1 (s(t))-x 2 (t)p 2 (s(t)),
dx 1 (t)/dt=x 1 (t)[p 1 (s(t-τ 1 ))-1],dx 2 (t)/dt=x 2 (t)[p 2 (s(t-τ 2 )-1],

where τ 1 , τ 2 0, s(t)=ϕ(t)0 on [-τ,0], τ=max(τ 1 ,τ 2 ), and x i (0)=x i0 0, i=1,2. After transforming the model, the authors give an analysis of a submodel, showing that the equilibrium in the interior of the plane may change its stability as a function of a delay parameter, and lead to a Hopf bifurcation. Numerical evidence is given. The work shows that in the chemostat a delay between nutrient uptake and reproduction can produce competitive coexistence.

Reviewer: Bingxi Li

MSC:
92D40Ecology
34D99Stability theory of ODE
92D25Population dynamics (general)
34C05Location of integral curves, singular points, limit cycles (ODE)
37-99Dynamic systems and ergodic theory (MSC2000)