1

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

Ashraf Benmebarek
  • 191
  • 6
  • Does $n$ denotes the time for the living bacteria? – Dr. Sonnhard Graubner Jan 11 '19 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 – Damien Jan 11 '19 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. – callculus42 Jan 11 '19 at 18:31
  • I've edited the question @Dr.SonnhardGraubner – Ashraf Benmebarek Jan 11 '19 at 18:32
  • @AshrafBenmebarek Is it a real life experiment starting with 100 bacteria? I don´t think so. – callculus42 Jan 11 '19 at 18:38
  • It's not @callculus I'm more focused on the mathematics behind modeling such experiments and not the experiments themselves. – Ashraf Benmebarek Jan 11 '19 at 18:44
  • @AshrafBenmebarek These are models are approximations and you have to check if the conditions for applying such models are fulfilled. For instance, is it reasonable to assume that the growth factor is $0,5$? – callculus42 Jan 11 '19 at 18:48
  • Indeed, I couldn't agree more. The thing is, I'm learning about sequences -just the basics- and I've encountered this question during my studies, so, I'm not really concerned with details. @callculus – Ashraf Benmebarek Jan 11 '19 at 19:03

1 Answers1

1

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 $n = 0$).

J P
  • 759
  • +1 I would stress even more that this is quite ridiculous presented as a "real world example" of exponential decay. – Ethan Bolker Jan 11 '19 at 19:39