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Population dynamics is the study of how and why populations change over time. Understanding these changes involves looking at factors like birth rates, death rates, and migration patterns. In the context of an exponential growth model, population dynamics focus mainly on how quickly a population grows under constant conditions. Exponential growth is characterized by the population increasing by a constant proportion (or rate, known as the common ratio) each time period.

In our problem, the population starts with an initial count of 11 individuals. The growth follows a consistent pattern where the population grows by 25% each period. This is captured in the formula given:

• The initial population is 11.
• The common ratio is 1.25, meaning the population increases by a factor of 1.25 each period.

The consistency in this growth allows us to predict future populations based on past counts and the growth factor.

Mathematical modelling is a powerful tool used to represent real-world phenomena through mathematical expressions and equations. It allows us to predict and analyze behaviors, trends, and outcomes under various conditions. By using mathematical models, we can simplify complex real-world processes into more manageable forms.

In the population growth task, we used an exponential growth model, which is one of the simplest forms of growth models. The key components of this model include:

• Initial population size \(P_0\) - The starting number of individuals in the population.
• Common ratio \(R\) - The constant factor by which the population grows each period.
• Number of time periods \(n\) - Defines how many periods the growth is measured.
• Future population \(P_n\) - The population at a given time in the future.

When these elements are combined into the formula \(P_{n} = P_{0} \cdot R^{n}\), it provides a clear representation of how the population grows over time, enabling predictions and deeper understanding of population dynamics.

Deriving an explicit formula means creating a direct equation to calculate results for any given input, without needing to calculate all previous results. In our population example, we derived the formula for calculating the population at any given time period \(N\) directly from the initial values.

Let's break down the derivation process:

• First, identify the initial conditions: \(P_{0} = 11\), which is the starting population.
• Next, specify the common ratio, \(R = 1.25\), representing the growth factor each period.
• Apply the general formula for exponential growth: \(P_{N} = P_{0} \cdot R^{N}\).

This formula, \(P_{N} = 11 \cdot 1.25^{N}\), allows us to compute the population at any time \(N\), whether it be the next period or several periods into the future. This explicit form is intuitive and efficient, reducing the need for step-by-step calculations whenever predicting future populations.