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)
2.1 Problem
Bounded growth for the rainbowfish
In the first module, we tried to predict the size of a rainbowfish population in an aquarium. Unfortunately, the growth of the population in the model of Section 1.4 was unbounded, and that of the improved model in Section 1.6 as well.
In this section, we will incorporate the limited capacity of the aquarium. When an aquarium gets crowded, the fish will lay fewer eggs, fewer young fish will survive and the fish may die earlier, so the growth of the population slows down. The problem statement is now:
An aquarium owner is breeding a population of rainbowfish. He starts with 30 rainbowfish and expects to sell 20 rainbowfish every day. The aquarium is large enough for a healthy population of 750. Predict the development of the size of the population.
Next, bounded growth will be incorporated into the model, you will investigate the long-term solutions and validate them.
Mathematical model
The number of rainbowfish in the aquarium will limit itself to the capacity of the aquarium of 750. In the next video this bounded growth is incorporated into the mathematical model.
Bounded growth in a mathematical model
In the next section you will investigate the solutions of the new differential equation
\(\dfrac{dP}{dt}=0.7(1-\dfrac{P(t)}{750}) \; P(t) - 20)\).
Calculation
In the mathematical model for the number of fish in the aquarium, the growth factor is now bounded:
\( \dfrac{dP}{dt}=0.7(1-\dfrac{P}{750}) - 20) \),
with the initial condition P(0)=30.
In the next exercises, you are going to calculate whether the resulting population stays bounded as well.
Validation
The initial value problem describing the fish population with bounded growth is:
\( \dfrac{dP}{dt}=0.7(1-\dfrac{P}{750}) - 20) \), \(P(0) = 30\).
The phase line of this autonomous differential equation is:
