#5 | Thinking from Mathematical Formulas! The Behind-the-Scenes of Medaka Keeping Not Taught in Elementary School

In the previous #4,
we introduced how to set up a tank while measuring water quality.

This time, we will take a step further and
think about the phenomena occurring inside the tank using mathematical formulas.

The formulas handled here are
empirical formulas (approximations) derived by our predecessors through repeated, extensive basic experiments to model the complex natural world.
Although the constants used vary depending on the environment, looking at the structure of the formulas helps you understand how to manage a tank!

First, we will start with the basics:
how to formulate chemical reactions.

We will use the following hydrogen combustion taught in junior high school as an example.

$${2\text{H}_2 + \text{O}_2 \rightarrow 2\text{H}_2\text{O}}$$

Although the actual hydrogen combustion process is much more complex,
we will treat it here as a simple chemical reaction.

At this time, if the rate at which water ($${\text{H}_2\text{O}}$$) is generated is expressed in molar concentration ($${[\text{mol/L}]}$$), it can be written as the following approximation.

$${\frac{d[\text{H}_2\text{O}]}{dt} = k [\text{H}_2]^2 [\text{O}_2]}$$

The rate at which ammonia ($${\text{NH}_3}$$) in the tank is decomposed can also be thought of in terms of molar concentration in the same way.

Since the counterpart is a living organism (bacteria), we use an approximation formula called the Monod type, which is also standard in fields such as wastewater treatment.

The rate at which ammonia is consumed can be formulated as follows.
※Please refer to article #2 for the chemical formula.

$${\frac{d[\text{NH}_3]}{dt} = - \frac{1}{Y} \cdot \left( \mu_{max} \frac{[\text{NH}_3]}{K_s + [\text{NH}_3]} \cdot \frac{[DO]}{K_O + [DO]} \right) \cdot X}$$

• $${[DO]}$$:
  Dissolved oxygen concentration (unit: mg/L)

• $${X}$$:
  Bacteria concentration (number)

• $${\mu_{max}, K_s, K_O, Y}$$:
  Constants derived from the experiments of our predecessors.
  These values differ depending on the conditions.

  • $${\mu_{max}}$$ (Maximum specific growth rate):
    The maximum processing speed that bacteria possess.

  • $${K_s}$$ (Half-saturation constant, ammonia):
    How well they latch onto ammonia.
    The smaller this value, the more it indicates that they are greedy bacteria that react sensitively and eat even when there is only a tiny amount of ammonia, thinking 'It's food!'

  • $${K_O}$$ (Half-saturation constant, oxygen):
    Resistance to low-oxygen conditions.
    The smaller this value, the more it indicates that the bacteria are energetic even with little oxygen.

  • $${Y}$$ (Yield coefficient):
    The size of the bacteria's stomach.
    The smaller this value, the more it indicates that they are big eaters that consume large amounts of ammonia to maintain the same amount of bacteria.

Looking at this formula, there are things you can understand.

• If there are zero bacteria, ammonia will not decrease:
  X (number of bacteria) is multiplied at the right end of the formula.
  You can see that in the early stages of setup where this X is close to zero, the result on the right side (the speed at which ammonia decreases) becomes almost zero, no matter how good the other conditions are.

• If there isn't enough oxygen, ammonia becomes harder to reduce:
  The formula includes [DO] (dissolved oxygen concentration) as a multiplier.
  You can see that if there is too little oxygen, this term approaches zero, making it harder for ammonia to decrease.

• There is a limit to how fast ammonia can decrease:
  No matter how high you make [NH3] or [DO], the fraction containing them simply approaches '1 (the maximum value)'.
  From this, we can see that the speed at which ammonia decreases has a set limit, known as μmax (maximum processing speed).

• If bacteria are big eaters, ammonia decreases faster:
  The 1/Y in the formula represents the size of the bacteria's stomach. The bigger the appetite of the bacteria, the faster they will reduce ammonia.
  In particular, this term shows that the type of bacteria is important.

I think there are other things you can understand as well.
Please take a look at the formula and think about it in various ways.

✔︎ Conclusion

By learning the formulas packed with the wisdom of our predecessors,
hasn't the way you look at your aquarium changed a little?

For example,
in an environment with few bacteria,
no matter how much you adjust the conditions, ammonia will not decrease as much as you might expect.

Also,
aeration plays a role in supporting specific terms.

To decide 'how to deal with it',
you need some kind of reason.

As one of those reasons,
why not try using these formulas packed with the wisdom of our predecessors?

Instead of trial and error in confusion,
you will be able to make decisions based on evidence!

From #1 to #5 (this article),
we have covered everything from failure cases to concrete setup methods.

The only things we haven't covered are daily feeding and water changes.

However, there are now many explanatory videos on YouTube.

I think it's a good idea to refer to those,
and improve upon them to find a method that works best for you.

Whether it's planted tanks or saltwater fish, it's a deep world if you get into it.

However,
the foundation for any aquarium
should be the content covered in this series.

First of all,
why not start one step at a time from here?

If you found this even a little helpful,
please give it a like or follow,
as it encourages me to create the next article!

• Mathematical theme: #Aquarium#Tank

• Subject: #Bacteria#NitrogenCycle#NitrogenCirculation#BiologicalFiltration#Biofiltration

• Usage Scenario: #IndependentResearch#Relearning#Reviewing#ClassroomTopics

• Topic: #ILoveLivingThings#ILoveInquiryBasedLearning