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
1.4 Population growth (Design a differential equation)
Problem definition
The problem that we are going to work on throughout this course is the following:
You want to breed rainbowfish to sell to pet stores. You start with a nice big aquarium and 30 fish, half of them male, half of them female. You want to predict the number of fish after a number of days, to see how many you can sell.
Mathematical model
In the previous section the question became:
Can you predict the number of rainbowfish in the aquarium after any number of days?
In the following video we are going to derive a mathematical model.
The first mathematical model
Calculation
In the previous section you have derived the differential equation and the intitial condition:
\(\dfrac{dP}{dt} = 0.7 P(t)\),
\(P(0) = 30\).
Is there a function P(t) that satisfies this initial value problem?
Validation
In the previous section you have seen that the solution of the differential equation
\(\dfrac{dP}{dt} = 0.7 P(t)\),
with the initial condition P(0)=30 is given by
\(P(t) = 30e^{0.7t}\).
Now you will validate this solution. Or in other words: does this solution represent a realistic solution to our problem?