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How to make a mathematical model of population growth or decline?

Understanding mathematical models is essential for analyzing real phenomena, such as population growth or decline. In this class, we will help you explore one of the most important and applied methods: the population growth or decline model. You will learn how to identify different rates of growth or decline and how to solve differential equations to predict how a population will evolve over time.

What is a population growth or decline model?

The population growth or decline model is based on analyzing a population over time, whether it is people, data or any other entity you wish to study. In this model, you must consider:

• A population to analyze.
• A time variable, which can be measured in seconds, years, etc.
• A growth rate, such as the number of births or friends added in social networks.
• A decay rate, such as the number of deaths or friends who decide to stop being friends.

The essence of the model is to discover how a population varies after a specific time interval.

How is this phenomenon mathematically modeled?

The mathematical model of population growth or decline is formulated starting with the initial quantities and rates of change. Here is a simple example for better understanding:

1. Initial population: Imagine a community of 100 people.
2. Growth rate: 20% annual increase.
3. Mortality rate: 5% annual decrease.
4. Time horizon: 5 years.

For a basic model, we calculate growth by multiplying the initial 100 people by 20% and then by the 5 years. In turn, the decrease is calculated by multiplying the 100 people by 5% for the 5 years. You can then combine these figures to model the population change.

How do you solve and fit such a model?

While a basic approach can provide an initial idea of the change, the optimal model uses smaller time intervals for greater accuracy. In mathematics, this is achieved through the derivative of the population with respect to time. In the example, the net growth constant and rates of change are considered, along with the derivative to achieve:

1. Integral of the left-hand side: The integral of the natural logarithm.
2. Right hand side integral: The growth constant multiplied by time plus a constant.

Finally, by applying exponentials to clear, we obtain a term that includes the initial population multiplied by a proportional constant and time, expressed as: ( P(t) = P_0 \cdot e^{kt} ).

This fitted model is crucial for interpreting how a population will grow or shrink over time, allowing you to predict changes over shorter periods and get a more accurate and useful analysis.

Next step: Practical applications.

You are now ready to apply this knowledge and concepts to real-world situations. From demographic studies to social network analysis, understanding these models will allow you to explore various scenarios and phenomena in everyday life. Get ready for our next class, where we will work with practical examples and consolidate these ideas in a more tangible context. The learning continues!