Problem 24 - Calculus for Biology and Medicine

Question

Let \(N(t)\) denote the size of a population at time \(t .\) Assume that the population exhibits exponential growth. (a) If you plot \(\log N(t)\) versus \(t\), what kind of graph do you get? (b) Find a differential equation that describes the growth of this population and sketch possible solution curves.

Step-by-Step Solution

Answer

(a) A straight line. (b) \(\frac{dN}{dt} = rN\), with exponential growth curves.

Step 1: Understand Exponential Growth

In exponential growth, the population size grows at a rate proportional to its current size. This is mathematically described by the equation \(N(t) = N_0 e^{rt}\), where \(N_0\) is the initial population size and \(r\) is the growth rate.

Step 2: Analyze Plot of \(\log N(t)\) vs. \(t\)

Taking the log of both sides of the exponential growth equation \(N(t) = N_0 e^{rt}\) gives \(\log N(t) = \log(N_0 e^{rt}) = \log N_0 + rt\). This is a linear equation in the form \(y = mx + c\), where the slope \(m = r\) and the y-intercept \(c = \log N_0\). Hence, the graph of \(\log N(t)\) versus \(t\) is a straight line.

Step 3: Construct the Differential Equation

To derive the differential equation for the population growth, differentiate \(N(t) = N_0 e^{rt}\) with respect to \(t\). We have \(\frac{dN}{dt} = rN_0 e^{rt} = rN(t)\). Therefore, the differential equation is \(\frac{dN}{dt} = rN\), indicating the rate of change of the population size is proportional to the current size.

Step 4: Sketch Possible Solution Curves

The general solution to the differential equation \(\frac{dN}{dt} = rN\) is the exponential growth function \(N(t) = N_0 e^{rt}\). If you sketch \(N(t)\) for various initial values \(N_0\) and positive growth rates \(r\), you will see a family of exponential growth curves beginning at different initial values and rising rapidly.

Key Concepts

Differential Equation

A differential equation is a type of equation that involves derivatives, which represent rates of change. In simple terms, it describes how one variable changes in relation to another. You'll often encounter these equations when dealing with things that grow or shrink, such as populations or investments.

In our context of exponential growth, \(\frac{dN}{dt} = rN\) is the key differential equation. Here, \(\frac{dN}{dt}\) represents the rate of change in population size over time, while \(r\) is a constant representing the growth rate. \(N\) denotes the population size at time \(t\).

Why is understanding this important?

• It gives insight into how quickly the population is changing.
• It serves as a foundation for solving more complex growth models.

The solution to such differential equations in the case of exponential growth is typically an exponential function, which we'll explore in another section.

Population Modeling

Population modeling is a mathematical way to represent how populations change over time. It's crucial for understanding biological, ecological, and even social dynamics. Exponential growth is one type of population model where the rate of population increase is proportional to the current population size.

This type of modeling helps answer questions such as:

• How fast is a species multiplying in a given environment?
• What will the population size be in the future?
• How do different factors, like resource availability, affect growth?

In our exercise, we assume a model where the population grows without any restrictions, leading to exponential growth. While real-world populations do face limits, this model serves as an essential starting point.

By understanding simple models, one can gradually layer complexity to simulate more realistic scenarios, like logistic growth, where growth slows as the population reaches the environment's carrying capacity.

Exponential Function

Exponential functions are a critical part of understanding exponential growth. They describe situations where the rate of growth accelerates over time. Mathematically, an exponential function with respect to time is written as \(N(t) = N_0 e^{rt}\), where \(N_0\) is the initial amount, \(e\) is Euler's number (approximately 2.718), and \(r\) is the growth rate.

Why are exponential functions crucial for this topic?

• They accurately capture how populations that grow unchecked will behave over time.
• They provide analytical solutions to differential equations representing growth, like the one we explored above.
• They're widespread and apply to many fields, from finance to biology.

When you plot an exponential function, you'll see it starts slowly and then shoots up quickly—this is the hallmark of exponential growth. Understanding this function helps predict and comprehend growth phenomena where growth rate accelerates proportionally to the current size.