• Use the properties of logarithms to solve exponential models for time
• Identify the carrying capacity in a logistic growth model
• Use a logistic growth model to predict growth

Exploring Exponential and Logistic Growth Models

In this ‘Apply It’ section, we explore the dynamic world of growth models through real-world scenarios. You will encounter diverse situations, from the population dynamics of endangered species to the spread of invasive plants, and even the user growth of a social media platform. Each scenario will challenge you to apply logarithmic and logistic growth models, enhancing your understanding of how these mathematical concepts are used to predict and analyze real-life phenomena. By delving into these examples, you’ll gain a deeper insight into the power and utility of exponential and logistic models in various contexts.

Scenario 1: Population Growth of an Endangered Species

Researchers are studying the population growth of an endangered bird species. The population has been growing exponentially, and they have collected data over the past decade. Below are the key points they have observed from their data:

• Initial population ([latex]P_0[/latex]): [latex]100[/latex] birds
• Growth rate ([latex]r[/latex]): [latex]0.05[/latex] ([latex]5\%[/latex] per year)
• Time period for data: [latex]10[/latex] years

After examining the exponential growth of an endangered bird species, let’s shift our focus to a different ecological challenge: the spread of invasive plants. This transition from animal to plant life demonstrates how logistic growth models can be applied to understand and predict the behavior of different biological populations, each with its unique set of challenges and environmental impacts.

Scenario 2: Spread of Invasive Plant Species

An invasive plant species has been spreading in a national park. Ecologists have determined that its growth follows a logistic model of [latex]P(t) = \frac{1000}{1 + 10e^{-0.08t}}[/latex].