This model is concerned with simulating a tank containing Talapia. The main focus will be dynamic modelling of the oxygen content of the aquarium. A constant temperature throughout the tank is assumed.
The setup:
In this simulation we have one aquarium which receives water at a set rate and loses water via overflow at the same rate. At the starting conditions the water is assumed to be in equilibrium with the atmosphere with concern to oxygen. This aquarium can be occupied by any number of fish. All fish start at a average weight of 3.8g.
Fish respiration:
While these fish are in the tank they respire at a certain rate which is dependent on the temperature, the weight of the fish and the current concentration of oxygen. This rate will primarily be limited by physiological factors and by oxygen concentration. The rate of respiration when in oxygen saturated water is considered the maximum rate of respiration. At a certain level of oxygen saturation the fish will begin to down regulate their activity to compensate for the low oxygen content. This saturation level will be denoted as the oxygen threshold. A final threshold is the oxygen level of no excess activity, which is also temperature dependent and below which the fish will do nothing but what is needed the just survive. This respiration rate is set to be 0.1 times their maximum respiration. Both the respiration, the oxygen level of no excess activity and the oxygen threshold are calculated from data of Fry and Hart 1948 and are temperature dependent.
Km:
Since respiration can be discribed through Michaelis-Menten kinetics, a Km (concentration at which respiration is half the maximum rate) has to be estimated. This is done by finding the relationship between temperature and oxygen threshold and dividing this value by 2 and multiplying by the oxygen solubility.
Fish Growth:
The fish grow at a set rate of 26.4 g/mol(O2) assuming a assimilation rate of 1 part C pr. part O2 respired. The more the fish grow, the higher the respiration and the faster the growth. If the oxygen level is below the level of no excess activity the growth is set to be 0.
Result:
The temperature dependent oxygen consumption through respiration is found to follow a power relationship with the following equation:
18.493*temperature^1.7926 µmol/kg/h
The Km was found to depend on temperature following a power relation with the following equation:
(0.0393*temperature^0.5472 )/2 µM
The level of no excess activity was found as a oxygen saturation adhering to the following power relation:
0.0055*temperature^0.9436µM
As fish grows they reach a point where their minimum respiration matches the input of oxygen in the tank and growth is thus halted and the concentration in the tank will equal the input (See the table).
Time Concentration - µM Biomass of fishes - kg
0 318.25 0.076
24 314.2664 0.181166
48 304.7993 0.431604
72 282.3204 1.026751
96 229.2151 2.433283
120 105.9264 5.699388
144 1.019153 12.50083
168 1.522708 12.50083
192 1.520291 12.50083
216 1.520303 12.50083
240 1.520303 12.50083
264 1.520303 12.50083
It can be seen from this that the maximum biomass supported by this system is 12.5 kg.
