CANADIAN APPLIED MATHEMATICS QUARTERLY 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 merry-go-round) are shown to exist for a mathe- matical model of competition between two microbial popula- tions 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 -+ m). This allows replacing a large periodic system of ordinary d~f - ferential equations by a periodic reaction-diffusion system on the circle. Rotating wave solutions of this system are approx- imated by a singular perturbation analysis.

0. Introduction. It is well known that if two different micro- bial populations compete for a single nutrient, supplied at a constant growth-limiting concentration, in a chemostat then the standard math- ematical model predicts that one of the populations will drive the other to extinction [7].However, if the nutrient concentration in the reservoir feeding the chemostat oscillates periodically at the correct amplitude and frequency, the model predicts that the two populations will coex- ist [8, 10-16, 61. In this case, the concentration of each population in the chemostat oscillates periodically at the driving frequency. In these two cases the populations compete in a single well-stirred vessel. Steady-state coexistence of two microbial populations is predicted to occur in the presence of a nutrient gradient, e.g., in a gradostat, where the limiting nutrient is supplied in constant concentration to one end of a linear array of vessels, each connected to its nearest neighbor to allow flow between vessels [9, 17, 18, 191.So far, there has been little study of the simultaneous effects of both spatial and temporal variation in

Research supported in part b NSF Grant DMS 9141550 Received by the editors on dbruary 26, 1991,and in revised form on January

Copyright 0 1 9 9 3 Rocky Mountain Mathematics Consortium

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84 H. SMITH

nutrient concentration on the outcome of two-population competition for a single nutrient in a continuous culture device. See [12] for work on single-species growth in periodically forced gradostats. The purpose of this article is t a study the effect of spatial and temporal variation in nu- trient concentration on a mathematical model of competition between two microbial populations in a gradostat.

One of the most interesting ways that a temporal and spatial nutrient variation can be combined is in the form of a rotating wave of high and low nutrient concentration. If the gradostat is configured as a circular array of vessels rather than a linear array, then a permanent rotating wave of high and low nutrient concentration, moving around the array, can easily be achieved. Each vessel of the gradostat would, in its turn, alternately experience a period of high nutrient concentration followed by a period of low nutrient concentration. Think of ponies on a merry- go-round.

A schematic diagram of the gradostat considered here appears in Figure 1. It is a circularly configured array of n vessels (n = 6 in Figure I), each fed by a reservoir, labeled R, which contains growth medium and nutrient. A collection vessel, labeled C, collects runoff from each vessel. The arrows indicate flow between vessels and all flow rates (volltime) are identical. Thus, each vessel exchanges material only with its two nearest neighbors in the array, receiving inflow from its own reservoir and sending runoff to the collecting vessel so that the volume of each vessel is maintained constant in time. Each vessel is well-stirred and contains growth medium, nutrient and the competing microbial populations. The populations compete simply by consuming nutrient. The growth medium contains all other nutrients required for the growth and maintenance of the microbial populations, at sufficient concentration so as not to be growth-limiting.

In order to fix the physical environment in which the competition takes place, it is only necessary to specify the concentration, Ci, of the nutrient in the reservoir Ri. As our aim is to create a spatially extended environment, each vessel of which receives an identical, periodically varying concentration of nutrient, the phase of which advances as we