Volume 1, Number 1, Winter 1993
PERIODIC ROTATING WAVES
IN A MODEL OF MICROBIAL COMPETITION
IN A CIRCULAR GRADOSTAT
HAL L. SMITH
Abstract. Periodic rotating waves of discrete type
(ponies on a merrygoround) are shown to exist for a mathematical
model of competition between two microbial populations
for a single nutrient in a circularly configured array of n
vessels, a gradostat, in response to a periodic rotating wave of
nutrient concentration in the reservoirs feeding the vessels of
the gradostat. These periodic rotating waves of nutrient and
microbial population concentrations represent coexistence of
the two populations. For gradostats consisting of a moderate
to a large number n of vessels, the rotating wave solutions are
approximated by passage to the continuum limit (n → ∞).
This allows replacing a large periodic system of ordinary differential
equations by a periodic reactiondiffusion system on
the circle. Rotating wave solutions of this system are approximated
by a singular perturbation analysis.
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