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)
Validation
We have worked with the system of two differential equations for the rainbowfish and gouramis.
\( \begin{cases}\frac{d P}{d t}=0.7 P-0.007 P^2- \alpha PG, & \text {where α = 0.04} , \\ \frac{d G}{d t}=-0.25 G+ \beta PG, & \text {where β = 0.008}.\end{cases} \)
You have learned how to approximate solutions numerically with Euler's method. In this section, we have used analytical methods to characterize the solutions beforehand. First the equilibrium points (P0,G0) were calculated:
\((0,0), (100, 0)\) and \( (\dfrac{1}{4 \beta} , \dfrac{0.7}{\alpha} (1 - \dfrac{1}{400 \beta}) ) \).
Then the differential equations were linearised around the three equilibrium points:
\(\begin{gathered}
\text { For } \vec{M}(t)=\left[\begin{array}{c}
P(t)-P_0 \\
G(t)-G_0
\end{array}\right], \\
\frac{d \vec{M}}{d t} \approx\left[\begin{array}{cc}
0.7-0.014 P_0-\alpha G_0 & -\alpha P_0 \\
\beta G_0 & -0.25+\beta P_0
\end{array}\right] \vec{M}(t) .
\end{gathered}\)
By calculating the eigenvalues of the three matrices, the types of the three equilibrium points were determined. In the following exercises, you are going to review whether these analytical calculations were usefull.