## Limits to Growth

If E. coli can double its population every 20 min, why has the world not yet been overrun by E. coli? The answer lies in resource limitation. To understand how resource limitation governs the population dynamics of microorganisms, we first focus on the growth of a single species on a single resource. Let us consider E. coli growing on fructose (Figure 1a). The growth rate of E. coli increases with fructose concentrations. However, at high fructose concentrations the growth rate reaches a maximum value. The bacterium cannot grow faster. This is known as the maximal specific growth rate of the species. The Nobel Prize winner Jacques Monod has captivated the growth rate of microorganisms by the following equation:

Here, p is the specific growth rate, pmax the maximum specific growth rate, R the concentration of the resource (fructose in our example), and K is known as the halfsaturation constant. When the resource concentration equals K, the growth rate equals half the maximum growth rate. This equation is now known as the Monod equation.

Fructose availability (ig l-1)

Fructose availability (ig l-1)

 Population \ f \ / y \ Jv Fructose Time Figure 1 Population dynamics of the bacterium Escherichia coli growing on fructose. (a) Specific growth rate (solid line) and specific loss rate (dashed line) of E. coli as a function of fructose availability. The point R* indicates fructose availability at equilibrium. (b) Time course of E. coli (solid line) and fructose availability (dashed line). Naturally, microbial populations also face losses by cell death, grazing, virus attack, etc. The net increase of a population depends on the balance between growth and losses: where N is the population density, t is time, and m is the specific loss rate. Things get a little more complicated, if we consider that the concentration of the resource (e.g., fructose) is likely to change in time as well. In general, a resource is supplied at a certain supply rate and diminished through natural losses (e.g., washout, sedimentation, and chemical transformations) and through consumption by species. Hence, a simple model of the dynamics of the resource might look like this: where S is the supply rate, D the dilution rate, and Y the yield (i.e., the number of individuals produced per unit of resource consumed). Equations [2] and [3] reveal the close link between species dynamics and resource dynamics; the growth rate of a species depends on resource availability. At the same time, resource availability is diminished through consumption by the species. Figure 1b shows the population dynamics predicted by eqns [2] and [3] for E. coli growing on fructose. Initially, fructose concentrations are high, and therefore, the population density of E. coli increases. During growth, E. coli consumes fructose. When fructose concentrations diminish, the growth rate of E. coli declines. At a certain point, E. coli has depleted fructose to such low levels that the growth rate of E. coli has been reduced to its loss rate. The population of E. coli cannot increase any further. An equilibrium (also called 'steady state') has been reached. The resource concentration at this equilibrium is called the critical resource concentration, abbreviated as R*. For resource concentrations above R*, there is sufficient fructose and the E. coli population will increase. For resource concentrations below R*, E. coli lacks food, and the population will decline.