Refining and Optimizing Analytical Models
This section gives an example of how the model developed earlier could be extended to account for additional knowledge of the system's structure. Again, if you have not yet been exposed to this particular type of mathematical analysis, simply note that all models can be refined as we gain more knowledge about the system. Think back to the model you identified in the last section. How could you extend this model to make it a better reflection of reality?
Long-term behaviour (Equilibrium, Phase space)
Rainbowfish and gouramis
In section Interaction (Euler's method for Systems), you encountered the following situation:
A client of the fish farm wants an aquarium containing both
rainbowfish and gouramis. He has a budget for 20 rainbowfish, 5 gouramis
and an aquarium which has a maximum capacity for 100 rainbowfish.
A model was constructed and solutions were calculated with Euler's method. According to the model, both populations start out to grow fast to triple their initial sizes. Then first the number of rainbowfish decreases again, and the gouramis follow 6 days later. Both populations keep oscillating in a damped oscillation, and in a month or so both populations have settled in their end states of approximately 31 and 12 fishes respectively.
In this section we are going to investigate this problem further to understand the model better.
What would happen with different initial numbers of both kinds of fish? There are three equilibrium points. One is the end state. What is going to happen with the populations when we start near one of the other equilibrium points? Can we calculate some results analytically, so we can understand better how the parameters we have chosen for the model influence the results? And maybe then we have more information to chose realistic values for the parameters.
The mathematical model for the populations of rainbowfish and gourami is the same as in Interaction (Euler's method for Systems), so on the next page we skip immediately to the calculation part of the modelling cycle.