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

Freedman, H. I.; So, Joseph W.-H.; Waltman, Paul

SIAM J. Appl. Math. 49, No. 3, 859-870 (1989).

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

Keywords:

limit cycles; discrete time delays; microbial populations; chemostat; equilibrium; stability; Hopf bifurcation; nutrient uptake; reproduction; competitive coexistence