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?
1. Introducing Mathematical Modelling
1.6 Removal (Phase line, Unstable equilibrium)
Problem
Back to the rainbowfish for a second modelling cycle.
Selling rainbowfish
In section 1.4 you encountered the problem of predicting the number of rainbowfish when you are breeding them.
A mathematical model was formulated in the form of a differential equation and a solution was found. However, this solution is unrealistic: the number of rainbowfish grows very large very quickly.
In this section we will add one sentence to the problem, in the hope that this will limit the growth. We have chosen the following addition:
The aquarium owner expects to sell 20 rainbowfish every day.
In the following pages of this section, you will incorporate this into the model, investigate the long-term solutions and validate them.
Mathematical model
is the number of rainbowfish in the aquarium, with t in days.
The birth rate of the fish is b=0.7 per day. The death rate is ignored: d≈0.
Now, 20 fish per day are sold.
In the exercises, you will derived the second mathematical model.
Calculation 1: Direction field & Equilibrium solution
In the previous section you have derived the differential equation
\(\dfrac{dP}{dt} = 0.7 P(t) - 20\),
still with the initial condition P(0)=30.
Can you estimate the number of rainbowfish P(t) over time?
To answer this question, we will first construct a direction field, which is a special type of graph. In the next video, you will see how you can do this. In the video also some solutions are estimated. See for yourself!
Direction fields
Calculation 2: Phase line & Stability
Now that you can find equilibrium solutions of a differential equation, it is time to investigate what kinds of equilibrium solutions can occur. In the next video you will learn about this.
Note: in the video an example of an unstable equilibrium point is given. However, the definition of an unstable equilibrium point is not in the video. That you can find below, in the text.
Phase line & Stability of equilibrium points
Validation
In the previous sections you have seen the differential equation
\(\dfrac{dP}{dt} = 0.7 P(t) - 20\),
the initial condition P(0)=30. The phase line for this differential equation is
and the direction field with some possible solutions is like this:
Now you will validate these solutions. Do these solutions represent realistic solutions to our problem?