Model Validation
Read this material on mathematical modeling. Try to complete the exercise at the end of the chapters. Don't worry if you have not yet learned the math necessary to complete the exercises. Rather, notice how many physical systems can be represented by mathematical models. We can then implement these models using programming languages like Python and R. We will discuss those later. Think about the types of systems you interact with, personally or in business situations. How might you use a mathematical model to help you understand this system?
2. Bounded growth (Phase line, Stable equilibrium)
2.3 Calculation (accuracy)
Now you have working code, it is time to investigate the stepsize further.
In the next figure you can see the approximate solutions for Δt=1, Δt=12 and Δt=14 day. As you can see the graphs of the biggest two stepsizes differ more than the graphs of the two smallest stepsizes. This behavior is called numerical convergence: The smaller you take Δt, the better your approximation becomes and it will look more and more like the exact solution.
In the figure you can also see that the largest differences in the value of P occur around 13 days:
So when you want to know the number of fish at 13 days, the stepsize is very important. The difference between Δt=1 day and Δt=0.5 day is more than 112 fish! And between the two smaller stepsizes, the difference is still more than 81 fish. So the results clearly change a lot, when the stepsize is halved.
Error of an approximation with Euler's method
We will not give any theoretical background here for the estimation of the error of a result with Euler's method. That would go too far for this course. So, without any proof:
For
differential equation \(\dfrac{dy}{dt} = f(t, y (t))\) (and an initial
condition), the exact solution at a specific time t is y(t). With
Euler's method and step-sizes Δt and 2Δt, that exact solution can be
approximated by wΔt and w2Δt, respectively. Than EΔt, the error in wΔt, can estimated as the difference between wΔt and w2Δt:
\( E_{Δt} = y(t) - w_{Δt} \approx w _{Δt} - w_{2Δt} \).
So an estimate of the error in the result with Δt=0.5 is 281.64−169.24=112.40 fish. And an estimate of the error in the result for Δt=0.25 is 81.45 fish.
Validation
In the previous section you have found an answer to the problem
Find teq such that P(teq)=720, where
\( \left\{\begin{aligned}\dfrac{dP}{dt} & =0.7P\;(1-\dfrac{P}{750}) - 20, \\P(0) & =30\end{aligned}\right. \).
which turned out to be teq=24.63. You did this by using Euler's Method with Δt=132.
We can now state that for our problem, the rainbowfish population is close to the equilibrium size within 25 days.