Tropical Fish Keeping - Aquarium fish care and resources » Freshwater Fish and Aquariums » Beginner Freshwater Aquarium » Growth Curves and Accuracy?

Growth Curves and Accuracy?
01-11-2010, 11:13 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
Attached Images
 curve2.jpg (4.9 KB, 31 views)

 01-12-2010, 02:37 AM #12 iamntbatman 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.
 01-12-2010, 02:13 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
01-12-2010, 05:02 PM   #15
iamntbatman

Quote:
 Originally Posted by 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
Ah ok, now it makes complete sense (including the shape of the function).

 01-12-2010, 05:59 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
 01-13-2010, 03:28 PM #17 bettababy 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 03:33 PM.. Reason: missed a few words
 01-13-2010, 07:02 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