×

zbMATH — the first resource for mathematics

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: \[ (1)\quad ds(t)/dt=1-s(t)-x_ 1(t)p_ 1(s(t))-x_ 2(t)p_ 2(s(t)), \] \[ dx_ 1(t)/dt=x_ 1(t)[p_ 1(s(t-\tau_ 1))-1],\quad dx_ 2(t)/dt=x_ 2(t)[p_ 2(s(t-\tau_ 2)-1], \] where \(\tau_ 1\), \(\tau_ 2\geq 0\), \(s(t)=\phi (t)\geq 0\) on [-\(\tau\),0], \(\tau =\max (\tau_ 1,\tau_ 2)\), and \(x_ i(0)=x_{i0}\geq 0\), \(i=1,2\). 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:
92D40 Ecology
34D99 Stability theory for ordinary differential equations
92D25 Population dynamics (general)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37-XX Dynamical systems and ergodic theory
PDF BibTeX XML Cite
Full Text: DOI