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Growth Curves and Accuracy?

This is a discussion on Growth Curves and Accuracy? within the Beginner Freshwater Aquarium forums, part of the Freshwater Fish and Aquariums category; --> Okay, I got some help from some other users and they suggested that I use a combination of formulas to eliminate the time factor ...

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Growth Curves and Accuracy?
Old 01-11-2010, 10:13 PM   #11
MOA
 
Okay,

I got some help from some other users and they suggested that I use a combination of formulas to eliminate the time factor that was limiting my calculations. Their suggestion morked and the result is as attached.

The formulas that resulted in this formula are both commonly used in aquaculture and have been used for many years. The first is the logistic formula:

dM/dt = k*M*(1-M/n) Where dM/dt is the derrivative of Mass with respect to time, M is mass, t is time, k is the growth rate, and n is the limiting size (the maximum mass the animal can achieve).

The second is the von Bertalanffy equation of length approximation:

dL/dt = c*(m-L) Where dm/dt is the derivative of length with regard to time, c is a growth coefficient, m is the maximum length of the animal, and L is the current length of the animal.

When these two curves are combined, the result is the graph shown.

MOA
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Old 01-12-2010, 01:37 AM   #12
 
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I'm not sure I understand that graph. I'm assuming the x-axis is time, but what's your y-axis? I guess I'm just confused as to how you're combining the equations since one deals with dm/dt and the other dL/dt.
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Old 01-12-2010, 01:13 PM   #13
MOA
 
Oops, Sorry.

Same axes as before: horizontal is length, vertical is biomass. As to how time is eliminated, the two differential equations are solved explicitly and then one is substituted into the other. The two equations come out to be:

M = n*C1*e^(k*t)/(C1*e^(k*t)-1) where C1 is a constant that makes an initial condition true and e is Euler's constant (2.718...).

L = m+c2*e^(-c*t) where C2 is a constant that makes an initial condition true and e is Euler's constant.

If one of these is solved for time (I used the second one), the result can be substituted into the other to replace time. This will substitute one independent variable for another. Subsequently, what was once two time-dependent formulas can become one length-dependent formulas.

Sorry for not explaining.

MOA
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Old 01-12-2010, 01:24 PM   #14
 
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MOA,
I just wanted to pop in quick and let you know I havne't forgotten you or your thread, I just have not had time to sit and write up what I had intended for you about your web page and the misleading and a few incorrect statements I found in it.
In catching up on the thread I have to agree with Pasfur that this information is very difficult to understand and not very interesting to read. Knowing that a great many of beginner fish keepers are under the age of 18 would concern me if that was presented to them. I would expect very few to comprehend the information.

I will get the info to you as soon as my time schedule allows. Since there is so much I would like to point out to you, is it possible for me to bring just a few things here to the forum and then to compose a document page that is more thorough that can be emailed directly to you?

I appreciate your patience! Please know you are on my list of priorities, but medical emergencies have to come first.
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Old 01-12-2010, 04:02 PM   #15
 
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Quote:
Originally Posted by MOA View Post
Oops, Sorry.

Same axes as before: horizontal is length, vertical is biomass. As to how time is eliminated, the two differential equations are solved explicitly and then one is substituted into the other. The two equations come out to be:

M = n*C1*e^(k*t)/(C1*e^(k*t)-1) where C1 is a constant that makes an initial condition true and e is Euler's constant (2.718...).

L = m+c2*e^(-c*t) where C2 is a constant that makes an initial condition true and e is Euler's constant.

If one of these is solved for time (I used the second one), the result can be substituted into the other to replace time. This will substitute one independent variable for another. Subsequently, what was once two time-dependent formulas can become one length-dependent formulas.

Sorry for not explaining.

MOA
Ah ok, now it makes complete sense (including the shape of the function).
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Old 01-12-2010, 04:59 PM   #16
MOA
 
iamntbatman,

Yeah, I kinda got ahead of myself. No wonder it didn't make sense. :(


bettababy,

Of course I understand that other things do come first (I hope the world doesn't revolve around fish, lol). When you are ready, you can post here on the forum or you can email me. Also, I am aware that most of my stuff is not much fun to read as the intricacies of stocking dynamics are not very exciting in and of themselves. I did not try to make something that was fun. Instead, it serves to let people know why I am doing what it is I am doing. No less, if you have suggestions on how to curtail the boring stuff without losing information I am all ears :).

As to comprehension level of younger aquarists, my stuff is definitely beyond their scope. My career is teaching, actually, so I am very aware that most of my material is not suitable for casual underage aquarists. I have actually thought about a MOA's: For Kids" section but found the idea to be a little contradictory to my goals. Again, I am open to suggestions in this regard.

As before, I look forward to your input with anticipation.

MOA
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Old 01-13-2010, 02:28 PM   #17
 
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Ok, my first contributions to the web page I read entirely are as follows... (please bear with me as I post a statement or 2 at a time, along with my input, there were quite a few I felt needed commenting on)

"Yes, some fish can survive without filters, but there are extremely few species that can live out their full life expectancy without a filter."

This is untrue. Whether we add a "filter" or not, there is always biological filtration happening in an aquatic environment. There are 3 types of filtration. There is biological, mechanical, and chemical filtration. Canister filters such as Eheim focus filtration on biological, which is a natural form of filtration, but allow for the use of filter medias to add chemical and mechanical filtration. If a tank is not over crowded and maintenance is sufficient, there are many species of fish that can thrive without mechanical and chemical filtration. If you check into the old German methods of filtration, their focus for many yrs was entirely on biological filtration and today still stresses it over other methods. A fish’s natural environment also relies entirely on biological filtration.

Filtration and circulation should also be defined here as two separate functions.



"Bacteria take in the wastes generated by the fish and convert them into less harmful substances; the bacteria do the work of keeping the aquarium water relatively clean (though bacteria do not remove all wastes). Therefore, it stands to reason that the better suited the environment is to the needs of filter bacteria, the more fish waste an aquarium can handle."

This is a very misleading statement. While bacteria do convert waste, what they convert it to is not a less harmful substance. Ammonia is converted to nitrite, which is more toxic to the animals than ammonia. Nitrite is converted to nitrate, and is only less harmful if in small amounts. There is a limit to how much bacteria any given aquarium can maintain based on surface area and food supply. When a system is overloaded with waste and the build up is beyond what the maximum bacteria culture in that system can handle, ammonia begins to build. If given enough time the present bacteria culture can break it all down, but the process takes longer, causing spikes in ammonia. These spikes in ammonia will cause spikes in nitrite as the ammonia is broken down, and the same on to nitrate. These spikes are highly toxic to fish even in an established aquarium that has been apparently healthy for many years.

Bacteria also cannot break down all forms of waste, such as organics. Fish waste but also food and plant waste will contribute organics to any aquarium system. It would be impossible to accurately predict the total amount of waste in any system before that system exists and is tested regularly. An aquarium that contains 2 fish and heavy feedings can produce as much or more waste than a same sized tank containing 5 of the same kind of fish that are very lightly fed or offered different foods. Even a tank kept in a dusty house will differ from the exact same set up in a house that has less dust in the air. Just as no 2 people, no 2 dogs, no 2 any living creature produce the exact same amount of waste in a given period, nor do fish. At best, predicting something of this nature is simply an educated guess, never an exact.


I will contribute more as I find the time. Please don't take this as anything more than "constructive criticism" meant to do two things... contribute to your information content and explain my stance on your overall goal, and the fact that at present, evolution and technology what they are, making such exact predictions is not possible. There may come a time in the distant future where technology exists to measure the many factors that would need to be taken into account, but to date it does not exist. Thus, my overall impression of what you will have as an end result is no different than other prediction methods for aquarium keeping. Each has vast limits and needs common sense applied to be even close to accurate, and each individual situation should be handled as such... the individual situation that it is.



I do want to note that I applaud your hard work and diligence in contributing to scientific research that may some day allow us such abilities. I hope my contributions can help as well.


Have a good day! I will be back with more as I find time.

Last edited by bettababy; 01-13-2010 at 02:33 PM.. Reason: missed a few words
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Old 01-13-2010, 06:02 PM   #18
MOA
 
Don't have time to read all of this now, but it looks like you have provided some awesome stuff. be back soon to read it all.

MOA
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Old 01-14-2010, 12:49 AM   #19
MOA
 
bettababy,

I am very impressed with your own diligence. Your remarks are proving to be most thorough--a truly delightful chance from my common experiences. I will now offer a defense of my statements, but please be aware that I apologize for placing you at an unfair disadvantage. You see, most of my defense, if one could call it that, centers on the entire scope of my website and not on individual statements. As such, I do realize that my defense is artificial in that I could not possibly expect you to be familiar with all of the pages and components yet. As such, your remarks, while they may not indicate a lack of understanding on my part, do showcase one of my greatest flaws: fractured presentation. Some of the data that would clarify my statements are, unfortunately, on other pages of the website, doing absolutely no good to the reader (an error I will correct shortly). For the time being, I will offer this diminutive defense of my inept actions (with all due apologies and thankfulness for your wonderful insights):


Point 1) "Whether we add a 'filter' or not, there is always biological filtration happening in an aquatic environment. There are 3 types of filtration. There is biological, mechanical, and chemical filtration."

I have to disagree, but only on a technicality. You see, the fundamental truth is that biological activity will always occur, but what is debatable is whether or not it can be called filtration. Imagine, if you will, a tank without any device designed to filter the water in any way. This means that even circulation devices would not be present (why is a matter I will address shortly). In such an environment, barring odd occurrences, the water would stagnate. Now, stagnant water is not without life and life forms will indeed start converting the raw organic matter into various less complex compounds. This fact is not in dispute, but the nature of what these compounds will be is. Stagnant water tends to have low levels of dissolved oxygen and microbes without oxygen tend to produce sulfur-based compounds. Unfortunately, sulfur-based compounds are toxic and our imaginary fish tank would not be a good candidate for supporting fish unless some mechanism was implemented to either remove the noxious compounds or to introduce oxygen. If oxygen can be supplied to the system, then the bacteria will switch over to aerobic processes and the net result will be a reduction in the lethality of the substances present.

Using this little example it can be seen that filtration, barring chemical filtration for the moment (as mechanical does little for the fish as most fish are used to living in water with a relatively high concentration of particulate matter), is directly connected to the concept of circulation. In fact, without circulation to transport oxygen and deliver nutrients, filtration that is beneficial to the fish would be nigh impossible. Consequently, it is technically an error to state that filtration and circulation should be separate concepts: circulation is filtration. This is also true of chemical filtration (though I do not much care for chemical methods) in that circulation is required to deliver the water that will be treated.

Despite my odd rebuttal, my original statement is still untrue in that I have left out a very major consideration that can trump any sort of filtration. That is, wastes can always be removed directly by large water changes. If water changes are large enough and frequent enough, then filtration of any sort would not be needed at all. At the same time, I did not feel it prudent to advise new aquarists of this approach as it is frough with problems and relies on the aquarist's unwavering dedication.

Much of the details of these intracacies are left to my "Biological Nitrogenous Cycle" page. It would be prudent of me to link them so that readers can explore the idea more thoroughly.


Part 2 to come soon.

MOA

P.S., I will try to change the wording to be a bit more reflective of your concerns, which are exceptionally valid and helpful.
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Old 01-14-2010, 01:47 AM   #20
MOA
 
Part 2) "While bacteria do convert waste, what they convert it to is not a less harmful substance."

This is probably more of a matter of point of view than of fact. Let me illustrate: It is true that seat belts save lives but it is also true that seat belts have been responsible for several deaths. No less, people tend to use the statistic that seat belts more often save a life than end it as justification for always wearing a seat belt. The difference is whether or not you are concerned with a general or specific trend. Specifically, the conversion of specific groups of bacteria do differ in lethality. Sometimes a conversion does not have a reliable ratio of lethality (for some species of animals, nitrate is more toxic than ammonia but the opposite can be true for other animals), but a prolonged series of conversions often does. The net effect of a healthy and properly balanced population of bacteria in a home aquarium is nitrate (in rare cases, nitrogen) and it is much less lethal than the initial culprit substance (ammonia).

"Nitrite is converted to nitrate, and is only less harmful if in small amounts."

Sorry, but I am going to have to see a source for this statement. Last time I checked, the LD50 (lethal dose for 50 percent of a sample) of nitrite and ammonia are both less than 20ppm (closer to 5-10ppm in many cases) while the LD50 of nitrate can be over 400ppm (not 40, 400). As a result of this ratio, it is inaccurate to say that nitrate is only less harmful in small amounts. In truth (unless you have a source, which you may), nitrate is always less harmful than nitrite (or ammonia) since its LD50 is so high. Toxicity is always measured as a ratio yet your statement implies that the amount of nitrate has to be smaller than the amount of nitrite despite that the opposite is generally true. The amount of nitrate seen in what are often considered healthy aquariums is not small, often exceeding 20ppm. I hope that aquarists keep their nitrate levels low, but idealism does not change nitrate's toxicity level.

"When a system is overloaded with waste and the build up is beyond what the maximum bacteria culture in that system can handle, ammonia begins to build. If given enough time the present bacteria culture can break it all down, but the process takes longer, causing spikes in ammonia. These spikes in ammonia will cause spikes in nitrite as the ammonia is broken down, and the same on to nitrate. These spikes are highly toxic to fish even in an established aquarium that has been apparently healthy for many years."

I do not disagree with this statement in general, though I do have to wonder how germane it is to the subject matter. My explanation on that page was simply meant to tell people why my filtration formulas use medium, aeration, and turnover as methods to judge whether or not a system is adequate for the bioload placed on it. Additionally, my statements on that page imply that the resources available to the bacteria are used a s a limiting factor, meaning that it would be relatively difficult to create a situation in which ammonia or nitrite levels would spontaneously spike; spikes are, as you stated, generally only present when when the bioload exceeds the capacity of the system--the very thing my stocking calculator tries to prevent. Moreover, most of my stocking calculators are designed with the end-game in mind and do not account for sudden jumps in and of themselves but my introductory page states:

"Fish are added to the aquarium in small groups over the course of several weeks. Generally, new aquarists do not need to add any more than one quarter of all the fish they plan on keeping in a period of two weeks. This means that most aquariums will take a couple months to completely stock with fish. The only notable exception to this guideline is shoaling fish, which do best if added as an entire shoal rather than fragments. It should also be noted that some cycling techniques do allow for complete stocking in as little as one week, but such techniques may not be suitable for beginners."

This statement is there to help keep people from assuming that rapid jumps are okay. As to overfeeding (which can also cause spikes), the spreadsheets (the new ones at least) require feeding information to ensure that the amounts do not generally exceed demand.

"Bacteria also cannot break down all forms of waste, such as organics [BTW, ammonia, nitrite, and nitrate are considered organic and thus are a glaring exception to this statement]. Fish waste but also food and plant waste will contribute organics to any aquarium system. It would be impossible to accurately predict the total amount of waste in any system before that system exists and is tested regularly. An aquarium that contains 2 fish and heavy feedings can produce as much or more waste than a same sized tank containing 5 of the same kind of fish that are very lightly fed or offered different foods[spreadsheets have a section for these types of issues too]. Even a tank kept in a dusty house will differ from the exact same set up in a house that has less dust in the air. Just as no 2 people, no 2 dogs, no 2 any living creature produce the exact same amount of waste in a given period, nor do fish. At best, predicting something of this nature is simply an educated guess, never an exact."

First point, predicting the total amount of possible waste is relatively easy. There is a formula that the spreadsheet uses called the RWI (Remaining Waste Index) which accounts for everything that could possibly enter the aquarium. No less, it is much more difficult to parse exactly what the remaining waste index is comprised of. Is it mostly nitrates? Mostly inert? GIH? Very hard to tell, but the limiting factors that are moreso obvious, as well as the index itself, tend to be much better predictors of the "safety zone" than outlying factors. In a sense, my spreadsheets do not try to see where your maximum limit is, but rather describe the safe limit. I'm not interested in letting new aquarists push the boundaries of fishkeeping, so my spreadsheets correct for most of those types of possibilities by finding a point below the exact limit. As long as my spreadsheet aim below the maximum, there is no point in figuring out exactly where that maximum is; I have no intention of being exact as exact is dangerous and irresponsible.

MOA
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