Exponential growth, when describing populations that double periodically (for example, bacteria), can be formulated once three parameters have been defined [NOTE that the values put onto these parameters in the discussion below are chosen arbitrarily to generate the graphs used as illustrations]:

(1) Total time of growth (mins) [t ≡ 0 .. 1000]

(2) Initial population size (where x_o is any real, positive integer)[x₀ = 1]

(3) Number of divisions per minute [r = 0.00001]

Hence, the size of the population, x, at time, t can be calculated as: x_t+1 = x_t.2^rt (equation 1)

_{which is shown in graph 1 (showing how population size varies with time)...}

In real situations this mode of population growth cannot continue indefinitely, though. There is usually some limiting factor, such as space or nutrients, that curbs the growth rate (=number of divisions per minute, r).

Monod (1949) was the first to consider this in his mathematical descriptions of bacterial growth. He assumed that it was limitation of nutrients and not cell death that ended the exponential phase of growth, and that growth is proportional to the concentration of the nutrient when looking at individual cells..

To model this it is necessary to evaluate how the growth rate varies with respect to the  concentration of limiting nutient.

This is determined experimentally and results in a graph like graph 2, described by equation 2, where:

The maximum rate of growth, R, in an unlimited concentration of nutrient is determined:   [R ≡ 0.001]

The concentration of nutrient in the system, C, varies between: [C = 0..1000]

And c is the concentration at which r = r_max/2; c = 30

Hence, the relationship can be described with a hyperbolic equation:

_{which is shown in graph 2 (which shows change in growth rate with nutrient concentration)...}

In order to link these two descriptions, we need to define how the concentration of the limiting nutrient varies with respect to time. This is not a simple matter as the concentration of the nutrient is dependent on the number of individuals in the system which is dependent on the rate of growth of each individual which is dependent on the concentration of the nutrient. Therefore, it is necessary to approximate the rate of decrease of nutrient concentration independently of the number of individuals consuming it. For this approximation we can assume that the concentration of the nutrient decreases in a exponential way.

If we define the initial concentration of the nutient:  C₀ ≡ 10000

And the rate at which it is consumed [0,1]: λ ≡ 0.99

The exponential decease in concentration can be calculated: C_t+1 = λC_t

_{which is illustrated in graph 3 (which shows change in nutrient concentration with time)...}

Then, equation 2 can be re-written with respect to t:

_{which is shown in graph 3 (which shows change in growth rate with time as nutrient is depleted)...}

And finally equation 1 can be re-written so that the growth rate is made to vary with time according to equation 3 and the population size is described by:

Thus, a curve is generated that describes the population growing at an increasing exponential rate initially, then reaching a steady state of exponential increase, and finally slowing as the concentration of nutrient becomes limiting (note that the y-axis is a log scale).

While Monod developed these descriptions for bacterial growth, they can be equally well applied to biomass evolution in fungal mycelia (where the increase in hyphal branches is equivalent to the increase in bacterial cells).

The model of hyphal branching and biomass evolution developed by Prosser and Trinci (1979) is based on the mathematics outlined above. 

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