# Mathematical modeling of bacteria decay

If our mathematical model for the decay of a bacteria colony is:

$$100\left (\frac {1} {2} \right) ^ {n-1}$$ =Number of Bacteria

n is the number of the elapsed hours starting with 100 bacteria in a Petri dish, Such that $$n\geq 1$$

Then according to our model, after 6hrs our Petri dish should only contain 3.125 live bacteria. However this doesn't make much sense; we should only have a whole number of live bacteria at any given moment. How does this real world constraint on the mathematical model help solve:

$$100\left (\frac {1} {2} \right) ^ {n-1}$$ = 0

• Does $n$ denotes the time for the living bacteria? Jan 11, 2019 at 17:47
• Generally, such model represents the pourcentage of still living bacteria. It must be multiplied by the (huge) number of initial bacteria. Moreover, it is a model, an approximation (evaluation) of real number Jan 11, 2019 at 18:07
• Practically it is irrelevant if the number of bacteria is $1.4632\cdot 10^{6}+0.321$ or $1.4632\cdot 10^{6}$ at a particular time. Jan 11, 2019 at 18:31
• I've edited the question @Dr.SonnhardGraubner Jan 11, 2019 at 18:32
• @AshrafBenmebarek Is it a real life experiment starting with 100 bacteria? I don´t think so. Jan 11, 2019 at 18:38

In general a mathematical model is not able to completely describe every aspect of a system with $$100\%$$ accuracy. The idea of mathematical modeling is to make a model that is "good enough" for whatever it is modeling. For the case of modeling bacteria, note that there are usually enormous numbers of bacteria in a petri dish, perhaps on the order of billions. So when a biologist wants to know how many bacteria are in the dish, it makes very little difference whether there are say $$1,000,000,000$$ or $$1,000,000,001$$. Thus the choice was made to represent the number of bacteria using a continuous rather than discrete value, even though there can never be a fractional amount of bacteria. Now you could do an arithmetic operation such as a floor or ceiling function to give yourself an integer number of bacteria, but this is not really relevant or necessary when "ballparking" a solution by using the model.
As for the equation $$100(\frac{1}{2})^{n-1} = 0$$, note that this is never zero for any positive value of $$n$$. However $$\lim_{n \to \infty} 100(\frac{1}{2})^{n-1} = 0$$, and you can interpret this as all of the bacteria dying after a long amount of time.
The reason that this example seems strange is that this model is being applied to a tiny sample of bacteria. However also notice that if $$n$$ is the number of elapsed hours, then the initial number of bacteria is actually $$100*(\frac{1}{2})^{-1} = 200$$, not $$100$$. (Just plug in $$n = 0$$).